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</div><h2>HL Paper 3</h2><div class="specification">
<p class="p1">A relation \(S\) is defined on \(\mathbb{R}\) by \(aSb\) if and only if \(ab > 0\).</p>
</div>
<div class="specification">
<p class="p1">A relation \(R\) is defined on a non-empty set \(A\). \(R\) is symmetric and transitive but not reflexive.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(S\) is</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>not reflexive;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>symmetric;</p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>transitive.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why there exists an element \(a \in A\) <span class="s1">that is not related to itself.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence prove that there is at least one element of \(A\) <span class="s1">that is not related to any other element of \(A\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> \(0S0\)</span> </span>is not true so \(S\) is not reflexive <span class="Apple-converted-space"> </span><strong><em>A1AG</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(aSb \Rightarrow ab > 0 \Rightarrow ba > 0 \Rightarrow bSa\)</span> so \(S\) is symmetric <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></p>
<p class="p1">(iii) <span class="Apple-converted-space"> \(aSb\)</span> and \(bSc \Rightarrow ab > 0\) and \(bc > 0 \Rightarrow a{b^2}c > 0 \Rightarrow ac > 0\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">since \({b^2} > 0\) (as \(b\) <span class="s1">could not be 0) \( \Rightarrow aSc\) </span>so \(S\) is transitive <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <em>R1 </em></strong>is for indicating that \({b^2} > 0\).</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">since \(R\) is not reflexive there is at least one element \(a\) belonging to \(A\) such that \(a\) is not related to \(a\) <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">argue by contradiction: suppose that \(a\) is related to some other element \(b\), <em>ie</em>, \(aRb\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">since \(R\) is symmetric \(aRb\) implies \(bRa\) <span class="Apple-converted-space"> </span><strong><em>R1A1</em></strong></p>
<p class="p1">since \(R\) is transitive \(aRb\) and \(bRa\) implies \(aRa\) <span class="Apple-converted-space"> </span><strong><em>R1A1</em></strong></p>
<p class="p1">giving the required contradiction <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">hence there is at least one element of \(A\) that is not related to any other member of \(A\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f:G \to H\) be a homomorphism between groups \(\{ G,{\text{ }} * \} \) and \(\{ H,{\text{ }} \circ \} \) with identities \({e_G}\) and \({e_H}\) respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that \(f({e_G}) = {e_H}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that \({\text{Ker}}(f)\) is a subgroup of \(\{ G,{\text{ }} * \} \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>let \(a \in G\) and \(f(a) \in H\)</p>
<p>\(f\) is a homomorphism so \(f(a * {e_G}) = f(a) \circ f({e_G})\) <strong><em>(M1)</em></strong></p>
<p>\(f(a) = f(a) \circ f({e_G})\) <strong><em>A1</em></strong></p>
<p>\({e_H} = f({e_G})\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>from part (a) \({e_G} \in {\text{Ker}}(f)\) and associativity follows from G <strong><em>R1</em></strong></p>
<p>let \(a,{\text{ }}b \in {\text{Ker}}(f)\)</p>
<p>\(f(a * b) = f(a) \circ f(b) = {e_H} \circ {e_H} = {e_H}\) <strong><em>A1</em></strong></p>
<p>hence closed since \(a * b \in {\text{Ker}}(f)\)</p>
<p>\({e_H} = f({a^{ - 1}} * a) = f({a^{ - 1}}) \circ f(a) = f({a^{ - 1}}) \circ {e_H} = f({a^{ - 1}})\) <strong><em>M1A1</em></strong></p>
<p>hence \({a^{ - 1}} \in {\text{Ker}}(f)\) <strong><em>R1</em></strong></p>
<p>hence \({\text{Ker}}(f)\) is subgroup of \(G\) <strong><em>AG</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>\(A\), \(B\) and \(C\) are three subsets of a universal set.</p>
</div>
<div class="specification">
<p>Consider the sets \(P = \{ 1,{\text{ }}2,{\text{ }}3\} ,{\text{ }}Q = \{ 2,{\text{ }}3,{\text{ }}4\} \) and \(R = \{ 1,{\text{ }}3,{\text{ }}5\} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Represent the following set on a Venn diagram,</p>
<p>\(A\Delta B\), the symmetric difference of the sets \(A\) and \(B\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Represent the following set on a Venn diagram,</p>
<p>\(A \cap (B \cup C)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For sets \(P\), \(Q\) and \(R\), verify that \(P \cup (Q\Delta R) \ne (P \cup Q)\Delta (P \cup R)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of the distributive law, describe what the result in part (b)(i) illustrates.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-10_om_13.37.16.png" alt="N17/5/MATHL/HP3/ENG/TZ0/SG/M/02.a.i"> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<p> </p>
<p><strong><em>Note: </em></strong><em>Accept alternative set configurations</em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-10_om_13.41.32.png" alt="N17/5/MATHL/HP3/ENG/TZ0/SG/M/02.a.ii"> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept alternative set configurations.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>LHS:</p>
<p>\(Q\Delta R = \{ 1,{\text{ }}2,{\text{ }}4,{\text{ }}5\} \) <strong><em>(A1)</em></strong></p>
<p>\(P \cup (Q\Delta R) = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5\} \) <strong><em>A1</em></strong></p>
<p>RHS:</p>
<p>\(P \cup Q = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4\} \) and \(P \cup R = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}5\} \) <strong><em>(A1)</em></strong></p>
<p>\((P \cup Q)\Delta (P \cup R) = \{ 4,{\text{ }}5\} \) <strong><em>A1</em></strong></p>
<p>hence \(P \cup (Q\Delta R) \ne (P \cup Q)\Delta (P \cup R)\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the result shows that union is not distributive over symmetric difference <strong><em>A1R1</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Award <strong><em>A1 </em></strong>for “union is not distributive” and <strong><em>R1 </em></strong>for “over symmetric difference”. Condone use of \( \cup \) and \(\Delta \).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The function \(f\,{\text{: }}\mathbb{Z} \to \mathbb{Z}\) is defined by \(f\left( n \right) = n + {\left( { - 1} \right)^n}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that \(f \circ f\) is the identity function.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f\) is injective.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f\) is surjective.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>\(\left( {f \circ f} \right)\left( n \right) = n + {\left( { - 1} \right)^n} + {\left( { - 1} \right)^{n + {{\left( { - 1} \right)}^n}}}\) <em><strong>M1A1</strong></em></p>
<p>\( = n + {\left( { - 1} \right)^n} + {\left( { - 1} \right)^n} \times {\left( { - 1} \right)^{{{\left( { - 1} \right)}^n}}}\) <em><strong>(A1)</strong></em></p>
<p>considering \({\left( { - 1} \right)^n}\) for even and odd \(n\) <em><strong>M1</strong></em></p>
<p>if \(n\) is odd, \({\left( { - 1} \right)^n} = - 1\) and if \(n\) is even, \({\left( { - 1} \right)^n} = 1\) and so \({\left( { - 1} \right)^{ \pm 1}} = - 1\) <em><strong>A1</strong></em></p>
<p>\( = n + {\left( { - 1} \right)^n} - {\left( { - 1} \right)^n}\) <em><strong>A1</strong></em></p>
<p>= \(n\) and so \(f \circ f\) is the identity function <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>\(\left( {f \circ f} \right)\left( n \right) = n + {\left( { - 1} \right)^n} + {\left( { - 1} \right)^{n + {{\left( { - 1} \right)}^n}}}\) <em><strong>M1A1</strong></em></p>
<p>\( = n + {\left( { - 1} \right)^n} + {\left( { - 1} \right)^n} \times {\left( { - 1} \right)^{{{\left( { - 1} \right)}^n}}}\) <em><strong>(A1)</strong></em></p>
<p>\( = n + {\left( { - 1} \right)^n} \times \left( {1 + {{\left( { - 1} \right)}^{{{\left( { - 1} \right)}^n}}}} \right)\) <em><strong>M1</strong></em></p>
<p>\({\left( { - 1} \right)^{ \pm 1}} = - 1\) <em><strong>R1</strong></em></p>
<p>\(1 + {\left( { - 1} \right)^{{{\left( { - 1} \right)}^n}}} = 0\) <em><strong>A1</strong></em></p>
<p>\(\left( {f \circ f} \right)\left( n \right) = n\) and so \(f \circ f\) is the identity function <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>\(\left( {f \circ f} \right)\left( n \right) = f\left( {n + {{\left( { - 1} \right)}^n}} \right)\) <em><strong>M1</strong></em></p>
<p>considering even and odd \(n\) <em><strong>M1</strong></em></p>
<p>if \(n\) is even, \(f\left( n \right) = n + 1\) which is odd <em><strong>A1</strong></em></p>
<p>so \(\left( {f \circ f} \right)\left( n \right) = f\left( {n + 1} \right) = \left( {n + 1} \right) - 1 = n\) <em><strong>A1</strong></em></p>
<p>if \(n\) is odd, \(f\left( n \right) = n - 1\) which is even <em><strong>A1</strong></em></p>
<p>so \(\left( {f \circ f} \right)\left( n \right) = f\left( {n - 1} \right) = \left( {n - 1} \right) + 1 = n\) <em><strong>A1</strong></em></p>
<p>\(\left( {f \circ f} \right)\left( n \right) = n\) in both cases</p>
<p>hence \(f \circ f\) is the identity function <em><strong> AG</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>suppose \(f\left( n \right) = f\left( m \right)\) <em><strong>M1</strong></em></p>
<p>applying \(f\) to both sides \( \Rightarrow n = m\) <em><strong>R1</strong></em></p>
<p>hence \(f\) is injective <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(m = f\left( n \right)\) has solution \(n = f\left( m \right)\) <em><strong>R1</strong></em></p>
<p>hence surjective <em><strong>AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(\{ G,{\text{ }} \circ \} \) be the group of all permutations of \(1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6\) under the operation of composition of permutations.</p>
</div>
<div class="specification">
<p class="p1">Consider the following Venn diagram, where \(A = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4\} ,{\text{ }}B = \{ 3,{\text{ }}4,{\text{ }}5,{\text{ }}6\} \).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-02_om_09.47.40.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SG/01.f"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span>Write the permutation \(\alpha = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 3&4&6&2&1&5 \end{array}} \right)\) </span>as a composition of disjoint cycles.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>State the order of \(\alpha \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span>Write the permutation \(\beta = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 6&4&3&5&1&2 \end{array}} \right)\) </span>as a composition of disjoint cycles.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>State the order of \(\beta \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write the permutation \(\alpha \circ \beta \) as a composition of disjoint cycles.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write the permutation \(\beta \circ \alpha \) as a composition of disjoint cycles.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the order of \(\{ G,{\text{ }} \circ \} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Find the number of permutations in \(\{ G,{\text{ }} \circ \} \) </span>which will result in \(A\), \(B\) and \(A \cap B\) remaining unchanged.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \((1{\text{ }}3{\text{ }}6{\text{ }}5)(2{\text{ }}4)\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1A1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>4 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: </strong>In (b) (c) and (d) single cycles can be omitted.</p>
<p class="p2"> </p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \((1{\text{ }}6{\text{ }}2{\text{ }}4{\text{ }}5)(3)\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>5 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(\left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 5&2&6&1&3&4 \end{array}} \right) = (1{\text{ }}5{\text{ }}3{\text{ }}6{\text{ }}4)(2)\) </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(\left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 3&5&2&4&6&1 \end{array}} \right) = (1{\text{ }}3{\text{ }}2{\text{ }}5{\text{ }}6)(4)\) </span><strong><em>(M1)A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A2A0 </em></strong>for (c) and (d) combined, if answers are the wrong way round.</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(6! = 720\) </span><span class="s1"><strong><em>A2</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">any composition of the cycles (1 2), (3 4) and (5 6) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">so \({2^3} = 8\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operations \( \odot \) and \( * \) are defined on \({\mathbb{R}^ + }\) by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[a \odot b = \sqrt {ab} {\text{ and }}a * b = {a^2}{b^2}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether or not</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \odot \) is commutative;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( * \) is associative;</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( * \) is distributive over \( \odot \) ;</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \odot \) has an identity element.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a \odot b = \sqrt {ab} = \sqrt {ba} = b \odot a\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(a \odot b = b \odot a\) it follows that \( \odot \) is commutative <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a * (b * c) = a * {b^2}{c^2} = {a^2}{b^4}{c^4}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((a * b) * c = {a^2}{b^2} * c = {a^4}{b^4}{c^2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">these are different, therefore \( * \) is not associative <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept numerical counter-example.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a * (b \odot c) = a * \sqrt {bc} = {a^2}bc\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((a * b) \odot (a * c) = {a^2}{b^2} \odot {a^2}{c^2} = {a^2}bc\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">these are equal so \( * \) is distributive over \( \odot \) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the identity e would have to satisfy</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a \odot e = a\) for all <em>a</em> <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">now \(a \odot e = \sqrt {ae} = a \Rightarrow e = a\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore there is no identity element <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\{ G,{\text{ }} * \} \) be a finite group that contains an element <em>a</em> (that is not the identity element) and \(H = \{ {a^n}|n \in {\mathbb{Z}^ + }\} \), where \({a^2} = a * a,{\text{ }}{a^3} = a * a * a\) etc.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since <em>G</em> is closed, <em>H</em> will be a subset of <em>G</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">closure: \(p,{\text{ }}q \in H \Rightarrow p = {a^r},{\text{ }}q = {a^s},{\text{ }}r,{\text{ }}s \in {\mathbb{Z}^ + }\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p * q = {a^r} * {a^s} = {a^{r + s}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r + s \in {\mathbb{Z}^ + } \Rightarrow p * q \in H\) hence <em>H</em> is closed <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">associativity follows since \( * \) is associative on <em>G</em> <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">identity: let the order of <em>a</em> in <em>G</em> be \(m \in {\mathbb{Z}^ + },{\text{ }}m \geqslant 2\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then \({a^m} = e \in H\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">inverses: \({a^{m - 1}} * a = e \Rightarrow {a^{m - 1}}\) is the inverse of <em>a</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({({a^{m - 1}})^n} * {a^n} = e\), showing that \({a^n}\) has an inverse in <em>H</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>H</em> is a subgroup of <em>G</em> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \((G,{\text{ }} * )\) is a finite group, and <em>H</em> is a non-empty closed subset of <em>G</em>, then \((H,{\text{ }} * )\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a subgroup of \((G,{\text{ }} * )\) <strong><em>R4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> To receive the <strong><em>R4</em></strong>, the candidate must explicitly state the theorem, <em>i.e.</em> the three given conditions, and conclusion.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally answered very poorly, if attempted at all. Candidates failed to realize that the property of closure needed to be properly proved. Others used negative indices when the question specifically states that the indices are positive integers.</span></p>
</div>
<br><hr><br><div class="specification">
<p>The set \(A\) contains all positive integers less than 20 that are congruent to 3 modulo 4.</p>
<p>The set \(B\) contains all the prime numbers less than 20.</p>
</div>
<div class="specification">
<p>The set \(C\) is defined as \(C = \{ 7,{\text{ }}9,{\text{ }}13,{\text{ }}19\} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down all the elements of \(A\) and all the elements of \(B\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the symmetric difference, \(A\Delta B\), of the sets \(A\) and \(B\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down all the elements of \(A \cap B,{\text{ }}A \cap C\) and \(B \cup C\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence by considering \(A \cap (B \cup C)\), verify that in this case the operation \( \cap \) is distributive over the operation \( \cup \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>the elements of \(A\) are: 3, 7, 11, 15, 19 <strong><em>A1</em></strong></p>
<p>the elements of \(B\) are 2, 3, 5, 7, 11, 13, 17, 19 <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(A = \{ 3,{\text{ }}7,{\text{ }}11,{\text{ }}15,{\text{ }}19\} \) and \(B = \{ 2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19\} \)</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to determine \(A\backslash B \cup B\backslash A\) or \((A \cup B) \cap (A \cap B)'\) <strong><em>(M1)</em></strong></p>
<p>symmetric difference \( = \{ 2,{\text{ }}5,{\text{ }}13,{\text{ }}15,{\text{ }}17\} \) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Allow <strong><em>(M1)A1FT</em></strong>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the elements of \(A \cap B\) are 3, 7, 11 and 19 <strong><em>A1</em></strong></p>
<p>the elements of \(A \cap C\) are 7 and19 <strong><em>A1</em></strong></p>
<p>the elements of \(B \cup C\) are 2, 3, 5, 7, 9, 11, 13, 17 and 19 <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(A \cap B = \{ 3,{\text{ }}7,{\text{ }}11,{\text{ }}19\} ,{\text{ }}A \cap C = \{ 7,{\text{ }}19\} \) and \(B \cup C = \{ 2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}9,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19\} \).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>we need to show that</p>
<p>\(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) <strong><em>(M1)</em></strong></p>
<p>\(A \cap (B \cup C) = \{ 3,{\text{ }}7,{\text{ }}11,{\text{ }}19\} \) <strong><em>A1</em></strong></p>
<p>\((A \cap B) \cup (A \cap C) = \{ 3,{\text{ }}7,{\text{ }}11,{\text{ }}19\} \) <strong><em>A1</em></strong></p>
<p>hence showing the required result</p>
<p> </p>
<p><strong>Note:</strong> Allow <strong><em>(M1)A1FTA1FT</em></strong>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The relation <em>R</em> is defined on \(\mathbb{R} \times \mathbb{R}\) such that \(({x_1},{\text{ }}{y_1})R({x_2},{\text{ }}{y_2})\) if and only if \({x_1}{y_1} = {x_2}{y_2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em>R</em> is an equivalence relation.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the equivalence class of <em>R</em> containing the element \((1,{\text{ }}2)\) and illustrate this graphically.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>R</em> is an equivalence relation if</p>
<p><em>R</em> is reflexive, symmetric and transitive <strong><em>A1</em></strong></p>
<p>\({x_1}{y_1} = {x_1}{y_1} \Rightarrow ({x_1},{\text{ }}{y_1})R({x_1},{\text{ }}{y_1})\) <strong><em>A1</em></strong></p>
<p>so <em>R</em> is reflexive</p>
<p>\(({x_1},{\text{ }}{y_1})R({x_2},{\text{ }}{y_2}) \Rightarrow {x_1}{y_1} = {x_2}{y_2} \Rightarrow {x_2}{y_2} = {x_1}{y_1} \Rightarrow ({x_2},{\text{ }}{y_2})R({x_1},{\text{ }}{y_1})\) <strong><em>A1</em></strong></p>
<p>so <em>R</em> is symmetric</p>
<p>\(({x_1},{\text{ }}{y_1})R({x_2},{\text{ }}{y_2})\) and \(({x_2},{\text{ }}{y_2})R({x_3},{\text{ }}{y_3}) \Rightarrow {x_1}{y_1} = {x_2}{y_2}\) and \({x_2}{y_2} = {x_3}{y_3}\) <strong><em>M1</em></strong></p>
<p>\( \Rightarrow {x_1}{y_1} = {x_3}{y_3} \Rightarrow ({x_1},{\text{ }}{y_1})R({x_3},{\text{ }}{y_3})\) <strong><em>A1</em></strong></p>
<p>so <em>R</em> is transitive</p>
<p><em>R</em> is an equivalence relation <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\((x,{\text{ }}y)R(1,{\text{ }}2)\) <strong><em>(M1)</em></strong></p>
<p>the equivalence class is \(\{ (x,{\text{ }}y)|xy = 2\} \) <strong><em>A1</em></strong></p>
<p><img src="images/Schermafbeelding_2018-02-10_om_14.24.38.png" alt="N17/5/MATHL/HP3/ENG/TZ0/SG/M/03.b"></p>
<p>correct graph <strong><em>A1</em></strong></p>
<p>\((1,{\text{ }}2)\) indicated on the graph <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award last <strong><em>A1 </em></strong>only if plotted on a curve representing the class.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The group \(\{ G,{\text{ }}{ \times _7}\} \) is defined on the set {1, 2, 3, 4, 5, 6} where \({ \times _7}\) denotes multiplication modulo 7.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the Cayley table for \(\{ G,{\text{ }}{ \times _7}\} \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine whether or not \(\{ G,{\text{ }}{ \times _7}\} \) is cyclic.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the subgroup of <em>G</em> of order 3, denoting it by <em>H</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Identify the element of order 2 in <em>G</em> and find its coset with respect to <em>H</em> .</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The group \(\{ K,{\text{ }} \circ \} \) is defined on the six permutations of the integers 1, 2, 3 and \( \circ \) denotes composition of permutations.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(\{ K,{\text{ }} \circ \} \) is non-Abelian.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Giving a reason, state whether or not \(\{ G,{\text{ }}{ \times _7}\} \) and \(\{ K,{\text{ }} \circ \} \) are isomorphic.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) the Cayley table is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img src="data:image/png;base64,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" alt><em><strong><span style="font-family: 'times new roman',times; font-size: medium;"> A3</span></strong></em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Deduct 1 mark for each error up to a maximum of 3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) by considering powers of elements, </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that 3 (or 5) is of order 6 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so the group is cyclic <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) we see that 2 and 4 are of order 3 so the subgroup of order 3 is {1, 2, 4} <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) the element of order 2 is 6 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the coset is {3, 5, 6} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) consider for example</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&3 \\ <br> 2&1&3 <br>\end{array}} \right) \circ \left( {\begin{array}{*{20}{c}}<br> 1&2&3 \\ <br> 2&3&1 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1&2&3 \\ <br> 1&3&2 <br>\end{array}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&3 \\ <br> 2&3&1 <br>\end{array}} \right) \circ \left( {\begin{array}{*{20}{c}}<br> 1&2&3 \\ <br> 2&1&3 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1&2&3 \\ <br> 3&2&1 <br>\end{array}} \right)\) <em><strong>M1A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A1M1A0</em></strong> if both compositions are done in the wrong order.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A1M0A0</em></strong> if the two compositions give the same result, if no further attempt is made to find two permutations which are not commutative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">these are different so the group is not Abelian </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1AG</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) they are not isomorphic because \(\{ G,{\text{ }}{ \times _7}\} \) is Abelian and \(\{ K,{\text{ }} \circ \} \) is not <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">The set of all permutations of the elements \(1,{\text{ }}2,{\text{ }} \ldots 10\) </span>is denoted by \(H\) and the binary operation \( \circ \) represents the composition of permutations.</p>
<p class="p1">The permutation \(p = (1{\text{ }}2{\text{ }}3{\text{ }}4{\text{ }}5{\text{ }}6)(7{\text{ }}8{\text{ }}9{\text{ }}10)\) <span class="s1">generates the subgroup \(\{ G,{\text{ }} \circ \} \) of the group \(\{ H,{\text{ }} \circ \} \)</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the order of \(\{ G,{\text{ }} \circ \} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the identity element in \(\{ G,{\text{ }} \circ \} \).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(p \circ p\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the inverse of \(p \circ p\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the maximum possible order of an element in \(\{ H,{\text{ }} \circ \} \).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Give an example of an element with this order.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">the order of \((G,{\text{ }} \circ )\) is \({\text{lcm}}(6,{\text{ }}4)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = 12\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\left( 1 \right){\rm{ }}\left( 2 \right){\rm{ }}\left( 3 \right){\rm{ }}\left( 4 \right){\rm{ }}\left( 5 \right){\rm{ }}\left( 6 \right){\rm{ }}\left( 7 \right){\rm{ }}\left( 8 \right){\rm{ }}\left( 9 \right){\rm{ }}\left( {10} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept ( ) or a word description.</p>
<p class="p3"><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(p \circ p = (1{\text{ }}3{\text{ }}5)(2{\text{ }}4{\text{ }}6)(7{\text{ }}9)(810)\) <span class="Apple-converted-space"> </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><span class="s1">(ii) <span class="Apple-converted-space"> </span>its inverse </span>\( = (1{\text{ }}5{\text{ }}3)(2{\text{ }}6{\text{ }}4)(7{\text{ }}9)(810)\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p3"><span class="s2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A1 </em></strong></span>for cycles of 2<span class="s2">, <strong><em>A1 </em></strong></span>for cycles of 3<span class="s2">.</span></p>
<p class="p3"><em><strong><span class="s2">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span>considering LCM </span>of length of cycles with length \(2\), \(3\) and \(5\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(30\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><span class="s2">(ii) <span class="Apple-converted-space"> </span><em>eg</em></span>\(\;\;\;(1{\text{ }}2)(3{\text{ }}4{\text{ }}5)(6{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}10)\) <span class="Apple-converted-space"> </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p3"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>allow FT as long as the length of cycles adds to \(10\) and their LCM is consistent with answer to part (i).</p>
<p class="p3"> </p>
<p class="p1"><strong>Note: </strong>Accept alternative notation for each part</p>
<p class="p1"><em><strong>[3 marks]</strong></em></p>
<p class="p1"><em><strong>Total [10 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R </em>is defined on the set \(\mathbb{N}\) such that for \(a{\text{ }},{\text{ }}b \in \mathbb{N}{\text{ }},{\text{ }}aRb\) if and only if \({a^3} \equiv {b^3}(\bmod 7)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>R </em>is an equivalence relation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equivalence class containing 0.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="line-height: normal; font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">Denote the equivalence class containing <em>n</em> by C<em><sub>n</sub></em> .</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">List the first six elements of \({C_1}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="line-height: normal; font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">Denote the equivalence class containing <em>n</em> by C<em><sub>n</sub></em> .</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \({C_n} = {C_{n + 7}}\) for all \(n \in \mathbb{N}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>reflexive</em>: \({a^3} - {a^3} = 0{\text{ }},{\text{ }} \Rightarrow R\) is reflexive <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>symmetric</em>: if \({a^3} \equiv {b^3}(\bmod 7)\) , then \({b^3} \equiv {a^3}(\bmod 7)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow R\) is symmetric <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>transitive</em>: \({a^3} = {b^3} + 7n\) and \({b^3} = {c^3} + 7m\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">then \({a^3} = {c^3} + 7(n + m)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {a^3} \equiv {c^3}(\bmod 7)\) <strong> <em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow R\) is transitive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">and is an equivalence relation <strong><em>AG</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Allow arguments that use \({a^3} - {b^3} \equiv 0(\bmod 7)\) <em>etc.</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em> </em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[6 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ 0,{\text{ }}7,{\text{ }}14,{\text{ }}21,{\text{ }}...\} \) <strong> <em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ 1,{\text{ }}2,{\text{ }}4,{\text{ }}8,{\text{ }}9,{\text{ }}11\} \) <strong> <em>A3</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Deduct 1 mark for each error or omission.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">consider \({(n + 7)^3} = {n^3} + 21{n^2} + 147n + 343 = {n^3} + 7N\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {n^3} \equiv {(n + 7)^3}(\bmod 7) \Rightarrow n\) and \(n + 7\) are in the same equivalence class <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates were mostly aware of the conditions required to show an equivalence relation although many seemed unsure as to the degree of detail required to show that the different conditions are met for the example in this question. In part (b) many candidates found the correct set although a number were unable to write down the set correctly, including or excluding elements that were not part of the equivalence class. Part (c) saw candidate being less successful than (b) and relatively few candidates were able to prove the equivalence class in part (d) although there were a number of very good solutions.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates were mostly aware of the conditions required to show an equivalence relation although many seemed unsure as to the degree of detail required to show that the different conditions are met for the example in this question. In part (b) many candidates found the correct set although a number were unable to write down the set correctly, including or excluding elements that were not part of the equivalence class. Part (c) saw candidate being less successful than (b) and relatively few candidates were able to prove the equivalence class in part (d) although there were a number of very good solutions.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates were mostly aware of the conditions required to show an equivalence relation although many seemed unsure as to the degree of detail required to show that the different conditions are met for the example in this question. In part (b) many candidates found the correct set although a number were unable to write down the set correctly, including or excluding elements that were not part of the equivalence class. Part (c) saw candidate being less successful than (b) and relatively few candidates were able to prove the equivalence class in part (d) although there were a number of very good solutions.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates were mostly aware of the conditions required to show an equivalence relation although many seemed unsure as to the degree of detail required to show that the different conditions are met for the example in this question. In part (b) many candidates found the correct set although a number were unable to write down the set correctly, including or excluding elements that were not part of the equivalence class. Part (c) saw candidate being less successful than (b) and relatively few candidates were able to prove the equivalence class in part (d) although there were a number of very good solutions.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The function \(f\) is defined by \(f:{\mathbb{R}^ + } \times {\mathbb{R}^ + } \to {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) where \(f(x,{\text{ }}y) = \left( {\sqrt {xy} ,{\text{ }}\frac{x}{y}} \right)\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(f\) is an injection.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Prove that \(f\) is a surjection.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence, or otherwise, write down the inverse function \({f^{ - 1}}\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">let \((a,{\text{ }}b)\) and \((c,{\text{ }}d) \in {\mathbb{R}^ + } \times {\mathbb{R}^ + }\)</span></p>
<p class="p1">suppose that \(f(a,{\text{ }}b) = f(c,{\text{ }}d)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">so that \(\sqrt {ab} = \sqrt {cd} \) <span class="s2">and \(\frac{a}{b} = \frac{c}{d}\) <span class="Apple-converted-space"> </span></span><strong><em>A1</em></strong></p>
<p class="p1">leading to either \({a^2} = {c^2}\) <span class="s3">or \({b^2} = {d^2}\) </span>or equivalent <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">state \(a = c\) <span class="s2">and \(b = d\) <span class="Apple-converted-space"> </span></span><strong><em>A1</em></strong></p>
<p class="p1">this shows that \(f\) is an injection since \(f(a,{\text{ }}b) = f(c,{\text{ }}d) \Rightarrow (a,{\text{ }}b) = (c,{\text{ }}d)\) <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept final statement seen anywhere for <strong><em>R1</em></strong>.</p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span>now let \((u,{\text{ }}v) \in {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) </span>and suppose that \(f(x,{\text{ }}y) = (u,{\text{ }}v)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2">then, \(u = \sqrt {xy} ,{\text{ }}v = \frac{x}{y}\) <span class="Apple-converted-space"> </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p1">attempt to eliminate \(x\) or \(y\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \Rightarrow x = u{v^{1/2}};{\text{ }}y = u{v^{ - 1/2}}\) </span><strong><em>A1A1</em></strong></p>
<p class="p2"><span class="s2">this shows that \(f\) </span>is a surjection since, given \((u,{\text{ }}v)\), there exists \((x,{\text{ }}y)\) <span class="s2">such that \(f(x,{\text{ }}y) = (u,{\text{ }}v)\) <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></span></p>
<p class="p1"><strong>Note: </strong>Accept final statement, seen anywhere, for <strong><em>R1</em></strong>.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> \({f^{ - 1}}(x,{\text{ }}y) = (x{y^{1/2}},{\text{ }}x{y^{ - 1/2}})\)</span> <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Those candidates who formulated their responses in terms of the basic mathematical definitions of injectivity and surjectivity were usually successful. Otherwise, verbal attempts such as ‘\(f\) is one-to-one \( \Rightarrow f\) is injective’ or ‘\(g\) is surjective because its range equals its codomain’, received no credit.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) Those candidates who formulated their responses in terms of the basic mathematical definitions of injectivity and surjectivity were usually successful. Otherwise, verbal attempts such as ‘\(f\) is one-to-one \( \Rightarrow f\) is injective’ or ‘\(g\) is surjective because its range equals its codomain’, received no credit.</p>
<p class="p1">(ii) It was surprising to see that some candidates were unable to relate what they had done in part (b)(i) to this part.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The relation \(R=\) is defined on \({\mathbb{Z}^ + }\) such that \(aRb\) if and only if \({b^n} - {a^n} \equiv 0(\bmod p)\) where \(n,{\text{ }}p\) <span class="s1">are fixed positive integers greater than 1.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(R\) is an equivalence relation.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(n = 2\) and \(p = 7\), determine the first four members of each of the four equivalence classes of \(R\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">since \({a^n} - {a^n} = 0\), <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">it follows that \((aRa)\) and \(R\) is reflexive <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">if \(aRb\) so that \({b^n} - {a^n} \equiv 0(\bmod p)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">then, \({a^n} - {b^n} \equiv 0(\bmod p)\) so that \(bRa\) and \(R\) is symmetric <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">if \(aRb\) and \(bRc\) so that \({b^n} - {a^n} \equiv 0(\bmod p)\) and \({c^n} - {b^n} \equiv 0(\bmod p)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">adding, (it follows that \({c^n} - {a^n} \equiv 0(\bmod p)\)) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">so that \(aRc\) and \(R\) is transitive <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Only accept the correct use of the terms “reflexive, symmetric, transitive”.</p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">we are now given that \(aRb\) if \({b^2} - {a^2} \equiv 0(\bmod 7)\)</p>
<p class="p1">attempt to find at least one equivalence class <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2">the equivalence classes are</p>
<p class="p2"><span class="Apple-converted-space">\(\{ 1,{\text{ }}6,{\text{ }}8,{\text{ }}13,{\text{ }} \ldots \} \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(\{ 2,{\text{ }}5,{\text{ }}9,{\text{ }}12,{\text{ }} \ldots \} \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(\{ 3,{\text{ }}4,{\text{ }}10,{\text{ }}11,{\text{ }} \ldots \} \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(\{ 7,{\text{ }}14,{\text{ }}21,{\text{ }}28,{\text{ }} \ldots \} \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates were familiar with the terminology of the required conditions to be satisfied for a relation to be an equivalence relation. The execution of the proofs was variable. It was grating to see such statements as \(R\) is symmetric because \(aRb = bRa\) or \(aRa = {a^n} - {a^n} = 0\), often without mention of \(\bmod p\)<span class="s1"><em>,</em> and such responses were not fully </span>rewarded.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was not well answered. Few candidates displayed a strategy to find the equivalence classes.</p>
<p class="p6"> </p>
<p class="p7"> </p>
<p class="p8"> </p>
<p class="p9"> </p>
<p class="p1"> </p>
<p class="p10"> </p>
<p class="p7"> </p>
<p class="p11"> </p>
<p class="p8"> </p>
<p class="p8"> </p>
<p class="p7"> </p>
<p class="p8"> </p>
<p class="p12"> </p>
<p class="p8"> </p>
<p class="p8"> </p>
<p class="p8"> </p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>c </em>be a positive, real constant. Let <em>G </em>be the set \(\{ \left. {x \in \mathbb{R}} \right| - c < x < c\} \) . The binary </span><span style="font-family: 'times new roman', times; font-size: medium;">operation \( * \) is defined on the set <em>G </em>by \(x * y = \frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 44.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Simplify \(\frac{c}{2} * \frac{{3c}}{4}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">State the identity element for <em>G </em>under \( * \).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">For \(x \in G\) find an expression for \({x^{ - 1}}\) (the inverse of <em>x </em>under \( * \)).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the binary operation \( * \) is commutative on <em>G </em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the binary operation \( * \) is associative on <em>G </em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) If \(x,{\text{ }}y \in G\) explain why \((c - x)(c - y) > 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence show that \(x + y < c + \frac{{xy}}{c}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>G </em>is closed under \( * \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why \(\{ G, * \} \) is an Abelian group.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{c}{2} * \frac{{3c}}{4} = \frac{{\frac{c}{2} + \frac{{3c}}{4}}}{{1 + \frac{1}{2} \cdot \frac{3}{4}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\frac{{5c}}{4}}}{{\frac{{11}}{8}}} = \frac{{10c}}{{11}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">identity is 0 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">inverse is –<em>x</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x * y = \frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}},{\text{ }}y * x = \frac{{y + x}}{{1 + \frac{{yx}}{{{c^2}}}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(since ordinary addition and multiplication are commutative)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x * y = y * x{\text{ so }} * \) is commutative </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Accept arguments using symmetry.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"> </p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\((x * y) * z = \frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}} * z = \frac{{\left( {\frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}}} \right) + z}}{{1 + \left( {\frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}}} \right)\frac{z}{{{c^2}}}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\frac{{\left( {x + y + z + \frac{{xyz}}{{{c^2}}}} \right)}}{{\left( {1 + \frac{{xy}}{{{c^2}}}} \right)}}}}{{\frac{{\left( {1 + \frac{{xy}}{{{c^2}}} + \frac{{xz}}{{{c^2}}} + \frac{{yz}}{{{c^2}}}} \right)}}{{\left( {1 + \frac{{xy}}{{{c^2}}}} \right)}}}} = \frac{{\left( {x + y + z + \frac{{xyz}}{{{c^2}}}} \right)}}{{\left( {1 + \left( {\frac{{xy + xz + yz}}{{{c^2}}}} \right)} \right)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x * (y * z) = x * \left( {\frac{{y + z}}{{1 + \frac{{yz}}{{{c^2}}}}}} \right) = \frac{{x + \left( {\frac{{y + z}}{{1 + \frac{{yz}}{{{c^2}}}}}} \right)}}{{1 + \frac{x}{{{c^2}}}\left( {\frac{{y + z}}{{1 + \frac{{yz}}{{{c^2}}}}}} \right)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\frac{{\left( {x + \frac{{xyz}}{{{c^2}}} + y + z} \right)}}{{\left( {1 + \frac{{yz}}{{{c^2}}}} \right)}}}}{{\frac{{\left( {1 + \frac{{yz}}{{{c^2}}} + \frac{{xy}}{{{c^2}}} + \frac{{xz}}{{{c^2}}}} \right)}}{{\left( {1 + \frac{{yz}}{{{c^2}}}} \right)}}}} = \frac{{\left( {x + y + z + \frac{{xyz}}{{{c^2}}}} \right)}}{{\left( {1 + \left( {\frac{{xy + xz + yz}}{{{c^2}}}} \right)} \right)}}\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">since both expressions are the same \( * \) is associative <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note</strong><span style="font-family: 'times new roman', times; font-size: medium;">: After the initial </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;">, correct arguments using symmetry also gain full marks.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(c > x{\text{ and }}c > y \Rightarrow c - x > 0{\text{ and }}c - y > 0 \Rightarrow (c - x)(c - y) > 0\) <strong><em>R1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({c^2} - cx - cy + xy > 0 \Rightarrow {c^2} + xy > cx + cy \Rightarrow c + \frac{{xy}}{c} > x + y{\text{ (as }}c > 0)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(x + y < c + \frac{{xy}}{c}\) <strong><em>M1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">if \(x,{\text{ }}y \in G{\text{ then }} - c - \frac{{xy}}{c} < x + y < c + \frac{{xy}}{c}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">thus \( - c\left( {1 + \frac{{xy}}{{{c^2}}}} \right) < x + y < c\left( {1 + \frac{{xy}}{{{c^2}}}} \right){\text{ and }} - c < \frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}} < c\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(({\text{as }}1 + \frac{{xy}}{{{c^2}}} > 0){\text{ so }} - c < x * y < c\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">proving that <em>G </em>is closed under \( * \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">as \(\{ G, * \} \) is closed, is associative, has an identity and all elements have an inverse <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">it is a group <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">as \( * \) is commutative <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">it is an Abelian group <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to answer part (a) indicating preparation in such questions. Many students failed to identify the command term “state” in parts (b) and (c) and spent a lot of time – usually unsuccessfully - with algebraic methods. Most students were able to offer satisfactory solutions to part (d) and although most showed that they knew what to do in part (e), few were able to complete the proof of associativity. Surprisingly few managed to answer parts (f) and (g) although many who continued to this stage, were able to pick up at least one of the marks for part (h), regardless of what they had done before. Many candidates interpreted the question as asking to prove that the group was Abelian, rather than proving that it was an Abelian group. Few were able to fully appreciate the significance in part (i) although there were a number of reasonable solutions.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to answer part (a) indicating preparation in such questions. Many students failed to identify the command term “state” in parts (b) and (c) and spent a lot of time – usually unsuccessfully - with algebraic methods. Most students were able to offer satisfactory solutions to part (d) and although most showed that they knew what to do in part (e), few were able to complete the proof of associativity. Surprisingly few managed to answer parts (f) and (g) although many who continued to this stage, were able to pick up at least one of the marks for part (h), regardless of what they had done before. Many candidates interpreted the question as asking to prove that the group was Abelian, rather than proving that it was an Abelian group. Few were able to fully appreciate the significance in part (i) although there were a number of reasonable solutions.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to answer part (a) indicating preparation in such questions. Many students failed to identify the command term “state” in parts (b) and (c) and spent a lot of time – usually unsuccessfully - with algebraic methods. Most students were able to offer satisfactory solutions to part (d) and although most showed that they knew what to do in part (e), few were able to complete the proof of associativity. Surprisingly few managed to answer parts (f) and (g) although many who continued to this stage, were able to pick up at least one of the marks for part (h), regardless of what they had done before. Many candidates interpreted the question as asking to prove that the group was Abelian, rather than proving that it was an Abelian group. Few were able to fully appreciate the significance in part (i) although there were a number of reasonable solutions.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to answer part (a) indicating preparation in such questions. Many students failed to identify the command term “state” in parts (b) and (c) and spent a lot of time – usually unsuccessfully - with algebraic methods. Most students were able to offer satisfactory solutions to part (d) and although most showed that they knew what to do in part (e), few were able to complete the proof of associativity. Surprisingly few managed to answer parts (f) and (g) although many who continued to this stage, were able to pick up at least one of the marks for part (h), regardless of what they had done before. Many candidates interpreted the question as asking to prove that the group was Abelian, rather than proving that it was an Abelian group. Few were able to fully appreciate the significance in part (i) although there were a number of reasonable solutions.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to answer part (a) indicating preparation in such questions. Many students failed to identify the command term “state” in parts (b) and (c) and spent a lot of time – usually unsuccessfully - with algebraic methods. Most students were able to offer satisfactory solutions to part (d) and although most showed that they knew what to do in part (e), few were able to complete the proof of associativity. Surprisingly few managed to answer parts (f) and (g) although many who continued to this stage, were able to pick up at least one of the marks for part (h), regardless of what they had done before. Many candidates interpreted the question as asking to prove that the group was Abelian, rather than proving that it was an Abelian group. Few were able to fully appreciate the significance in part (i) although there were a number of reasonable solutions.</span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to answer part (a) indicating preparation in such questions. Many students failed to identify the command term “state” in parts (b) and (c) and spent a lot of time – usually unsuccessfully - with algebraic methods. Most students were able to offer satisfactory solutions to part (d) and although most showed that they knew what to do in part (e), few were able to complete the proof of associativity. Surprisingly few managed to answer parts (f) and (g) although many who continued to this stage, were able to pick up at least one of the marks for part (h), regardless of what they had done before. Many candidates interpreted the question as asking to prove that the group was Abelian, rather than proving that it was an Abelian group. Few were able to fully appreciate the significance in part (i) although there were a number of reasonable solutions.</span></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to answer part (a) indicating preparation in such questions. Many students failed to identify the command term “state” in parts (b) and (c) and spent a lot of time – usually unsuccessfully - with algebraic methods. Most students were able to offer satisfactory solutions to part (d) and although most showed that they knew what to do in part (e), few were able to complete the proof of associativity. Surprisingly few managed to answer parts (f) and (g) although many who continued to this stage, were able to pick up at least one of the marks for part (h), regardless of what they had done before. Many candidates interpreted the question as asking to prove that the group was Abelian, rather than proving that it was an Abelian group. Few were able to fully appreciate the significance in part (i) although there were a number of reasonable solutions.</span></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to answer part (a) indicating preparation in such questions. Many students failed to identify the command term “state” in parts (b) and (c) and spent a lot of time – usually unsuccessfully - with algebraic methods. Most students were able to offer satisfactory solutions to part (d) and although most showed that they knew what to do in part (e), few were able to complete the proof of associativity. Surprisingly few managed to answer parts (f) and (g) although many who continued to this stage, were able to pick up at least one of the marks for part (h), regardless of what they had done before. Many candidates interpreted the question as asking to prove that the group was Abelian, rather than proving that it was an Abelian group. Few were able to fully appreciate the significance in part (i) although there were a number of reasonable solutions.</span></p>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Below are the graphs of the two functions \(F:P \to Q{\text{ and }}g:A \to B\) .</span></p>
<p><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Determine, with reference to features of the graphs, whether the functions are injective and/or surjective.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given two functions \(h:X \to Y{\text{ and }}k:Y \to Z\) . </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) if both <em>h</em> and <em>k</em> are injective then so is the composite function \(k \circ h\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) if both <em>h</em> and <em>k</em> are surjective then so is the composite function \(k \circ h\) .</span></p>
<div> </div>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>f </em>is surjective because every horizontal line through <em>Q </em>meets the graph somewhere <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>f </em>is not injective because it is a many-to-one function <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>g </em>is injective because it always has a positive gradient <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(accept horizontal line test reasoning)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>g </em>is not surjective because a horizontal line through the negative part of <em>B </em>would not meet the graph at all <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \({x_1},{\text{ }}{x_2} \in X{\text{ and }}{y_1} = h({x_1}){\text{ and }}{y_2} = h({x_2})\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Then</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k \circ \left( {h({x_1})} \right) = k \circ \left( {h({x_2})} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow k({y_1}) = k({y_2})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {y_1} = {y_2}\,\,\,\,\,{\text{(}}k{\text{ is injective)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow h({x_1}) = h({x_2})\,\,\,\,\,\left( {h({x_1}) = {y_1}{\text{ and }}h({x_2}) = {y_2}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {x_1} \equiv {x_2}\,\,\,\,\,(h{\text{ is injective)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence \(k \circ h\) is injective <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{x}}_1},{\text{ }}{x_2} \in X,{\text{ }}{x_1} \ne {x_2}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since <em>h </em>is an injection \( \Rightarrow h({x_1}) \ne h({x_2})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(h({x_1}),{\text{ }}h({x_2}) \in Y\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since <em>k </em>is an injection \( \Rightarrow k\left( {h({x_1})} \right) \ne k\left( {h({x_2})} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k\left( {h({x_1})} \right),{\text{ }}k\left( {h({x_2})} \right) \in \mathbb{Z}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(k \circ h\) is an injection. <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <em>h </em>and <em>k </em>are surjections and let \(z \in \mathbb{Z}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Since <em>k </em>is surjective there exists \(y \in Y\) such that <em>k</em>(<em>y</em>) = <em>z </em><strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Since <em>h </em>is surjective there exists \(x \in X\) such that <em>h</em>(<em>x</em>) = <em>y </em><strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore there exists \(x \in X\) such that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k \circ h(x) = k\left( {h(x)} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = k(y)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = z\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">So \(k \circ h\) is surjective <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">‘Using features of the graph’ should have been a fairly open hint but too many candidates contented themselves with describing what injective and surjective meant rather than explaining which graph had which properties. Candidates found considerable difficulty with presenting a convincing argument in part (b).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">‘Using features of the graph’ should have been a fairly open hint but too many candidates contented themselves with describing what injective and surjective meant rather than explaining which graph had which properties. Candidates found considerable difficulty with presenting a convincing argument in part (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the group \(\{ G,{\text{ }}{ \times _{18}}\} \) defined on the set \(\{ 1,{\text{ }}5,{\text{ }}7,{\text{ }}11,{\text{ }}13,{\text{ }}17\} \) where \({ \times _{18}}\) denotes multiplication modulo 18. The group \(\{ G,{\text{ }}{ \times _{18}}\} \) is shown in the following Cayley table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-10_om_09.24.56.png" alt="N17/5/MATHL/HP3/ENG/TZ0/SG/01"></p>
</div>
<div class="specification">
<p>The subgroup of \(\{ G,{\text{ }}{ \times _{18}}\} \) of order two is denoted by \(\{ K,{\text{ }}{ \times _{18}}\} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the order of elements 5, 7 and 17 in \(\{ G,{\text{ }}{ \times _{18}}\} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State whether or not \(\{ G,{\text{ }}{ \times _{18}}\} \) is cyclic, justifying your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the elements in set \(K\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the left cosets of \(K\) in \(\{ G,{\text{ }}{ \times _{18}}\} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>considering powers of elements <strong><em>(M1)</em></strong></p>
<p>5 has order 6 <strong><em>A1</em></strong></p>
<p>7 has order 3 <strong><em>A1</em></strong></p>
<p>17 has order 2 <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(G\) is cyclic <strong><em>A1</em></strong></p>
<p>because there is an element (are elements) of order 6 <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept “there is a generator”; allow <strong><em>A1R0</em></strong>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\{ 1,{\text{ }}17\} \) <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>multiplying \(\{ 1,{\text{ }}17\} \) by each element of \(G\) <strong><em>(M1)</em></strong></p>
<p>\(\{ 1,{\text{ }}17\} ,{\text{ }}\{ 5,{\text{ }}13\} ,{\text{ }}\{ 7,{\text{ }}11\} \) <strong><em>A1A1A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A group \(\{ D,{\text{ }}{ \times _3}\} \) <span class="s1">is defined so that \(D = \{ 1,{\text{ }}2\} \) </span>and \({ \times _3}\) is multiplication modulo \(3\)<span class="s1">.</span></p>
<p class="p2">A function \(f:\mathbb{Z} \to D\) is defined as \(f:x \mapsto \left\{ {\begin{array}{*{20}{c}} {1,{\text{ }}x{\text{ is even}}} \\ {2,{\text{ }}x{\text{ is odd}}} \end{array}} \right.\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that the function \(f\) is a homomorphism from the group \(\{ \mathbb{Z},{\text{ }} + \} {\text{ to }}\{ D,{\text{ }}{ \times _3}\} \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the kernel of \(f\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(\{ {\text{Ker}}(f),{\text{ }} + \} \) is a subgroup of \(\{ \mathbb{Z},{\text{ }} + \} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">consider the cases, \(a\) and \(b\) both even, <em>one </em>is even and <em>one </em>is odd and \(a\) and \(b\) are both odd <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">calculating \(f(a + b)\) and \(f(a){ \times _3}f(b)\) in at least one case <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">if \(a\) is even and \(b\)<span class="Apple-converted-space"> </span>is even, then \(a + b\) is even</p>
<p class="p1">so\(\;\;\;f(a + b) = 1.\;\;\;f(a){ \times _3}f(b) = 1{ \times _3}1 = 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so\(\;\;\;f(a + b) = f(a){ \times _3}f(b)\)</p>
<p class="p1">if <em>one </em>is even and <em>the other </em>is odd, then \(a + b\) is odd</p>
<p class="p1">so\(\;\;\;f(a + b) = 2.\;\;\;f(a){ \times _3}f(b) = 1{ \times _3}2 = 2\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so\(\;\;\;f(a + b) = f(a){ \times _3}f(b)\)</p>
<p class="p1">if \(a\) is odd and \(b\) is odd, then \(a + b\) is even</p>
<p class="p1">so\(\;\;\;f(a + b) = 1.\;\;\;f(a){ \times _3}f(b) = 2{ \times _3}2 = 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so\(\;\;\;f(a + b) = f(a){ \times _3}f(b)\)</p>
<p class="p1">as\(\;\;\;f(a + b) = f(a){ \times _3}f(b)\;\;\;\)in all cases, so\(\;\;\;f:\mathbb{Z} \to D\) is a homomorphism <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">\(1\) </span>is the identity of \(\{ D,{\text{ }}{ \times _3}\} \) <span class="Apple-converted-space"> </span><strong><em>(M1)(A1)</em></strong></p>
<p class="p1"><span class="s1">so</span>\(\;\;\;{\text{Ker}}(f)\) is all even numbers <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">sum of any two even numbers is even so closure applies <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">associative as it is a subset of \(\{ \mathbb{Z},{\text{ }} + \} \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="s1">identity is \(0\)</span>, which is in the kernel <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">the inverse of any even number is also even <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">\({\text{ker}}(f) \ne \emptyset \)</p>
<p class="p1">\({b^{ - 1}} \in {\text{ker}}(f)\) for any \(b\)</p>
<p class="p1">\(a{b^{ - 1}} \in {\text{ker}}(f)\) for any \(a\) and \(b\)</p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Allow a general proof that the Kernel is always a subgroup.</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<p class="p1"><em><strong>Total [13 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Associativity and commutativity are two of the five conditions for a set <em>S </em>with the binary operation \( * \) to be an Abelian group; state the other three conditions.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The Cayley table for the binary operation \( \odot \) defined on the set <em>T </em>= {<em>p</em>, <em>q</em>, <em>r</em>, <em>s</em>, <em>t</em>} is given below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that exactly three of the conditions for {<em>T </em>, \( \odot \)} to be an Abelian group are satisfied, but that neither associativity nor commutativity are satisfied.</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the proper subsets of <em>T </em>that are groups of order 2, and comment on your result in the context of Lagrange’s theorem.</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the solutions of the equation \((p \odot x) \odot x = x \odot p\)<em> </em>.</span></p>
<div class="marks">[15]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">closure, identity, inverse <strong><em>A2</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for two correct properties, <strong><em>A0 </em></strong>otherwise.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) closure: there are no extra elements in the table <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">identity: <em>s </em>is a (left and right) identity <strong><em>R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">inverses: all elements are self-inverse <strong><em>R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">commutative: no, because the table is not symmetrical about the leading diagonal, or by counterexample <strong><em>R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">associativity: for example, \((pq)t = rt = p\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">not associative because \(p(qt) = pr = t \ne p\) <strong><em>R1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>M1A1 </em></strong>for 1 complete example whether or not it shows non-associativity.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(\{ s,\,p\} ,{\text{ }}\{ s,\,q\} ,{\text{ }}\{ s,\,r\} ,{\text{ }}\{ s,\,t\} \) <strong><em>A2</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for 2 or 3 correct sets.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">as 2 does not divide 5, Lagrange’s theorem would have been contradicted if <em>T </em>had been a group <strong><em>R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) any attempt at trying values <strong><em>(M1)<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">the solutions are <em>q</em>, <em>r</em>, <em>s </em>and <em>t </em><strong><em>A1A1A1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Deduct <strong><em>A1 </em></strong>if <em>p </em>is included.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[15 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was on the whole a well answered question and it was rare for a candidate not to obtain full marks on part (a). In part (b) the vast majority of candidates were able to show that the set satisfied the properties of a group apart from associativity which they were also familiar with. Virtually all candidates knew the difference between commutativity and associativity and were able to distinguish between the two. Candidates were familiar with Lagrange’s Theorem and many were able to see how it did not apply in the case of this problem. Many candidates found a solution method to part (iii) of the problem and obtained full marks.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">This was on the whole a well answered question and it was rare for a candidate not to obtain full marks on part (a). In part (b) the vast majority of candidates were able to show that the set satisfied the properties of a group apart from associativity which they were also familiar with. Virtually all candidates knew the difference between commutativity and associativity and were able to distinguish between the two. Candidates were familiar with Lagrange’s Theorem and many were able to see how it did not apply in the case of this problem. Many candidates found a solution method to part (iii) of the problem and obtained full marks.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The binary operation \( * \) is defined by</p>
<p style="text-align: center;">\(a * b = a + b - 3\) for \(a,{\text{ }}b \in \mathbb{Z}\).</p>
</div>
<div class="specification">
<p>The binary operation \( \circ \) is defined by</p>
<p style="text-align: center;">\(a \circ b = a + b + 3\) for \(a,{\text{ }}b \in \mathbb{Z}\).</p>
<p>Consider the group \(\{ \mathbb{Z},{\text{ }} \circ {\text{\} }}\) and the bijection \(f:\mathbb{Z} \to \mathbb{Z}\) given by \(f(a) = a - 6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\{ \mathbb{Z},{\text{ }} * \} \) is an Abelian group.</p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there is no element of order 2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a proper subgroup of \(\{ \mathbb{Z},{\text{ }} * \} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the groups \(\{ \mathbb{Z},{\text{ }} * \} \) and \(\{ \mathbb{Z},{\text{ }} \circ \} \) are isomorphic.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>closure: \(\{ \mathbb{Z},{\text{ }} * \} \) is closed because \(a + b - 3 \in \mathbb{Z}\) <strong><em>R1</em></strong></p>
<p>identity: \(a * e = a + e - 3 = a\) <strong><em>(M1)</em></strong></p>
<p>\(e = 3\) <strong><em>A1</em></strong></p>
<p>inverse: \(a * {a^{ - 1}} = a + {a^{ - 1}} - 3 = 3\) <strong><em>(M1)</em></strong></p>
<p>\({a^{ - 1}} = 6 - a\) <strong><em>A1</em></strong></p>
<p>associative: \(a * (b * c) = a * (b + c - 3) = a + b + c - 6\) <strong><em>A1</em></strong></p>
<p>\(\left( {a{\text{ }}*{\text{ }}b} \right){\text{ }}*{\text{ }}c{\text{ }} = \left( {a{\text{ }} + {\text{ }}b{\text{ }} - {\text{ }}3} \right)*{\text{ }}c{\text{ }} = {\text{ }}a{\text{ }} + {\text{ }}b{\text{ }} + {\text{ }}c{\text{ }} - {\text{ }}6\) <strong><em>A1</em></strong></p>
<p>associative because \(a * (b * c) = (a * b) * c\) <strong><em>R1</em></strong></p>
<p>\(b * a = b + a - 3 = a + b - 3 = a * b\) therefore commutative hence Abelian <strong><em>R1</em></strong></p>
<p>hence \(\{ \mathbb{Z},{\text{ }} * \} \) is an Abelian group <strong><em>AG</em></strong></p>
<p><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>if \(a\) is of order 2 then \(a * a = 2a - 3 = 3\) therefore \(a = 3\) <strong><em>A1</em></strong></p>
<p>which is a contradiction</p>
<p>since \(e = 3\) and has order 1 <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>R1 </em></strong>for recognising that the identity has order 1.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>for example \(S = \{ - 6,{\text{ }} - 3,{\text{ }}0,{\text{ }}3,{\text{ }}6 \ldots \} \) or \(S = \{ \ldots ,{\text{ }} - 1,{\text{ }}1,{\text{ }}3,{\text{ }}5,{\text{ }}7 \ldots \} \) <strong><em>A1R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>R1 </em></strong>for deducing, justifying or verifying that \(\left\{ {S, * } \right\}\) is indeed a proper subgroup.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>we need to show that \(f(a * b) = f(a) \circ f(b)\) <strong><em>R1</em></strong></p>
<p>\(f(a * b) = f(a + b - 3) = a + b - 9\) <strong><em>A1</em></strong></p>
<p>\(f(a) \circ f(b) = (a - 6) \circ (b - 6) = a + b - 9\) <strong><em>A1</em></strong></p>
<p>hence isomorphic <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>R1</em></strong> for recognising that \(f\) preserves the operation; award <strong><em>R1A0A0</em></strong> for an attempt to show that \(f(a \circ b) = f(a) * f(b)\).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The set \(S\) is defined as the set of real numbers greater than 1.</p>
<p>The binary operation \( * \) is defined on \(S\) by \(x * y = (x - 1)(y - 1) + 1\) for all \(x,{\text{ }}y \in S\).</p>
</div>
<div class="specification">
<p>Let \(a \in S\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(x * y \in S\) for all \(x,{\text{ }}y \in S\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the operation \( * \) on the set \(S\) is commutative.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the operation \( * \) on the set \(S\) is associative.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that 2 is the identity element.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that each element \(a \in S\) has an inverse.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(x,{\text{ }}y > 1 \Rightarrow (x - 1)(y - 1) > 0\) <strong><em>M1</em></strong></p>
<p>\((x - 1)(y - 1) + 1 > 1\) <strong><em>A1</em></strong></p>
<p>so \(x * y \in S\) for all \(x,{\text{ }}y \in S\) <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(x * y = (x - 1)(y - 1) + 1 = (y - 1)(x - 1) + 1 = y * x\) <strong><em>M1A1</em></strong></p>
<p>so \( * \) is commutative <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(x * (y * z) = x * \left( {(y - 1)(z - 1) + 1} \right)\) <strong><em>M1</em></strong></p>
<p>\( = (x - 1)\left( {(y - 1)(z - 1) + 1 - 1} \right) + 1\) <strong><em>(A1)</em></strong></p>
<p>\( = (x - 1)(y - 1)(z - 1) + 1\) <strong><em>A1</em></strong></p>
<p>\((x * y) * z = \left( {(x - 1)(y - 1) + 1} \right) * z\) <strong><em>M1</em></strong></p>
<p>\( = \left( {(x - 1)(y - 1) + 1 - 1} \right)(z - 1) + 1\)</p>
<p>\( = (x - 1)(y - 1)(z - 1) + 1\) <strong><em>A1</em></strong></p>
<p>so \( * \) is associative <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(2 * x = (2 - 1)(x - 1) + 1 = x,{\text{ }}x * 2 = (x - 1)(2 - 1) + 1 = x\) <strong><em>M1</em></strong></p>
<p>\(2 * x = x * 2 = 2{\text{ }}(2 \in S)\) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept reference to commutativity instead of explicit expressions.</p>
<p> </p>
<p>so 2 is the identity element <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(a * {a^{ - 1}} = 2 \Rightarrow (a - 1)({a^{ - 1}} - 1) + 1 = 2\) <strong><em>M1</em></strong></p>
<p>so \({a^{ - 1}} = 1 + \frac{1}{{a - 1}}\) <strong><em>A1</em></strong></p>
<p>since \(a - 1 > 0 \Rightarrow {a^{ - 1}} > 1{\text{ }}({a^{ - 1}} * a = a * {a^{ - 1}})\) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>R1 </em></strong>dependent on <strong><em>M1</em></strong>.</p>
<p> </p>
<p>so each element, \(a \in S\), has an inverse <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The elements of sets <em>P </em>and <em>Q </em>are taken from the universal set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. <em>P </em>= {1, 2, 3} and <em>Q </em>= {2, 4, 6, 8, 10}.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(R = (P \cap Q')'\) , list the elements of <em>R </em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">For a set <em>S </em>, let \({S^ * }\)<span style="font: 7.0px Times;"> </span>denote the set of all subsets of <em>S </em>,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) find \({P^ * }\)<span style="font: 7.0px Times;"> </span>;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) find \(n({R^ * })\)<span style="font: 12.5px Times;"> </span>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P = \{ 1,{\text{ }}2,{\text{ }}3\} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(Q' = \{ 1,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}9\} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(P \cap Q' = \{ 1,{\text{ }}3\} \) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">so \((P \cap Q')' = \{ 2,{\text{ }}4,{\text{ }}5,{\text{ }}6,{\text{ }}7,{\text{ }}8,{\text{ }}9,{\text{ }}10\} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({P^ * } = \left\{ {\{ 1\} ,{\text{ }}\{ 2\} ,{\text{ }}\{ 3\} ,{\text{ }}\{ 1,{\text{ }}2\} ,{\text{ }}\{ 2,{\text{ }}3\} ,{\text{ }}\{ 3,{\text{ }}1\} ,{\text{ }}\{ 1,{\text{ }}2,{\text{ }}3),{\text{ }}\emptyset } \right\}\) <strong><em>A2</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>if one error, <strong><em>A0 </em></strong>for two or more.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({R^ * }\) contains: the empty set (1 element); sets containing one element (8 elements); sets containing two elements <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 0 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 1 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 2 <br>\end{array}} \right) + ...\left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 8 <br>\end{array}} \right)\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {2^8}{\text{ }}( = 256)\) <strong> <em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>FT </em></strong>in (ii) applies if no empty set included in (i) and (ii).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was also a well answered question with many candidates obtaining full marks on both parts of the problem. Some candidates attempted to use a factorial rather than a sum of combinations to solve part (b) (ii) and this led to incorrect answers.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was also a well answered question with many candidates obtaining full marks on both parts of the problem. Some candidates attempted to use a factorial rather than a sum of combinations to solve part (b) (ii) and this led to incorrect answers.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The relation \(R\) is defined such that \(aRb\) if and only if \({4^a} - {4^b}\) is divisible by 7, where \(a,{\text{ }}b \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="specification">
<p>The equivalence relation \(S\) is defined such that \(cSd\) if and only if \({4^c} - {4^d}\) is divisible by 6, where \(c,{\text{ }}d \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(R\) is an equivalence relation.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the equivalence classes of \(R\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the number of equivalence classes of \(S\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>reflexive: \({4^a} - {4^a} = 0\) which is divisible by 7 (for all \(a \in \mathbb{Z}\)) <strong><em>R1</em></strong></p>
<p>so \(aRa\) therefore reflexive</p>
<p>symmetric: Let \(aRb\) so that \({4^a} - {4^b}\) is divisible by 7 <strong><em>M1</em></strong></p>
<p>it follows that \({4^b} - {4^a} = - ({4^a} - {4^b})\) is also divisible by 7 <strong><em>A1</em></strong></p>
<p>it follows that \(bRa\) therefore symmetric</p>
<p>transitive: let \(aRb\) and \(bRc\) so that \({4^a} - {4^b}\) and \({4^b} - {4^c}\) are divisible by 7 <strong><em>M1</em></strong></p>
<p>it follows that \({4^a} - {4^b} = 7M\) and \({4^b} - {4^c} = 7N\) so that \(({4^a} - {4^b}) + ({4^b} - {4^c}) = {4^a} - {4^c} = 7(M + N)\) <strong><em>A1</em></strong></p>
<p>therefore \(aRb\) and \(bRc \Rightarrow aRc\) <strong><em>R1</em></strong></p>
<p>so that \(R\) is transitive</p>
<p> </p>
<p><strong>Note:</strong> For transitivity, award <strong><em>A0 </em></strong>if the same variable is used to express the multiples of 7; <strong><em>R1 </em></strong>is dependent on the <strong><em>M </em></strong>mark.</p>
<p> </p>
<p>since \(R\) <em>R </em>is reflexive, symmetric and transitive, it is an equivalence relation <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>reflexive: \({4^a} - {4^a} \equiv 0\bmod 7\) (for all \(a \in \mathbb{Z}\)) <strong><em>R1</em></strong></p>
<p>so \(aRa\) therefore reflexive</p>
<p>symmetric: let \(aRb\). Then \({4^a} - {4^b} \equiv 0\bmod 7\) <strong><em>M1</em></strong></p>
<p>it follows that \({4^b} - {4^a} \equiv - ({4^a} - {4^b}) \equiv 0\bmod 7\) <strong><em>A1</em></strong></p>
<p>it follows that \(bRa\) therefore symmetric</p>
<p>transitive: let \(aRb\) and \(bRc\), <em>ie</em>, \({4^a} - {4^b} \equiv 0\bmod 7\) and \({4^b} - {4^c} \equiv 0\bmod 7\) <strong><em>M1</em></strong></p>
<p>so that \({4^a} - {4^c} \equiv ({4^a} - {4^b}) + ({4^b} - {4^c}) \equiv 0\bmod 7\) <strong><em>A1</em></strong></p>
<p>therefore \(aRb\) and \(bRc \Rightarrow aRc\) <strong><em>R1</em></strong></p>
<p>so \(R\) is transitive</p>
<p> </p>
<p><strong>Note:</strong> For transitivity, award <strong><em>A0 </em></strong>if mod 7 is omitted; <strong><em>R1 </em></strong>is dependent on the <strong><em>M </em></strong>mark.</p>
<p> </p>
<p>since \(R\) is reflexive, symmetric and transitive, it is an equivalence relation <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve \({4^a} \equiv 4\bmod 7\) or \({4^a} \equiv {4^2} \equiv 2\bmod 7\) or \({4^a} \equiv {4^3} \equiv 1\bmod 7\) <strong><em>(M1)</em></strong></p>
<p>the equivalence classes are</p>
<p>\(\{ 1,{\text{ }}4,{\text{ }}7,{\text{ }} \ldots \} ,{\text{ \{ }}2,{\text{ }}5,{\text{ }}8,{\text{ }} \ldots \} \) and \(\{ 3,{\text{ }}6,{\text{ }}9,{\text{ }} \ldots \} \) <strong><em>A2</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1)A1 </em></strong>for one or two correct equivalence classes.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>starting with 1, we find that 2, 3, 4, … all belong to the same equivalence class <strong>or</strong> \({4^c} - 4 \equiv 4({4^{c - 1}} - 1) \equiv 4({2^{c - 1}} - 1)({2^{c - 1}} - 1) \equiv 0\bmod 6\) or \({4^c} \equiv 4\bmod 6\) <strong><em>(M1)</em></strong></p>
<p>therefore there is one equivalence class <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">An Abelian group, \(\{ G,{\text{ }} * \} \), has <span class="s1">12 </span>different elements which are of the form \({a^i} * {b^j}\) where \(i \in \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4\} \) and \(j \in \{ 1,{\text{ }}2,{\text{ }}3\} \). The elements \(a\) and \(b\) satisfy \({a^4} = e\) and \({b^3} = e\) where \(e\) is the identity.</p>
</div>
<div class="specification">
<p class="p1">Let \(\{ H,{\text{ }} * \} \) be the proper subgroup of \(\{ G,{\text{ }} * \} \) having the maximum possible order.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">State the possible orders of an element of \(\{ G,{\text{ }} * \} \) </span>and for each order give an example of an element of that order.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State a generator for \(\{ H,{\text{ }} * \} \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the elements of \(\{ H,{\text{ }} * \} \).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Write down the elements of the coset of \(H\) containing \(a\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">orders are 1 2 3 4 6 12 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: <em>A1 </em></strong>for four or five correct orders.</p>
<p class="p2"> </p>
<p class="p3"><strong>Note: </strong>For the rest of this question condone absence of xxx <span class="s2">and accept equivalent expressions.</span></p>
<p class="p3"> </p>
<p class="p3">\(\begin{array}{*{20}{l}} {{\text{order:}}}&1&{{\text{element:}}}&2&{A1} \\ {}&2&{}&{{a^2}}&{A1} \\ {}&3&{}&{b{\text{ or }}{{\text{b}}^2}}&{A1} \\ {}&4&{}&{a{\text{ or }}{a^3}}&{A1} \\ {}&6&{}&{{a^2} * b{\text{ or }}{a^2} * {b^2}}&{A1} \\ {}&{12}&{}&{a * b{\text{ or }}a * {b^2}{\text{ or }}{a^3} * b{\text{ or }}{a^3} * {b^2}}&{A1} \end{array}\)</p>
<p class="p1"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(H\) has order 6 <span class="Apple-converted-space"> </span><strong><em>(R1)</em></strong></p>
<p class="p1">generator is \({a^2} * b\) or \({a^2} * {b^2}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(H = \left\{ {e,{\text{ }}{a^2} * b,{\text{ }}{b^2},{\text{ }}{a^2},{\text{ }}b,{\text{ }}{a^2} * {b^2}} \right\}\)</span> <span class="Apple-converted-space"> </span><strong><em>A3</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <em>A2 </em></strong>for 4 or 5 correct. <strong><em>A1 </em></strong>for 2 or 3 correct.</p>
<p class="p2"> </p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>required coset is \(Ha\) <span class="s1">(</span>or \(aH\)<span class="s1">) <span class="Apple-converted-space"> </span></span><strong><em>(R1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(Ha = \left\{ {a,{\text{ }}{a^3} * b,{\text{ }}a * {b^2},{\text{ }}{a^3},{\text{ }}a * b,{\text{ }}{a^3} * {b^2}} \right\}\) </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The relation \(R\) is defined such that \(xRy\) if and only if \(\left| x \right| + \left| y \right| = \left| {x + y} \right|\) for \(x\), \(y\), \(y \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(R\) is reflexive.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(R\) is symmetric.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show, by means of an example, that \(R\) is not transitive.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(for \(x \in \mathbb{R}\)), \(\left| x \right| + \left| x \right| = 2\left| x \right|\) <em><strong>A1</strong></em></p>
<p>and \(\left| x \right| + \left| x \right| = \left| {2x} \right| = 2\left| x \right|\) <em><strong>A1</strong></em></p>
<p>hence \(xRx\)</p>
<p>so \(R\) is reflexive <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1 </strong></em>for correct verification of identity for \(x\) > 0; <em><strong>A1 </strong></em>for correct verification for \(x\) ≤ 0.</p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>if \(xRy \Rightarrow \left| x \right| + \left| y \right| = \left| {x + y} \right|\)</p>
<p>\(\left| x \right| + \left| y \right| = \left| y \right| + \left| x \right|\) <em><strong>A1</strong></em></p>
<p>\(\left| {x + y} \right| = \left| {y + x} \right|\) <em><strong>A1</strong></em></p>
<p>hence \(yRx\)</p>
<p>so \(R\) is symmetric <em><strong> AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognising a condition where transitivity does not hold <em><strong>(M1)</strong></em></p>
<p>(<em>eg</em>, \(x\) > 0, \(y\) = 0 and \(z\) < 0)</p>
<p>for example, 1\(R\)0 and 0\(R\)(−1) <em><strong>A1</strong></em></p>
<p>however \(\left| 1 \right| + \left| { - 1} \right| \ne \left| {1 + - 1} \right|\) <em><strong>A1</strong></em></p>
<p>so 1\(R\)(−1) (for example) is not true <em><strong>R1</strong></em></p>
<p>hence \(R\) is not transitive <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The group <em>G </em>has a unique element, <em>h </em>, of order 2.</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(gh{g^{ - 1}}\) has order 2 for all \(g \in G\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Deduce that <em>gh </em>= <em>hg </em>for all \(g \in G\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) consider \({(gh{g^{ - 1}})^2}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = gh{g^{ - 1}}gh{g^{ - 1}} = g{h^2}{g^{ - 1}} = g{g^{ - 1}} = e\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(gh{g^{ - 1}}\) cannot be order 1 (= <em>e</em>) since <em>h</em> is order 2 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(gh{g^{ - 1}}\) has order 2 <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) but <em>h </em>is the unique element of order 2 <strong><em>R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \(gh{g^{ - 1}} = h \Rightarrow gh = hg\) <strong><em>A1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was by far the problem to be found most challenging by the candidates. Many were able to show that </span><span style="font-family: 'times new roman', times; font-size: medium;">\(gh{g^{ - 1}}\) </span><span style="font-family: 'times new roman', times; font-size: medium;">had order one or two although hardly any candidates also showed that the order was not one thus losing a mark. Part a (ii) was answered correctly by a few candidates who noticed the equality of </span><em style="font-family: 'times new roman', times; font-size: medium;">h</em><span style="font-family: 'times new roman', times; font-size: medium;"> and \(gh{g^{ - 1}}\). However, many candidates went into algebraic manipulations that led them nowhere and did not justify any marks. Part (b) (i) was well answered by a small number of students who appreciated the nature of the identity and element </span><em style="font-family: 'times new roman', times; font-size: medium;">h</em><span style="font-family: 'times new roman', times; font-size: medium;"> thus forcing the other two elements to have order four. However, (ii) was only occasionally answered correctly and even in these cases not systematically. It is possible that candidates lacked time to fully explore the problem. A small number of candidates “guessed” the correct answer.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two functions, <em>F</em> and <em>G</em> , are defined on \(A = \mathbb{R}\backslash \{ 0,{\text{ }}1\} \) by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[F(x) = \frac{1}{x},{\text{ }}G(x) = 1 - x,{\text{ for all }}x \in A.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that under the operation of composition of functions each function is its own inverse.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <em>F</em> and <em>G</em> together with four other functions form a closed set under the operation of composition of functions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find these four functions.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) the following two calculations show the required result</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(F \circ F(x) = \frac{1}{{\frac{1}{x}}} = x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(G \circ G(x) = 1 - (1 - x) = x\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) part (a) shows that the identity function defined by <em>I</em>(<em>x</em>) = <em>x</em> belongs to <em>S</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the two compositions of <em>F</em> and <em>G</em> are:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(F \circ G(x) = \frac{1}{{1 - x}};\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(G \circ F(x) = 1 - \frac{1}{x}\left( { = \frac{{x - 1}}{x}} \right)\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the final element is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(G \circ F \circ G(x) = 1 - \frac{1}{{1 - x}}\left( { = \frac{x}{{x - 1}}} \right)\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done. In part(a), the quickest answer involved showing that squaring the function gave the identity. Some candidates went through the more elaborate method of finding the inverse function in each case.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The binary operation \( * \) is defined for \(x,{\text{ }}y \in S = \{ 0,{\text{ }}1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6\} \) by</p>
<p class="p1">\[x * y = ({x^3}y - xy)\bmod 7.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the element \(e\) such that \(e * y = y\), for all \(y \in S\)<span class="s1">.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the least solution of \(x * x = e\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Deduce that \((S,{\text{ }} * )\) is not a group.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine whether or not \(e\) is an identity element.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to solve \({e^3}y - ey \equiv y\bmod 7\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">the only solution is \(e = 5\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>attempt to solve \({x^4} - {x^2} \equiv 5\bmod 7\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">least solution is \(x = 2\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>suppose \((S,{\text{ }} * )\) is a group with order 7 <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(2\) has order \(2\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">since \(2\) does not divide \(7\), Lagrange’s Theorem is contradicted <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">hence, \((S,{\text{ }} * )\) is not a group <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(\(5\) is a left-identity), so need to test if it is a right-identity:</p>
<p class="p1">ie, is \(y * 5 = y\)? <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(1 * 5 = 0 \ne 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so \(5\) is not an identity <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [10 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates were not sufficiently familiar with modular arithmetic to complete this question satisfactorily. In particular, some candidates completely ignored the requirement that solutions were required to be found modulo 7, and returned decimal answers to parts (a) and (b). Very few candidates invoked Lagrange's theorem in part (b)(ii). Some candidates were under the misapprehension that a group had to be Abelian, so tested for commutativity in part (b)(ii). It was pleasing that many candidates realised that an identity had to be both a left and right identity.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates were not sufficiently familiar with modular arithmetic to complete this question satisfactorily. In particular, some candidates completely ignored the requirement that solutions were required to be found modulo 7, and returned decimal answers to parts (a) and (b). Very few candidates invoked Lagrange's theorem in part (b)(ii). Some candidates were under the misapprehension that a group had to be Abelian, so tested for commutativity in part (b)(ii). It was pleasing that many candidates realised that an identity had to be both a left and right identity.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates were not sufficiently familiar with modular arithmetic to complete this question satisfactorily. In particular, some candidates completely ignored the requirement that solutions were required to be found modulo 7, and returned decimal answers to parts (a) and (b). Very few candidates invoked Lagrange's theorem in part (b)(ii). Some candidates were under the misapprehension that a group had to be Abelian, so tested for commutativity in part (b)(ii). It was pleasing that many candidates realised that an identity had to be both a left and right identity.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">All of the relations in this question are defined on \(\mathbb{Z}\backslash \{ 0\} \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow x + y > 7\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow - 2 < x - y < 2\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow xy > 0\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow \frac{x}{y} \in \mathbb{Z}\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">One of the relations from parts (a), (b), (c) and (d) is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">For this relation, state what the equivalence classes are.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) not reflexive <em>e.g.</em> 1 + 1 = 2 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric since <em>x </em>+ <em>y </em>= <em>y </em>+ <em>x </em><strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) <em>e.g.</em> 1 + 11 > 7, 11 + 2 > 7 but 1 + 2 = 3, so not transitive <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">For each </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">mark the correct decision and a valid reason must be given.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive since \(x - x = 0\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric since \(\left| {x - y} \right| = \left| {y - x} \right|\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) <em>e.g. </em>1R2, 2R3 but 1 − 3 = −2 , so not transitive <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">For each </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">mark the correct decision and a valid reason must be given.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive since \({x^2} > 0\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric since \(xy = yx\)<em> </em><strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(xy > 0{\text{ and }}yz > 0 \Rightarrow x{y^2}z > 0 \Rightarrow xz > 0{\text{ since }}{y^2} > 0\), so transitive <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">For each </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">mark the correct decision and a valid reason must be given.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive since \(\frac{x}{x} = 1\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) not symmetric <em>e.g. </em>\(\frac{2}{1} = 2{\text{ but }}\frac{1}{2} = 0.5\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(\frac{x}{y} \in \mathbb{Z}{\text{ and }}\frac{y}{z} \in \mathbb{Z} \Rightarrow \frac{{xy}}{{yz}} = \frac{x}{z} \in \mathbb{Z}\), so transitive <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">For each </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">mark the correct decision and a valid reason must be given.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"> </p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">only (c) is an equivalence relation <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the equivalence classes are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{1, 2, 3,…} and {−1,−2,−3,…} <em><strong>A1A1</strong></em><br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally this question was well answered, with students showing a sound knowledge of relations. There were a few candidates who mixed reflexive and symmetric qualities and marks were also lost because reasoning was either unclear or absent. Most students were able to offer counterexamples for transitivity in parts (a) and (b) but a number lost marks in failing to give adequate working to show transitivity in parts (c) and (d). That said, there were a pleasing number of good solutions here showing all the required rigour. Whilst most students were able to identify part (c) as an equivalence relation, surprisingly few gave the correct equivalence classes.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally this question was well answered, with students showing a sound knowledge of relations. There were a few candidates who mixed reflexive and symmetric qualities and marks were also lost because reasoning was either unclear or absent. Most students were able to offer counterexamples for transitivity in parts (a) and (b) but a number lost marks in failing to give adequate working to show transitivity in parts (c) and (d). That said, there were a pleasing number of good solutions here showing all the required rigour. Whilst most students were able to identify part (c) as an equivalence relation, surprisingly few gave the correct equivalence classes.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally this question was well answered, with students showing a sound knowledge of relations. There were a few candidates who mixed reflexive and symmetric qualities and marks were also lost because reasoning was either unclear or absent. Most students were able to offer counterexamples for transitivity in parts (a) and (b) but a number lost marks in failing to give adequate working to show transitivity in parts (c) and (d). That said, there were a pleasing number of good solutions here showing all the required rigour. Whilst most students were able to identify part (c) as an equivalence relation, surprisingly few gave the correct equivalence classes.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally this question was well answered, with students showing a sound knowledge of relations. There were a few candidates who mixed reflexive and symmetric qualities and marks were also lost because reasoning was either unclear or absent. Most students were able to offer counterexamples for transitivity in parts (a) and (b) but a number lost marks in failing to give adequate working to show transitivity in parts (c) and (d). That said, there were a pleasing number of good solutions here showing all the required rigour. Whilst most students were able to identify part (c) as an equivalence relation, surprisingly few gave the correct equivalence classes.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally this question was well answered, with students showing a sound knowledge of relations. There were a few candidates who mixed reflexive and symmetric qualities and marks were also lost because reasoning was either unclear or absent. Most students were able to offer counterexamples for transitivity in parts (a) and (b) but a number lost marks in failing to give adequate working to show transitivity in parts (c) and (d). That said, there were a pleasing number of good solutions here showing all the required rigour. Whilst most students were able to identify part (c) as an equivalence relation, surprisingly few gave the correct equivalence classes.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 39.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(A = \left\{ {a,{\text{ }}b} \right\}\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let the set of all these subsets be denoted by \(P(A)\) . The binary operation symmetric difference, \(\Delta\) , is defined on \(P(A)\) by \(X\Delta Y = (X\backslash Y) \cup (Y\backslash X)\) where \(X\) , \(Y \in P(A)\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let \({\mathbb{Z}_4} = \left\{ {0,{\text{ }}1,{\text{ }}2,{\text{ }}3} \right\}\) and \({ + _4}\) denote addition modulo \(4\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let \(S\) be any non-empty set. Let \(P(S)\) be the set of all subsets of \(S\) . For the following parts, you are allowed to assume that \(\Delta\), \( \cup \) and \( \cap \) are associative.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down all four subsets of <em>A </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Construct the Cayley table for \(P(A)\) under \(\Delta \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that </span><span style="font-family: 'times new roman', times; font-size: medium;">\(\left\{ {P(A),{\text{ }}\Delta } \right\}\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> is a group. You are allowed to assume that \(\Delta \) is associative.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Is \(\{ P(A){\text{, }}\Delta \} \) isomorphic to \(\{ {\mathbb{Z}_4},{\text{ }}{ + _4}\} \) ? Justify your answer.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the identity element for \(\{ P(S){\text{, }}\Delta \} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down \({X^{ - 1}}\) for \(X \in P(S)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence prove that \(\{ P(S){\text{, }}\Delta \} \) is a group.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why \(\{ P(S){\text{, }} \cup \} \) is not a group.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why \(\{ P(S){\text{, }} \cap \} \) is not a group.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\emptyset {\text{, \{ a\} , \{ b\} , \{ a, b\} }}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"> <em><strong><span style="font-family: 'times new roman', times; font-size: medium;">A3</span></strong></em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A2 </em></strong>for one error, <strong><em>A1 </em></strong>for two errors, <strong><em>A0 </em></strong>for more than two errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">closure is seen from the table above <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\emptyset \) is the identity <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">each element is self-inverse <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Showing each element has an inverse is sufficient.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">associativity is assumed so we have a group </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>AG</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">not isomorphic as in the above group all elements are self-inverse whereas in \(({\mathbb{Z}_4},{\text{ }}{ + _4})\) there is an element of order 4 (<em>e.g. </em>1) <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\emptyset \) is the identity <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({X^{ - 1}} = X\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) if <em>X </em>and <em>Y </em>are subsets of <em>S </em>then \(X\Delta Y\) (the set of elements that belong to <em>X </em>or <em>Y </em>but not both) is also a subset of <em>S</em>, hence closure is proved <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ P(S){\text{, }}\Delta \} \) is</span><span style="font-family: 'times new roman', times; font-size: medium;"> a group because it is closed, has an identity, all elements have inverses (and \(\Delta \) is associative) <strong><em>R1AG</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p> </p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">not a group because although the identity is \(\emptyset {\text{, if }}X \ne \emptyset \) it is impossible to find a set <em>Y </em>such that \(X \cup Y = \emptyset \), so there are elements without an inverse <strong><em>R1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">not a group because although the identity is <em>S</em>, if \(X \ne S\) is impossible to find a set <em>Y </em>such that \(X \cap Y = S\), so there are elements without an inverse <strong><em>R1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px Arial;"><span style="font-family: 'times new roman', times; font-size: small;">A surprising number of candidates were unable to answer part (a) and consequently were unable to access much of the rest of the question. Most candidates however, were successful in parts (a), (b) and (c), and it was pleasing to see the preparedness of candidates in these parts. There were also many good answers for parts (d) and (e) although the third part of (e) caused the most problems with candidates failing to provide sufficient reasoning. Few candidates managed good responses to parts (f) and (g).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to answer part (a) and consequently were unable to access much of the rest of the question. Most candidates however, were successful in parts (a), (b) and (c), and it was pleasing to see the preparedness of candidates in these parts. There were also many good answers for parts (d) and (e) although the third part of (e) caused the most problems with candidates failing to provide sufficient reasoning. Few candidates managed good responses to parts (f) and (g).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to answer part (a) and consequently were unable to access much of the rest of the question. Most candidates however, were successful in parts (a), (b) and (c), and it was pleasing to see the preparedness of candidates in these parts. There were also many good answers for parts (d) and (e) although the third part of (e) caused the most problems with candidates failing to provide sufficient reasoning. Few candidates managed good responses to parts (f) and (g).</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to answer part (a) and consequently were unable to access much of the rest of the question. Most candidates however, were successful in parts (a), (b) and (c), and it was pleasing to see the preparedness of candidates in these parts. There were also many good answers for parts (d) and (e) although the third part of (e) caused the most problems with candidates failing to provide sufficient reasoning. Few candidates managed good responses to parts (f) and (g).</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to answer part (a) and consequently were unable to access much of the rest of the question. Most candidates however, were successful in parts (a), (b) and (c), and it was pleasing to see the preparedness of candidates in these parts. There were also many good answers for parts (d) and (e) although the third part of (e) caused the most problems with candidates failing to provide sufficient reasoning. Few candidates managed good responses to parts (f) and (g).</span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to answer part (a) and consequently were unable to access much of the rest of the question. Most candidates however, were successful in parts (a), (b) and (c), and it was pleasing to see the preparedness of candidates in these parts. There were also many good answers for parts (d) and (e) although the third part of (e) caused the most problems with candidates failing to provide sufficient reasoning. Few candidates managed good responses to parts (f) and (g).</span></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to answer part (a) and consequently were unable to access much of the rest of the question. Most candidates however, were successful in parts (a), (b) and (c), and it was pleasing to see the preparedness of candidates in these parts. There were also many good answers for parts (d) and (e) although the third part of (e) caused the most problems with candidates failing to provide sufficient reasoning. Few candidates managed good responses to parts (f) and (g).</span></p>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The binary operation \( * \) is defined on the set \(T = \{ 0,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6\} \) by \(a * b = (a + b - ab)(\bmod 7),{\text{ }}a,{\text{ }}b \in T\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Copy and complete the following Cayley table for \(\{ T,{\text{ }} * \} \).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-21_om_14.49.34.png" alt></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(\{ T,{\text{ }} * \} \) forms an Abelian group.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the order of each element in \(T\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(\{ H,{\text{ }} * \} \) is the subgroup of \(\{ T,{\text{ }} * \} \) of order \(2\)<span class="s1">, partition \(T\) into the left cosets with respect to \(H\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Cayley table is</p>
<p class="p3"><img src="images/Schermafbeelding_2016-01-21_om_14.54.07.png" alt> <strong><em>A4</em></strong></p>
<p class="p3">award <strong><em>A4 </em></strong>for all 16 correct, <strong><em>A3 </em></strong>for up to 2 errors, <strong><em>A2 </em></strong>for up to 4 errors, <strong><em>A1 </em></strong>for up to 6 errors</p>
<p class="p3"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">closed as no other element appears in the Cayley table <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">symmetrical about the leading diagonal so commutative <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2">hence it is Abelian</p>
<p class="p1"><span class="s1">\(0\) </span>is the identity</p>
<p class="p1">as \(x * 0( = 0 * x) = x + 0 - 0 = x\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><span class="s1">\(0\)</span> and <span class="s1">\(2\)</span> are self inverse, <span class="s1">\(3\)</span> and <span class="s1">\(5\)</span> is an inverse pair, <span class="s1">\(4\)</span> and <span class="s1">\(6\)</span> <span class="s2">is an inverse pair <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></span></p>
<p class="p3"> </p>
<p class="p4"><span class="s3"><strong>Note: <span class="Apple-converted-space"> </span></strong></span>Accept “Every row and every column has a <span class="s1">\(0\)</span> so each element has an inverse”.</p>
<p class="p5"> </p>
<p class="p1">\((a * b) * c = (a + b - ab) * c = a + b - ab + c - (a + b - ab)c\) <span class="Apple-converted-space"> </span><span class="s3"><strong><em>M1</em></strong></span></p>
<p class="p1">\( = a + b + c - ab - ac - bc + abc\) <span class="Apple-converted-space"> </span><span class="s3"><strong><em>A1</em></strong></span></p>
<p class="p1">\(a * (b * c) = a * (b + c - bc) = a + b + c - bc - a(b + c - bc)\) <span class="Apple-converted-space"> </span><span class="s3"><strong><em>A1</em></strong></span></p>
<p class="p1">\( = a + b + c - ab - ac - bc + abc\)</p>
<p class="p1"><span class="s4">so </span>\((a * b) * c = a * (b * c)\) <span class="s3">and </span>\( * \)<span class="s3"> is associative</span></p>
<p class="p6"> </p>
<p class="p7"><strong>Note: <span class="Apple-converted-space"> </span></strong><span class="s4">Inclusion of mod 7 </span>may be included at any stage.</p>
<p class="p7"><em><strong>[7 marks]</strong></em></p>
<p class="p7"> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(0\) has order \(1\) and \(2\) has order <span class="Apple-converted-space">\(2\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\({3^2} = 4,{\text{ }}{3^3} = 2,{\text{ }}{3^4} = 6,{\text{ }}{3^5} = 5,{\text{ }}{3^6} = 0\) so \(3\) has order \(6\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\({4^2} = 6,{\text{ }}{4^3} = 0\) so \(4\) has order \(3\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\(5\) has order \(6\) and \(6\) has order \(3\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(H = \{ 0,{\text{ }}2\} \) <strong><em>A1</em></strong></p>
<p>\(0 * \{ 0,{\text{ }}2\} = \{ 0,{\text{ }}2\} ,{\text{ }}2 * \{ 0,{\text{ }}2\} = \{ 2,{\text{ }}0\} ,{\text{ }}3 * \{ 0,{\text{ }}2\} = \{ 3,{\text{ }}6\} ,{\text{ }}4 * \{ 0,{\text{ }}2\} = \{ 4,{\text{ }}5\} ,\)</p>
<p>\(5 * \{ 0,{\text{ }}2\} = \{ 5,{\text{ }}4\} ,{\text{ }}6 * \{ 0,{\text{ }}2\} = \{ 6,{\text{ }}3\} \) <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award the <strong><em>M1 </em></strong>if sufficient examples are used to find at least two of the cosets.</p>
<p> </p>
<p>so the left cosets are \(\{ 0,{\text{ }}2\} ,{\text{ }}\{ 3,{\text{ }}6\} ,{\text{ }}\{ 4,{\text{ }}5\} \) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p><strong><em>Total [18 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"> </p>
<p class="p2" style="text-align: center;"><strong><em> </em></strong></p>
<p class="p3"> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The function \(f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) is defined by \(f(x,{\text{ }}y) = (2{x^3} + {y^3},{\text{ }}{x^3} + 2{y^3})\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f\) is a bijection.</p>
<div class="marks">[12]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down the inverse function \({f^{ - 1}}(x,{\text{ }}y)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>for \(f\) to be a bijection it must be both an injection and a surjection <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award this <strong><em>R1 </em></strong>for stating this anywhere.</p>
<p> </p>
<p>suppose that \(f(a,{\text{ }}b) = f(c,{\text{ }}d)\) <strong><em>(M1)</em></strong></p>
<p>it follows that</p>
<p>\(2{a^3} + {b^3} = 2{c^3} + {d^3}\) and \({a^3} + 2{b^3} = {c^3} + 2{d^3}\) <strong><em>A1</em></strong></p>
<p>attempting to solve the two equations <strong><em>M1</em></strong></p>
<p>to obtain \(3{a^3} = 3{c^3}\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>only if a good attempt is made to solve the system.</p>
<p> </p>
<p>\( \Rightarrow a = c\) and therefore \(b = d\) <strong><em>A1</em></strong></p>
<p>\(f\) is an injection because \(f(a,{\text{ }}b) = f(c,{\text{ }}d) \Rightarrow (a,{\text{ }}b) = (c,{\text{ }}d)\) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award this <strong><em>R1 </em></strong>for stating this anywhere providing that an attempt is made to prove injectivity.</p>
<p> </p>
<p>let \((p,{\text{ }}q) \in \mathbb{R} \times \mathbb{R}\) and let \(f(r,{\text{ }}s) = (p,{\text{ }}q)\) <strong><em>(M1)</em></strong></p>
<p>then, \(p = 2{r^3} + {s^3}\) and \(q = {r^3} + 2{s^3}\) <strong><em>A1</em></strong></p>
<p>attempting to solve the two equations <strong><em>M1</em></strong></p>
<p>\(r = \sqrt[3]{{\frac{{2p - q}}{3}}}\) and \(s = \sqrt[3]{{\frac{{2q - p}}{3}}}\) <strong><em>A1A1</em></strong></p>
<p>\(f\) is a surjection because given \((p,{\text{ }}q) \in \mathbb{R} \times \mathbb{R}\), there exists \((r,{\text{ }}s) \in \mathbb{R} \times \mathbb{R}\) such that \(f(r,{\text{ }}s) = (p,{\text{ }}q)\) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award this <strong><em>R1 </em></strong>for stating this anywhere providing that an attempt is made to prove surjectivity.</p>
<p> </p>
<p><strong><em>[12 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\left( {{f^{ - 1}}(x,{\text{ }}y) = } \right)\,\,\,\left( {\sqrt[3]{{\frac{{2x - y}}{3},}}{\text{ }}\sqrt[3]{{\frac{{2y - x}}{3}}}} \right)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>A1 </em></strong>for correct expressions in \(x\) and \(y\), allow <strong><em>FT </em></strong>only if the expression is deduced in part (a).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Let \(A\) </span>be the set \(\{ x|x \in \mathbb{R},{\text{ }}x \ne 0\} \). Let \(B\) be the set \(\{ x|x \in ] - 1,{\text{ }} + 1[,{\text{ }}x \ne 0\} \).</p>
<p class="p1">A function \(f:A \to B\) is defined by \(f(x) = \frac{2}{\pi }\arctan (x)\).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Let \(D\) </span>be the set \(\{ x|x \in \mathbb{R},{\text{ }}x > 0\} \).</p>
<p class="p1">A function \(g:\mathbb{R} \to D\) is defined by \(g(x) = {{\text{e}}^x}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Sketch the graph of \(y = f(x)\) and hence justify whether or not \(f\) <span class="s1">is a bijection.</span></p>
<p class="p2"><span class="s2">(ii) <span class="Apple-converted-space"> </span>Show that \(A\) </span>is a group under the binary operation of multiplication.</p>
<p class="p2"><span class="s2">(iii) <span class="Apple-converted-space"> </span>Give a reason why \(B\) </span>is not a group under the binary operation of multiplication.</p>
<p class="p1">(iv) <span class="Apple-converted-space"> </span>Find an example to show that \(f(a \times b) = f(a) \times f(b)\) is not satisfied for all \(a,{\text{ }}b \in A\).</p>
<div class="marks">[13]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Sketch the graph of \(y = g(x)\) and hence justify whether or not \(g\) <span class="s1">is a bijection.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \(g(a + b) = g(a) \times g(b)\) for all \(a,{\text{ }}b \in \mathbb{R}\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Given that \(\{ \mathbb{R},{\text{ }} + \} \) and \(\{ D,{\text{ }} \times \} \) are both groups, explain whether or not they are isomorphic.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> <img src="images/Schermafbeelding_2017-03-02_om_12.36.51.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SG/M/02.a.i"></span> <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Notes: </strong>Award <strong><em>A1 </em></strong>for general shape, labelled asymptotes, and showing that \(x \ne 0\).</p>
<p class="p2"> </p>
<p class="p1">graph shows that it is injective since it is increasing or by the horizontal line test <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">graph shows that it is surjective by the horizontal line test <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Allow any convincing reasoning.</p>
<p class="p1"> </p>
<p class="p1">so \(f\) is a bijection <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>closed since non-zero real times non-zero real equals non-zero real <span class="Apple-converted-space"> </span><strong><em>A1R1</em></strong></p>
<p class="p1">we know multiplication is associative <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p3">identity is 1 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">inverse of \(x\) <span class="s2">is \(\frac{1}{x}(x \ne 0)\) <span class="Apple-converted-space"> </span></span><strong><em>A1</em></strong></p>
<p class="p1">hence it is a group <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>\(B\) does not have an identity <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1">hence it is not a group <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><span class="s2">(iv) <span class="Apple-converted-space"> \(f(1 \times 1) = f(1) = \frac{1}{2}\)</span> whereas \(f(1) \times f(1) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\) </span>is one counterexample <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1">hence statement is not satisfied <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[13 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><img src="images/Schermafbeelding_2017-03-02_om_12.52.46.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SG/M/02.b"></p>
<p class="p2"><span class="s1">award <strong><em>A1 </em></strong></span>for general shape going through (0, 1) <span class="s1">and with domain \(\mathbb{R}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></span></p>
<p class="p1">graph shows that it is injective since it is increasing or by the horizontal line test and graph shows that it is surjective by the horizontal line test <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p3"> </p>
<p class="p1"><strong>Note: </strong>Allow any convincing reasoning.</p>
<p class="p3"> </p>
<p class="p1">so \(g\)<span class="Apple-converted-space"> </span>is a bijection <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(g(a + b) = {{\text{e}}^{a + b}}\)</span> and \(g(a) \times g(b) = {{\text{e}}^a} \times {{\text{e}}^b} = {{\text{e}}^{a + b}}\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">hence \(g(a + b) = g(a) \times g(b)\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>since \(g\) is a bijection and the homomorphism rule is obeyed <span class="Apple-converted-space"> </span><strong><em>R1R1</em></strong></p>
<p class="p1">the two groups are isomorphic <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) defined by \(f(x,{\text{ }}y) = (2x + y,{\text{ }}x - y)\) is a bijection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the inverse of <em>f</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) we need to show that the function is both injective and surjective to be a bijection <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">suppose \(f(x,{\text{ }}y) = f(u,{\text{ }}v)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((2x + y,{\text{ }}x - y) = (2u + v,{\text{ }}u - v)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">forming a pair of simultaneous equations <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2x + y = 2u + v\) (i)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x - y = u - v\) (ii)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((i) + (ii) \Rightarrow 3x = 3u \Rightarrow x = u\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((i) - 2(ii) \Rightarrow 3y = 3v \Rightarrow y = v\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence function is injective <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(2x + y = s\) and \(x - y = t\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 3x = s + t\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x = \frac{{s + t}}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">also \(3y = s - 2t\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow y = \frac{{s - 2t}}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for any \((s,{\text{ }}t) \in \mathbb{R} \times \mathbb{R}\) there exists \((x,{\text{ }}y) \in \mathbb{R} \times \mathbb{R}\) and the function is surjective <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the inverse is \({f^{ - 1}}(x,{\text{ }}y) = \left( {\frac{{x + y}}{3},{\text{ }}\frac{{x - 2y}}{3}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [11 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many students were able to show that the expression was injective, but found more difficulty in showing it was subjective. As with question 1 part (e), a number of candidates did not realise that the answer to part (b) came directly from part (a), hence the reason for it being worth only one mark.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( * \) is defined on \(\mathbb{R}\) as follows. For any elements <em>a</em> , \(b \in \mathbb{R}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[a * b = a + b + 1.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \( * \) is commutative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the identity element.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the inverse of the element <em>a</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( \cdot \) is defined on \(\mathbb{R}\) as follows. For any elements <em>a</em> , \(b \in \mathbb{R}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a \cdot b = 3ab\) . The set <em>S</em> is the set of all ordered pairs \((x,{\text{ }}y)\) of real numbers and the binary operation \( \odot \) is defined on the set <em>S</em> as</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(({x_1},{\text{ }}{y_1}) \odot ({x_2},{\text{ }}{y_2}) = ({x_1} * {x_2},{\text{ }}{y_1} \cdot {y_2}){\text{ }}.\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether or not \( \odot \) is associative.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) if \( * \) is commutative \(a * b = b * a\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(a + b + 1 = b + a + 1\) , \( * \) is commutative <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) let <em>e</em> be the identity element</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a * e = a + e + 1 = a\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow e = - 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) let <em>a</em> have an inverse, \({a^{ - 1}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a * {a^{ - 1}} = a + {a^{ - 1}} + 1 = - 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {a^{ - 1}} = - 2 - a\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(({x_1},{\text{ }}{y_1}) \odot \left( {({x_2},{\text{ }}{y_2}) \odot ({x_3},{\text{ }}{y_3})} \right) = ({x_1},{\text{ }}{y_1}) \odot ({x_2} + {x_3} + 1,{\text{ }}3{y_2}{y_3})\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = ({x_1} + {x_2} + {x_3} + 2,{\text{ }}9{y_1}{y_2}{y_3})\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {({x_1},{\text{ }}{y_1}) \odot ({x_2},{\text{ }}{y_2})} \right) \odot ({x_3},{\text{ }}{y_3}) = ({x_1} + {x_2} + 1,{\text{ }}3{y_1}{y_2}) \odot ({x_3},{\text{ }}{y_3})\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = ({x_1} + {x_2} + {x_3} + 2,{\text{ }}9{y_1}{y_2}{y_3})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \( \odot \) is associative <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) of this question was the most accessible on the paper and was completed correctly by the majority of candidates.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (b) was completed by many candidates, but a significant number either did not understand what was meant by associative, confused associative with commutative, or were unable to complete the algebra.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Draw the Cayley table for the set of integers <em>G</em> = {0, 1, 2, 3, 4, 5} under addition modulo 6, \({ + _6}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that \(\{ G,{\text{ }}{ + _6}\} \) is a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the order of each element.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Show that \(\{ G,{\text{ }}{ + _6}\} \) is cyclic and state its generators.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Find a subgroup with three elements. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Find the other proper subgroups of \(\{ G,{\text{ }}{ + _6}\} \).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) </span><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A2 </em></strong>for 1 error, <strong><em>A1 </em></strong>for 2 errors and <strong><em>A0 </em></strong>for more than 2 errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The table is closed <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Identity element is 0 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Each element has a unique inverse (0 appears exactly once in each row and column) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Addition mod 6 is associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence \(\{ G,{\text{ }}{ + _6}\} \) forms a group <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) 0 has order 1 (0 = 0),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">1 has order 6 (1 + 1 + 1 + 1 + 1 + 1 = 0),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">2 has order 3 (2 + 2 + 2 = 0),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">3 has order 2 (3 + 3 = 0),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">4 has order 3 (4 + 4 + 4 = 0),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">5 has order 6 (5 + 5 + 5 + 5 + 5 + 5 = 0). <strong><em>A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman',times; font-size: medium;"><strong>Note</strong></span><span style="font-family: 'times new roman', times; font-size: medium;"><strong><span style="font-size: medium;">:</span> </strong>Award <strong><em>A2 </em></strong>for 1 error, <strong><em>A1 </em></strong>for 2 errors and <strong><em>A0 </em></strong>for more than 2 errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Since 1 and 5 are of order 6 (the same as the order of the group) every element can be written as sums of either 1 or 5. Hence the group is cyclic. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The generators are 1 and 5. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) A subgroup of order 3 is \(\left( {\{ 0,{\text{ }}2,{\text{ }}4\} ,{\text{ }}{ + _6}} \right)\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>if only {0, 2, 4} is seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Other proper subgroups are \(\left( {\{ 0\} { + _6}} \right),{\text{ }}\left( {\{ 0,{\text{ }}3\} { + _6}} \right)\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>if only {0}, {0, 3} is seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><em><strong><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span></strong></em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><em><strong><span style="font-family: 'times new roman', times; font-size: medium;">Total [16 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">The table was well done as was showing its group properties. The order of the elements in (b) was done well except for the order of 0 which was often not given. Finding the generators did not seem difficult but correctly stating the subgroups was not often done. The notion of a ‘proper’ subgroup is not well known.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f:[0,{\text{ }}\infty [ \to [0,{\text{ }}\infty [\) is defined by \(f(x) = 2{{\text{e}}^x} + {{\text{e}}^{ - x}} - 3\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find \(f'(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that <em>f</em> is a bijection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find an expression for \({f^{ - 1}}(x)\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(f'(x) = 2{{\text{e}}^x} - {{\text{e}}^{ - x}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <em>f</em> is an injection because \(f'(x) > 0\) for \(x \in [0,{\text{ }}\infty [\) <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(accept GDC solution backed up by a correct graph)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(f(0) = 0\) and \(f(x) \to \infty \) as \(x \to \infty \) , (and <em>f</em> is continuous) it is a surjection <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence it is a bijection <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) let \(y = 2{{\text{e}}^x} + {{\text{e}}^{ - x}} - 3\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(2{{\text{e}}^{2x}} - (y + 3){{\text{e}}^x} + 1 = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^x} = \frac{{y + 3 \pm \sqrt {{{(y + 3)}^2} - 8} }}{4}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \ln \left( {\frac{{y + 3 \pm \sqrt {{{(y + 3)}^2} - 8} }}{4}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(x \geqslant 0\) we must take the positive square root <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({f^{ - 1}}(x) = \ln \left( {\frac{{x + 3 + \sqrt {{{(x + 3)}^2} - 8} }}{4}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In many cases the attempts at showing that <em>f</em> is a bijection were unconvincing. The candidates were guided towards showing that <em>f</em> is an injection by noting that \(f'(x) > 0\) for all <em>x</em>, but some candidates attempted to show that \(f(x) = f(y) \Rightarrow x = y\) which is much more difficult. Solutions to (c) were often disappointing, with the algebra defeating many candidates.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The universal set contains all the positive integers less than 30. The set <em>A</em> contains all prime numbers less than 30 and the set <em>B</em> contains all positive integers of the form \(3 + 5n{\text{ }}(n \in \mathbb{N})\) that are less than 30. Determine the elements of</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>A</em> \ <em>B</em> ;</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 37.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A\Delta B\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>A</em> = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>B</em> = {3, 8, 13, 18, 23, 28} <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: <em>FT</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> on their </span><em style="font-family: 'times new roman', times; font-size: medium;">A</em><span style="font-family: 'times new roman', times; font-size: medium;"> and </span><em style="font-family: 'times new roman', times; font-size: medium;">B</em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><em style="font-family: 'times new roman', times; font-size: medium;"> </em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><em style="font-family: 'times new roman', times; font-size: medium;">A</em><span style="font-family: 'times new roman', times; font-size: medium;"> \ </span><em style="font-family: 'times new roman', times; font-size: medium;">B</em><span style="font-family: 'times new roman', times; font-size: medium;"> = {elements in </span><em style="font-family: 'times new roman', times; font-size: medium;">A</em><span style="font-family: 'times new roman', times; font-size: medium;"> that are not in </span><em style="font-family: 'times new roman', times; font-size: medium;">B</em><span style="font-family: 'times new roman', times; font-size: medium;">} </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= {2, 5, 7, 11, 17, 19, 29} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(B\backslash A\) = {8, 18, 28} <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A\Delta B = (A\backslash B) \cup (B\backslash A)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= {2, 5, 7, 8, 11, 17, 18, 19, 28, 29} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It was disappointing to find that many candidates wrote the elements of A and B incorrectly. The most common errors were the inclusion of 1 as a prime number and the exclusion of 3 in B. It has been suggested that some candidates use </span><em style="font-family: 'times new roman', times; font-size: medium;">N</em><span style="font-family: 'times new roman', times; font-size: medium;"> to denote the positive integers. If this is the case, then it is important to emphasise that the IB notation is that </span><em style="font-family: 'times new roman', times; font-size: medium;">N</em><span style="font-family: 'times new roman', times; font-size: medium;"> denotes the positive integers and zero and IB candidates should all be aware of that. Most candidates solved the remaining parts of the question correctly and follow through ensured that those candidates with incorrect A and/or B were not penalised any further.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It was disappointing to find that many candidates wrote the elements of A and B incorrectly. The most common errors were the inclusion of 1 as a prime number and the exclusion of 3 in B. It has been suggested that some candidates use </span><em style="font-family: 'times new roman', times; font-size: medium;">N </em><span style="font-family: 'times new roman', times; font-size: medium;">to denote the positive integers. If this is the case, then it is important to emphasise that the IB notation is that </span><em style="font-family: 'times new roman', times; font-size: medium;">N</em><span style="font-family: 'times new roman', times; font-size: medium;"> denotes the positive integers and zero and IB candidates should all be aware of that. Most candidates solved the remaining parts of the question correctly and follow through ensured that those candidates with incorrect A and/or B were not penalised any further.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( * \) is defined on \(\mathbb{N}\) by \(a * b = 1 + ab\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether or not \( * \)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is closed;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is commutative;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is associative;</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">has an identity element.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( * \) is closed <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">because \(1 + ab \in \mathbb{N}\) (when \(a,b \in \mathbb{N}\)) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a * b = 1 + ab = 1 + ba = b * a\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore \( * \) is commutative</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a * (b * c) = a * (1 + bc) = 1 + a(1 + bc){\text{ }}( = 1 + a + abc)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((a * b) * c = (1 + ab) * c = 1 + c(1 + ab){\text{ }}( = 1 + c + abc)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(these two expressions are unequal when \(a \ne c\)) so \( * \) is not associative <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">proof by counter example, for example</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 * (2 * 3) = 1 * 7 = 8\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((1 * 2) * 3 = 3 * 3 = 10\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(these two numbers are unequal) so \( * \) is not associative <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>e</em> denote the identity element; so that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a * e = 1 + ae = a\) gives \(e = \frac{{a - 1}}{a}\) (where \(a \ne 0\)) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then any valid statement such as: \(\frac{{a - 1}}{a} \notin \mathbb{N}\) or <em>e</em> is not unique <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">there is therefore no identity element <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award the final </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> only if the previous </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> is awarded.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For the commutative property some candidates began by setting \(a * b = b * a\) . For the identity element some candidates confused \(e * a\) and \(ea\) stating \(ea = a\) . Others found an expression for an inverse element but then neglected to state that it did not belong to the set of natural numbers or that it was not unique.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For the commutative property some candidates began by setting \(a * b = b * a\) . For the identity element some candidates confused \(e * a\) and \(ea\) stating \(ea = a\) . Others found an expression for an inverse element but then neglected to state that it did not belong to the set of natural numbers or that it was not unique.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For the commutative property some candidates began by setting \(a * b = b * a\) . For the identity element some candidates confused \(e * a\) and \(ea\) stating \(ea = a\) . Others found an expression for an inverse element but then neglected to state that it did not belong to the set of natural numbers or that it was not unique.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For the commutative property some candidates began by setting \(a * b = b * a\) . For the identity element some candidates confused \(e * a\) and \(ea\) stating \(ea = a\) . Others found an expression for an inverse element but then neglected to state that it did not belong to the set of natural numbers or that it was not unique.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A group with the binary operation of multiplication modulo 15 is shown in the following Cayley table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-13_om_06.41.36.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the values represented by each of the letters in the table.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the order of each of the elements of the group.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the three sets that form subgroups of order 2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the three sets that form subgroups of order 4.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(a = 1\;\;\;b = 8\;\;\;c = 4\)</p>
<p class="p1">\(d = 8\;\;\;e = 4\;\;\;f = 2\)</p>
<p class="p1">\(g = 4\;\;\;h = 2\;\;\;i = 1\) <span class="Apple-converted-space"> </span><strong><em>A3</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A3 </em></strong>for 9 correct answers, <strong><em>A2 </em></strong>for 6 or more, and <strong><em>A1 </em></strong>for 3 or more.</p>
<p class="p1"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space"><img src="images/Schermafbeelding_2015-12-13_om_06.47.49.png" alt> </span><strong><em>A3</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A3 </em></strong>for 8 correct answers, <strong><em>A2 </em></strong>for 6 or more, and <strong><em>A1 </em></strong>for 4 or more.</p>
<p class="p1"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\{ 1,{\text{ }}4\} ,{\text{ }}\{ 1,{\text{ }}11\} ,{\text{ }}\{ 1,{\text{ }}14\} \) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A1 </em></strong>for 1 correct answer and <strong><em>A2 </em></strong>for all 3 (and no extras).</p>
<p class="p1"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\{ 1,{\text{ }}2,{\text{ }}4,{\text{ }}8\} ,{\text{ }}\{ 1,{\text{ }}4,{\text{ }}7,{\text{ }}13\} ,\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1">\(\{ 1,{\text{ }}4,{\text{ }}11,{\text{ }}14\} \) <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<p class="p1"><strong><em>Total [12 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The first two parts of this question were generally well done. It was surprising to see how many difficulties there were with parts (c) and (d) with many answers given as {4}, {11} and {14} for example.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">The first two parts of this question were generally well done. It was surprising to see how many difficulties there were with parts (c) and (d) with many answers given as {4}, {11} and {14} for example.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">The first two parts of this question were generally well done. It was surprising to see how many difficulties there were with parts (c) and (d) with many answers given as {4}, {11} and {14} for example.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">The first two parts of this question were generally well done. It was surprising to see how many difficulties there were with parts (c) and (d) with many answers given as {4}, {11} and {14} for example.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>p</em> , <em>q</em> and <em>r</em> are elements of a group, prove the left-cancellation rule, <em>i.e.</em> \(pq = pr \Rightarrow q = r\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Your solution should indicate which group axiom is used at each stage of the proof.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the group <em>G</em> , of order 4, which has distinct elements <em>a</em> , <em>b</em> and <em>c</em> and the identity element <em>e</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Giving a reason in each case, explain why <em>ab</em> cannot equal <em>a</em> or <em>b</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that <em>c</em> is self inverse, determine the two possible Cayley tables for <em>G</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Determine which one of the groups defined by your two Cayley tables is isomorphic to the group defined by the set {1, −1, i, −i} under multiplication of complex numbers. Your solution should include a correspondence between <em>a</em>, <em>b</em>, <em>c</em>, <em>e</em> and 1, −1, i, −i .</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(pq = pr\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({p^{ - 1}}(pq) = {p^{ - 1}}(pr)\) , every element has an inverse <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(({p^{ - 1}}p)q = ({p^{ - 1}}p)r\) , Associativity <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Brackets in lines 2 and 3 must be seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(eq = er\), \({p^{ - 1}}p = e\), the identity </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q = r\), \(ea = a\) for all elements <em>a</em> of the group <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) let <em>ab</em> = <em>a</em> so <em>b</em> = <em>e</em> be which is a contradiction <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>ab</em> = <em>b</em> so <em>a</em> = <em>e</em> which is a contradiction <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore <em>ab</em> cannot equal either <em>a</em> or <em>b</em> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the two possible Cayley tables are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">table 1</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">table 2</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) the group defined by table 1 is isomorphic to the given group <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">because</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">both contain one self-inverse element (other than the identity) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">both contain an inverse pair <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">both are cyclic <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the correspondence is \(e \to 1\), \(c \to - 1\), \(a \to i\), \(b \to - i\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(or vice versa for the last two) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award the final <strong><em>A2</em></strong> only if the correct group table has been identified.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to (a) were often poor with inadequate explanations often seen. It was not uncommon to see \(pq = pr\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({p^{ - 1}}pq = {p^{ - 1}}pr\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q = r\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">without any mention of associativity. Many candidates understood what was required in (b)(i), but solutions to (b)(ii) were often poor with the tables containing elements such as <em>ab</em> and <em>bc</em> without simplification. In (b)(iii), candidates were expected to determine the isomorphism by noting that the group defined by {1, –1, i, –i} under multiplication is cyclic or that –1 is the only self-inverse element apart from the identity, without necessarily writing down the Cayley table in full which many candidates did. Many candidates just stated that there was a bijection between the two groups without giving any justification for this.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to (a) were often poor with inadequate explanations often seen. It was not uncommon to see \(pq = pr\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({p^{ - 1}}pq = {p^{ - 1}}pr\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q = r\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">without any mention of associativity. Many candidates understood what was required in (b)(i), but solutions to (b)(ii) were often poor with the tables containing elements such as <em>ab</em> and <em>bc</em> without simplification. In (b)(iii), candidates were expected to determine the isomorphism by noting that the group defined by {1, –1, i, –i} under multiplication is cyclic or that –1 is the only self-inverse element apart from the identity, without necessarily writing down the Cayley table in full which many candidates did. Many candidates just stated that there was a bijection between the two groups without giving any justification for this.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A binary operation is defined on {−1, 0, 1} by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[A \odot B = \left\{ {\begin{array}{*{20}{c}}<br> { - 1,}&{{\text{if }}\left| A \right| < \left| B \right|} \\ <br> {0,}&{{\text{if }}\left| A \right| = \left| B \right|} \\ <br> {1,}&{{\text{if }}\left| A \right| > \left| B \right|{\text{.}}} <br>\end{array}} \right.\]<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Construct the Cayley table for this operation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Giving reasons, determine whether the operation is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) closed;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) commutative;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) associative.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) the Cayley table is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\begin{gathered}<br> \begin{array}{*{20}{c}}<br> {}&{ - 1}&0&1 <br>\end{array} \\<br> \begin{array}{*{20}{c}}<br> { - 1} \\ <br> 0 \\ <br> 1 <br>\end{array}\left( {\begin{array}{*{20}{c}}<br> 0&1&0 \\ <br> { - 1}&0&{ - 1} \\ <br> 0&1&0 <br>\end{array}} \right) \\ <br>\end{gathered} \) <strong><em>M1A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> Award <strong><em>M1</em></strong> for setting up a Cayley table with labels.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Deduct <strong><em>A1</em></strong> for each error or omission.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) closed <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">because all entries in table belong to {–1, 0, 1} <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) not commutative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">because the Cayley table is not symmetric, or counter-example given <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) not associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for example because <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 \odot ( - 1 \odot 0) = 0 \odot 1 = - 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">but</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((0 \odot - 1) \odot 0 = - 1 \odot 0 = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">or alternative counter-example</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done, with the exception of part(b)(iii), showing that the operation is non-associative.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sets <em>X </em>and <em>Y </em>are defined by \({\text{ }}X = \left] {0,{\text{ }}1} \right[;{\text{ }}Y = \{ 0,{\text{ }}1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5\} \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Sketch the set \(X \times Y\) in the Cartesian plane.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Sketch the set \(Y \times X\) in the Cartesian plane.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) State \((X \times Y) \cap (Y \times X)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the function \(f:X \times Y \to \mathbb{R}\) defined by \(f(x,{\text{ }}y) = x + y\) and the function \(g:X \times Y \to \mathbb{R}\) defined by \(g(x,{\text{ }}y) = xy\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the range of the function <em>f</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the range of the function <em>g</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Show that \(f\) is an injection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Find \({f^{ - 1}}(\pi )\), expressing your answer in exact form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(v) Find all solutions to \(g(x,{\text{ }}y) = \frac{1}{2}\).</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) </span><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-10_om_14.48.07.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">correct horizontal lines <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">correctly labelled axes <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">clear indication that the endpoints are not included <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) </span><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-10_om_14.48.48.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">fully correct diagram <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman'; min-height: 20.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Do not penalize the inclusion of endpoints twice.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica; min-height: 20.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) the intersection is empty <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) range \((f) = \left] {0,{\text{ 1}}} \right[ \cup \left] {1,{\text{ 2}}} \right[ \cup \rm{L} \cup \left] {5,{\text{ 6}}} \right[\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>A1 </em></strong>for six intervals and <strong><em>A1 </em></strong>for fully correct notation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> Accept \(0 < x < 6,{\text{ }}x \ne 0{\text{, 1, 2, 3, 4, 5, 6}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) range \((g) = \left[ {0,{\text{ 5}}} \right[\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Attempt at solving</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f({x_1},{\text{ }}{y_1}) = f({x_2},{\text{ }}{y_2})\) <em><strong>M1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(x,{\text{ }}y) \in \left] {y,{\text{ }}y + 1} \right[ \Rightarrow {y_1} = {y_2}\) <em><strong>M1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and then \({x_1} = {x_2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(f\) is injective <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) </span><span style="font-family: 'times new roman', times; font-size: medium;">\({f^{ - 1}}(\pi ) = (\pi - 3,{\text{ }}3)\) <em><strong>A1A1</strong></em></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(v) solutions: (0.5, 1), (0.25, 2), \(\left( {\frac{1}{6},{\text{ 3}}} \right)\), (0.125, 4), (0.1, 5) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> <strong><em>A2 </em></strong>for all correct, <strong><em>A1 </em></strong>for 2 correct.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f:G \to H\) be a homomorphism of finite groups.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \(f({e_G}) = {e_H}\), where \({e_G}\) is the identity element in \(G\) and \({e_H}\) is the identity</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">element in \(H\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Prove that the kernel of \(f,{\text{ }}K = {\text{Ker}}(f)\), is closed under the group operation.</p>
<p>(ii) Deduce that \(K\) is a subgroup of \(G\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Prove that \(gk{g^{ - 1}} \in K\) for all \(g \in G,{\text{ }}k \in K\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Deduce that each left coset of <em>K </em>in <em>G </em>is also a right coset.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(g) = f({e_G}g) = f({e_G})f(g)\) for \(g \in G\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow f({e_G}) = {e_H}\) <em><strong>AG</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(i) closure: let \({k_1}\) and \({k_2} \in K\), then </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f({k_1}{k_2}) = f({k_1})f({k_2})\) </span><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>M1A1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = {e_H}{e_H} = {e_H}\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"> hence </span><span style="font-family: 'times new roman', times; font-size: medium;">\({k_1}{k_2} \in K\) <em><strong>R1</strong></em></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <em>K </em>is non-empty because \({e_G}\) belongs to <em>K </em><strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> a closed non-empty subset of a finite group is a subgroup <strong><em>R1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(f(gk{g^{ - 1}}) = f(g)f(k)f({g^{ - 1}})\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = f(g){e_H}f({g^{ - 1}}) = f(g{g^{ - 1}})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = f({e_G}) = {e_H}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow gk{g^{ - 1}} \in K\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) clear definition of both left and right cosets, seen somewhere. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">use of part (i) to show \(gK \subseteq Kg\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">similarly \(Kg \subseteq gK\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \(gK = Kg\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(X\) and \(Y\) be sets. The functions \(f:X \to Y\) and \(g:Y \to X\) are such that \(g \circ f\) is the identity function on \(X\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that: </p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(f\) is an injection,</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(g\) is a surjection.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(X = {\mathbb{R}^ + } \cup \{ 0\} \) and \(Y = \mathbb{R}\), choose a suitable pair of functions \(f\) and \(g\) to show that \(g\) is not necessarily a bijection.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>to test injectivity, suppose \(f({x_1}) = f({x_2})\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">apply \(g\) to both sides \(g\left( {f({x_1})} \right) = g\left( {f({x_2})} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( \Rightarrow {x_1} = {x_2}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so \(f\) is injective <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Do not accept arguments based on “\(f\) has an inverse”.</p>
<p class="p2"> </p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>to test surjectivity, suppose \(x \in X\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">define \(y = f(x)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">then \(g(y) = g\left( {f(x)} \right) = x\) <span class="Apple-converted-space"> </span>A1</p>
<p class="p1">so \(g\) is surjective <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">choose, for example, \(f(x) = \sqrt x \) and \(g(y) = {y^2}\) <span class="Apple-converted-space"> </span><em><strong>A1</strong></em></p>
<p class="p1">then \(g \circ f(x) = {\left( {\sqrt x } \right)^2} = x\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">the function \(g\) is not injective as \(g(x) = g( - x)\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [9 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Those candidates who formulated the questions in terms of the basic definitions of injectivity and surjectivity were usually sucessful. Otherwise, verbal attempts such as '\(f{\text{ is one - to - one }} \Rightarrow f{\text{ is injective}}\)' or '\(g\) is surjective because its range equals its codomain', received no credit. Some candidates made the false assumption that \(f\) and \(g\) were mutual inverses.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Few candidates gave completely satisfactory answers. Some gave functions satisfying the mutual identity but either not defined on the given sets or for which \(g\) was actually a bijection.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \((H,{\text{ }} * {\text{)}}\) be a subgroup of the group \((G,{\text{ }} * {\text{)}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the relation \(R\) defined in \(G\) by \(xRy\) if and only if \({y^{ - 1}} * x \in H\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(R\) is an equivalence relation on \(G\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine the equivalence class containing the identity element.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(R\) is reflexive as \({x^{ - 1}} * x = e \in H \Rightarrow xRx\) for any \(x \in G\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">if \(xRy\) then \({y^{ - 1}} * x = h \in H\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">but \(h \in H \Rightarrow {h^{ - 1}} \in H\), <em>ie</em>, </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\(\underbrace {{{({y^{ - 1}} * x)}^{ - 1}}}_{{x^{ - 1}}{\text{*}}y} \in H\) </span></span></span><strong style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><em>M1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore \(yRx\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(R\) is symmetric <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">if \(xRy\) then \({y^{ - 1}} * x = h \in H\) and if \(yRz\) then \({z^{ - 1}} * y = k \in H\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k * h \in H\), <em>ie</em>, \(\underbrace {({z^{ - 1}} * y) * ({y^{ - 1}} * x)}_{{z^{ - 1}} * x} \in H\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore \(xRz\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(R\) is transitive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(R\) is an equivalence relation on \(G\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(xRe \Leftrightarrow {e^{ - 1}} * x \in H\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x \in H\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\([e] = H\) <strong><em>A1 N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was fairly well answered by many candidates. They knew how to apply the equivalence relations axioms in this particular example. Part (b) however proved to be very challenging and hardly any correct answers were seen.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the set \(A\) consisting of all the permutations of the integers \(1,2,3,4,5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Two members of \(A\) are given by \(p = (1{\text{ }}2{\text{ }}5)\) and \(q = (1{\text{ }}3)(2{\text{ }}5)\).</p>
<p class="p1">Find the single permutation which is equivalent to \(q \circ p\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State a permutation belonging to<em> </em>\(A\) of order</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(4\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(6\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(P = \) {all permutations in \(A\) where exactly two integers change position},</p>
<p>and \(Q = \) {all permutations in \(A\) where the integer \(1\) changes position}.</p>
<p>(i) List all the elements in \(P \cap Q\).</p>
<p>(ii) Find \(n(P \cap Q')\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(q \circ p = (1{\text{ }}3)(2{\text{ }}5)(1{\text{ }}2{\text{ }}5)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\( = (1{\text{ }}5{\text{ }}3)\) <span class="Apple-converted-space"> </span><strong><em>M1A1A1</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span><em>M1 </em></strong>for an answer consisting of disjoint cycles, <strong><em>A1 </em></strong>for \((1{\text{ }}5{\text{ }}3)\),</p>
<p class="p1"><strong><em>A1 </em></strong>for either \((2)\) or \((2)\) omitted.</p>
<p class="p1"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Allow \(\left( {\begin{array}{*{20}{c}} 1&2&3&4&5 \\ 5&2&1&4&3 \end{array}} \right)\)</p>
<p class="p1">If done in the wrong order and obtained \((1{\text{ }}3{\text{ }}2)\), award <strong><em>A2</em></strong><em>.</em></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>any cycle with length \(4\) <em>eg </em>(\(1234\)) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>any permutation with \(2\) disjoint cycles one of length \(2\) and one of length \(3\) <em>eg </em>(\(1\) \(2\)) (\(3\) \(4\) \(5\)) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1A0 </em></strong>for any permutation with \(2\) non-disjoint cycles one of length \(2\) and one of length \(3\).</p>
<p class="p1">Accept non cycle notation.</p>
<p class="p1"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>(\(1\), \(2\)), (\(1\), \(3\)), (\(1\), \(4\)), (\(1\), \(5\)) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>(\(2\) \(3\)), (\(2\) \(4\)), (\(2\) \(5\)), (\(3\) \(4\)), (\(3\) \(5\)), (\(4\) \(5\)) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">6 <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1 </em></strong>for at least one correct cycle.</p>
<p class="p3"><em><strong>[4 marks]</strong></em></p>
<p class="p3"><em><strong>Total [11 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many students were unable to start the question, seemingly as they did not understand the cyclic notation. Many of those that did understand found it quite straightforward to obtain good marks on this question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many students were unable to start the question, seemingly as they did not understand the cyclic notation. Many of those that did understand found it quite straightforward to obtain good marks on this question.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many students were unable to start the question, seemingly as they did not understand the cyclic notation. Many of those that did understand found it quite straightforward to obtain good marks on this question.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Given the sets \(A\) and \(B\), use the properties of sets to prove that \(A \cup (B' \cup A)' = A \cup B\), justifying each step of the proof.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">\(A \cup (B' \cup A)' = A \cup (B \cap A')\) <span class="Apple-converted-space"> </span>De Morgan <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">\( = (A \cup B) \cap (A \cup A')\) <span class="Apple-converted-space"> </span>Distributive property <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">\( = (A \cup B) \cap U\) <span class="Apple-converted-space"> </span>(Union of set and its complement) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\( = A \cup B\) <span class="Apple-converted-space"> </span>(Intersection with the universal set) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not accept proofs using Venn diagrams unless the properties are clearly stated.</p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept double inclusion proofs: <strong><em>M1A1 </em></strong>for each inclusion, final <strong><em>A1 </em></strong>for conclusion of equality of sets.</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Write down why the table below is a Latin square.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\begin{gathered}<br> \begin{array}{*{20}{c}}<br> {}&d&e&b&a&c <br>\end{array} \\<br> \begin{array}{*{20}{c}}<br> d \\ <br> e \\ <br> b \\ <br> a \\ <br> c <br>\end{array}\left[ {\begin{array}{*{20}{c}}<br> c&d&e&b&a \\ <br> d&e&b&a&c \\ <br> a&b&d&c&e \\ <br> b&a&c&e&d \\ <br> e&c&a&d&b <br>\end{array}} \right] \\ <br>\end{gathered} \]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Use Lagrange’s theorem to show that the table is not a group table.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Each row and column contains all the elements of the set. <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) There are 5 elements therefore any subgroup must be of an order that is a factor of 5 <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">But there is a subgroup \(\begin{gathered}<br> \begin{array}{*{20}{c}}<br> {}&e&a <br>\end{array} \\<br> \begin{array}{*{20}{c}}<br> e \\ <br> a <br>\end{array}\left( {\begin{array}{*{20}{c}}<br> e&a \\ <br> a&e <br>\end{array}} \right) \\ <br>\end{gathered} \) of order 2 so the table is not a group table <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>R0R2 </em></strong>for “<em>a </em>is an element of order 2 which does not divide the order of the group”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) presented no problem but finding the order two subgroups (Lagrange’s theorem was often quoted correctly) was beyond some candidates. Possibly presenting the set in non-alphabetical order was the problem.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(p = {2^k} + 1,{\text{ }}k \in {\mathbb{Z}^ + }\) be a prime number and let <em>G </em>be the group of integers 1, 2, ..., <em>p </em>− 1 under multiplication defined modulo <em>p</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">By first considering the elements \({2^1},{\text{ }}{2^2},{\text{ ..., }}{2^k}\) and then the elements \({2^{k + 1}},{\text{ }}{2^{k + 2}},{\text{ …,}}\) show that the order of the element 2 is 2<em>k</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Deduce that \(k = {2^n}{\text{ for }}n \in \mathbb{N}\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The identity is 1. <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({2^1},{\text{ }}{2^2},{\text{ }}{2^3},{\text{ ..., }}{2^k}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({2^k} = p - 1\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore all the above powers of two are different <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Now consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({2^{k + 1}} \equiv 2p - 2(\bmod p) = p - 2\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({2^{k + 2}} \equiv 2p - 4(\bmod p) = p - 4\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({2^{k + 3}} = p - 8\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>etc.</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({2^{2k - 1}} = p - {2^{k - 1}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({2^{2k}} = p - {2^k}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and this is the first power of 2 equal to 1. <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The order of 2 is therefore 2<em>k</em>. <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using Lagrange’s Theorem, it follows that 2<em>k </em>is a factor of \({2^k}\) , the order of the group, in which case <em>k </em>must be as given. <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[12 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Few solutions were seen to this question with many candidates unable even to start.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \((A \cap B)\backslash (A \cap C) = A \cap (B\backslash C)\) where <em>A</em>, <em>B</em> and <em>C</em> are three subsets of the universal set <em>U</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((A \cap B)\backslash (A \cap C) = (A \cap B) \cap (A \cap C)'\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (A \cap B) \cap (A' \cup C')\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (A \cap B \cap A') \cup (A \cap B \cap C')\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (A \cap A' \cap B) \cup (A \cap B \cap C')\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (\emptyset \cap B) \cup (A \cap B \cap C')\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \emptyset \cup (A \cap B \cap C')\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {A \cap (B \cap C')} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = A \cap (B\backslash C)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not accept proofs by Venn diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Venn diagram ‘proof’ are not acceptable. Those who used de Morgan’s laws usually were successful in this question.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\{ G,{\text{ }} * \} \) be a finite group and let <em>H</em> be a non-empty subset of <em>G</em> . Prove that \(\{ H,{\text{ }} * \} \) is a group if <em>H</em> is closed under \( * \).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the associativity property carries over from <em>G</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">closure is given <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(h \in H\) and let <em>n</em> denote the order of <em>h</em>, (this is finite because <em>G</em> is finite) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that \({h^n} = e\), the identity element <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and since <em>H</em> is closed, \(e \in H\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(h * {h^{n - 1}} = e\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that \({h^{n - 1}}\) is the inverse, \({h^{ - 1}}\), of <em>h</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and since <em>H</em> is closed, \({h^{ - 1}} \in H\) so each element of <em>H</em> has an inverse element <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the four requirements for <em>H</em> to be a group are therefore satisfied <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The group \(\{ G,{\rm{ }} * {\rm{\} }}\) has identity \({e_G}\) and the group \(\{ H,{\text{ }} \circ \} \) has identity \({e_H}\). A homomorphism \(f\) is such that \(f:G \to H\). It is given that \(f({e_G}) = {e_H}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that for all \(a \in G,{\text{ }}f({a^{ - 1}}) = {\left( {f(a)} \right)^{ - 1}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ H,{\text{ }} \circ \} \) be the cyclic group of order seven, and let \(p\) be a generator.</p>
<p class="p1">Let \(x \in G\) such that \(f(x) = {p^{\text{2}}}\).</p>
<p class="p1">Find \(f({x^{ - 1}})\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(f(x * y) = p\), find \(f(y)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(f({e_G}) = {e_H} \Rightarrow f(a * {a^{ - 1}}) = {e_H}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(f\) is a homomorphism so \(f(a * {a^{ - 1}}) = f(a) \circ f({a^{ - 1}}) = {e_H}\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">by definition \(f(a) \circ {\left( {f(a)} \right)^{ - 1}} = {e_H}\) so \(f({a^{ - 1}}) = {\left( {f(a)} \right)^{ - 1}}\) <span class="s1">(by the left-cancellation law) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">from (a) \(f({x^{ - 1}}) = {\left( {f(x)} \right)^{ - 1}}\)</p>
<p class="p1">hence \(f({x^{ - 1}}) = {({p^2})^{ - 1}} = {p^5}\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(f(x * y) = f(x) \circ f(y)\;\;\;\)<span class="s1">(homomorphism) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></span></p>
<p class="p1">\({p^2} \circ f(y) = p\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(f(y) = {p^5} \circ p\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = {p^6}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<p class="p1"><strong><em>Total [10 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was well answered by those who understood what a homomorphism is. However many candidates simply did not have this knowledge and consequently could not get into the question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was well answered, even by those who could not do (a). However, there were many who having not understood what a homomorphism is, made no attempt on this easy question part. Understandably many lost a mark through not simplifying \({p^{ - 2}}\) to \({p^5}\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Those who knew what a homomorphism is generally obtained good marks in part (c).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>H</em> and <em>K</em> are subgroups of a group <em>G</em>. By considering the four group axioms, prove that \(H \cap K\) is also a subgroup of <em>G</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">closure: let \(a,{\text{ }}b \in H \cap K\), so that \(a,{\text{ }}b \in H\) and \(a,{\text{ }}b \in K\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore \(ab \in H\) and \(ab \in K\) so that \(ab \in H \cap K\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">associativity: this carries over from <em>G</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">identity: the identity \(e \in H\) and \(e \in K\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore \(e \in H \cap K\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">inverse:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a \in H \cap K\) implies \(a \in H\) and \(a \in K\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that \({a^{ - 1}} \in H\) and \({a^{ - 1}} \in K\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and therefore that \({a^{ - 1}} \in H \cap K\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the four group axioms are therefore satisfied <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question presented the most difficulty for students. Overall the candidates showed a lack of ability to present a formal proof. Some gained points for the proof of the identity element in the intersection and the statement that the associative property carries over from the group. However, the vast majority gained no points for the proof of closure or the inverse axioms.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that set difference is not associative.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we are trying to prove \((A\backslash B)\backslash C \ne A\backslash (B\backslash C)\) <strong><em>M1(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{LHS}} = (A \cap B')\backslash C\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (A \cap B') \cap C'\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{RHS}} = A\backslash (B \cap C')\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = A \cap (B \cap C')'\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = A \cap (B' \cup C)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">as LHS does not contain any element of <em>C</em> and RHS does, \({\text{LHS}} \ne {\text{RHS}}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence set difference is not associative <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept answers which use a proof containing a counter example.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was found difficult by a large number of candidates, but a number of correct solutions were seen. A number of candidates who understood what was required failed to gain the final reasoning mark. Many candidates seemed to be ill-prepared to deal with this style of question.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Define \(f:\mathbb{R}\backslash \{ 0.5\} \to \mathbb{R}\) by \(f(x) = \frac{{4x + 1}}{{2x - 1}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that <em>\(f\) </em>is an injection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that <em>\(f\) </em>is not a surjection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">\(f(x) = f(y) \Rightarrow \frac{{4x + 1}}{{2x - 1}} = \frac{{4y + 1}}{{2y - 1}}\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">for attempting to cross multiply and simplify <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\((4x + 1)(2y - 1) = (2x - 1)(4y + 1)\)</p>
<p class="p1">\( \Rightarrow 8xy + 2y - 4x - 1 = 8xy + 2x - 4y - 1 \Rightarrow 6y = 6x\)</p>
<p class="p1">\( \Rightarrow x = y\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">hence an injection <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">\(f'(x) = \frac{{4(2x - 1) - 2(4x + 1)}}{{{{(2x - 1)}^2}}} = \frac{{ - 6}}{{{{(2x - 1)}^2}}}\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">\( < 0\;\;\;{\text{(for all }}x \ne 0.5{\text{)}}\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">therefore the function is decreasing on either side of the discontinuity</p>
<p class="p1">and \(f(x) < 2\) and \(x < 0.5\) for \(f(x) > 0.5\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">hence an injection <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>If a correct graph of the function is shown, and the candidate states this is decreasing in each part (or horizontal line test) and hence an injection, award <strong><em>M1A1R1</em></strong>.</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">attempt to solve \(y = \frac{{4x + 1}}{{2x - 1}}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(y(2x - 1) = 4x + 1 \Rightarrow 2xy - y = 4x + 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(2xy - 4x = 1 + y \Rightarrow x = \frac{{1 + y}}{{2y - 4}}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">no value for \(y = 2\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">hence not a surjection <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">consider \(y = 2\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">attempt to solve \(2 = \frac{{4x + 1}}{{2x - 1}}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(4x - 2 = 4x + 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">which has no solution <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">hence not a surjection <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>If a correct graph of the function is shown, and the candidate states that because there is a horizontal asymptote at \(y = 2\) then the function is not a surjection, award <strong><em>M1R1</em></strong>.</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<p class="p1"><em><strong>Total [8 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most students indicated an understanding of the concepts of Injection and Surjection, but many did not give rigorous proofs. Even where graphs were used, it was very common for a sketch to be so imprecise with no asymptotes marked that it was difficult to award even partial credit. Some candidates mistakenly stated that the function was not surjective because 0.5 was not in the domain.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most students indicated an understanding of the concepts of Injection and Surjection, but many did not give rigorous proofs. Even where graphs were used, it was very common for a sketch to be so imprecise with no asymptotes marked that it was difficult to award even partial credit. Some candidates mistakenly stated that the function was not surjective because 0.5 was not in the domain.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the sets</p>
<p class="p1">\[G = \left\{ {\frac{n}{{{6^i}}}|n \in \mathbb{Z},{\text{ }}i \in \mathbb{N}} \right\},{\text{ }}H = \left\{ {\frac{m}{{{3^j}}}|m \in \mathbb{Z},{\text{ }}j \in \mathbb{N}} \right\}.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \((G,{\text{ }} + )\) forms a group where \( + \) denotes addition on \(\mathbb{Q}\). Associativity may be assumed.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Assuming that \((H,{\text{ }} + )\) forms a group, show that it is a proper subgroup of \((G,{\text{ }} + )\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The mapping \(\phi :G \to G\) is given by \(\phi (g) = g + g\), for \(g \in G\).</p>
<p class="p1">Prove that \(\phi \) is an isomorphism.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">closure: \(\frac{{{n_1}}}{{{6^{{i_1}}}}} + \frac{{{n_2}}}{{{6^{{i_2}}}}} = \frac{{{6^{{i_2}}}{n_1} + {6^{{i_1}}}{n_2}}}{{{6^{{i_1} + {i_2}}}}} \in G\) <span class="Apple-converted-space"> </span><strong><em>A1R1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Award <strong><em>A1</em></strong> for RHS of equation. <strong><em>R1</em></strong> is for the use of two different, but not necessarily most general elements, and the result \( \in G\) or equivalent.</p>
<p class="p1">identity: \(0\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">inverse: \(\frac{{ - n}}{{{6^i}}}\) <span class="Apple-converted-space"> </span><em><strong>A1</strong></em></p>
<p class="p1">since associativity is given, \((G,{\text{ }} + )\) forms a group <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>The <em><strong>R1</strong></em> is for considering closure, the identity, inverses and associativity.</p>
<p class="p1"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">it is required to show that \(H\) is a proper subset of \(G\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">let \(\frac{n}{{{3^i}}} \in H\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">then \(\frac{n}{{{3^i}}} = \frac{{{2^i}n}}{{{6^i}}} \in G\) hence \(H\) is a subgroup of \(G\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(H \ne G\) since \(\frac{1}{6} \in G\) but \(\frac{1}{6} \notin H\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>The final <strong><em>A1</em></strong> is only dependent on the first <strong><em>M1</em></strong>.</p>
<p class="p2"> </p>
<p class="p1">hence, \(H\) is a proper subgroup of \(G\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">consider \(\phi ({g_1} + {g_2}) = ({g_1} + {g_2}) + ({g_1} + {g_2})\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( = ({g_1} + {g_1}) + ({g_2} + {g_2}) = \phi ({g_1}) + \phi ({g_2})\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(hence \(\phi \) is a homomorphism)</p>
<p class="p1">injectivity: let \(\phi ({g_1}) = \phi ({g_2})\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">working within \(\mathbb{Q}\) we have \(2{g_1} = 2{g_2} \Rightarrow {g_1} = {g_2}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">surjectivity: considering even and odd numerators <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(\phi \left( {\frac{n}{{{6^i}}}} \right) = \frac{{2n}}{{{6^i}}}\) and \(\phi \left( {\frac{{3(2n + 1)}}{{{6^{i + 1}}}}} \right) = \frac{{2n + 1}}{{{6^i}}}\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1">hence \(\phi \) is an isomorphism <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
<p class="p1"><strong><em>Total [16 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This part was generally well done. Where marks were lost, it was usually because a candidate failed to choose two different elements in the proof of closure.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Only a few candidates realised that they did not have to prove that \(H\) is a group - that was stated in the question. Some candidates tried to invoke Lagrange's theorem, even though \(G\) is an infinite group.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates showed that the mapping is injective. Most attempts at proving surjectivity were unconvincing. Those candidates who attempted to establish the homomorphism property sometimes failed to use two different elements.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following functions</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(f:\left] {1,{\text{ }} + \infty } \right[ \to {\mathbb{R}^ + }\) where \(f(x) = (x - 1)(x + 2)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(g:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) where \(g(x,{\text{ }}y) = \left( {\sin (x + y),{\text{ }}x + y} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(h:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) where \(h(x,{\text{ }}y) = (x + 3y,{\text{ }}2x + y)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(f\) is bijective.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine, with reasons, whether</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(g\) is injective;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(g\)<em> </em>is surjective.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find an expression for \({h^{ - 1}}(x,{\text{ }}y)\) and hence justify that \(h\)<em> </em>has an inverse function.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <strong>Method 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">sketch of the graph of \(f\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-17_om_18.25.22.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">range of \(f = \) co-domain, therefore \(f\) is surjective <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">graph of \(f\) passes the horizontal line test, therefore \(f\) is injective <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore \(f\) is bijective <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Other explanations may be given (<em>eg </em>use of derivative or description of parabola).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Method 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Injective: \(f(a) = f(b) \Rightarrow a = b\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\((a - 1)(a + 2) = (b - 1)(b + 2)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({a^2} + a = {b^2} + b\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">solving for \(a\) by completing the square, or the quadratic formula, <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = b\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">surjective: for all \(y \in {\mathbb{R}^ + }\) there exists \(x \in \left] {1,{\text{ }}\infty } \right[\) such that \(f(x) = y\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">solving \(y = {x^2} + x - 2\) for x, \(x = \frac{{\sqrt {4y + 9} - 1}}{2}\). For all positive real \(y\), the minimum value for \(\sqrt {4y + 9} \) is \(3\). Hence, \(x \geqslant 1\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">since \(f\) is both injective and surjective, \(f\) is bijective. <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Method 3</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f\) is bijective if and only if \(f\) has an inverse <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving \(y = {x^2} + x - 2\) for \(x,{\text{ }}x = \frac{{\sqrt {4y + 9} - 1}}{2}\). For all positive real \(y\), the minimum value for \(\sqrt {4y + 9} \) is \(3\). Hence, \(x \geqslant 1\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({f^{ - 1}}(x) = \frac{{\sqrt {4x + 9} - 1}}{2}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f\) has an inverse, hence \(f\) is bijective <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) attempt to find counterexample <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(g(x,{\text{ }}y) = g(y,{\text{ }}x),{\text{ }}x \ne y\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>g </em>is not injective <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \( - 1 \leqslant \sin (x + y) \leqslant 1\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">range of \(g\) is \(\left[ { - 1,{\text{ 1}}} \right] \times \mathbb{R} \ne \mathbb{R} \times \mathbb{R}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(g\) is not surjective <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) let \(h(x,{\text{ }}y) = (u,{\text{ }}v)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">then \(u = x + 3y\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = 2x + y\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneous equations <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\left( \begin{array}{l}x\\y\end{array} \right) = {\left( {\begin{array}{*{20}{c}}1&3\\2&1\end{array}} \right)^{ - 1}}\left( \begin{array}{l}u\\v\end{array} \right) \Rightarrow \left( \begin{array}{l}x\\y\end{array} \right) = - \frac{1}{5}\left( {\begin{array}{*{20}{c}}1&{ - 3}\\{ - 2}&1\end{array}} \right)\left( \begin{array}{l}u\\v\end{array} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \frac{{ - u + 3v}}{5}y = \frac{{2u - v}}{5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">hence \({h^{ - 1}}(x,{\text{ }}y) = \left( {\frac{{ - x + 3y}}{5},{\text{ }}\frac{{2x - y}}{5}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">as this expression is defined for any values of \((x,{\text{ }}y) \in \mathbb{R} \times \mathbb{R}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">the inverse of \(h\) exists <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks] </em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">For part (a), given the command term ‘show that’ and the number of marks for this part, the best approach is a graphical one, i.e., an informal approach. Many candidates chose an algebraic approach and generally made correct statements for injective and surjective. However, they often did not follow through with the necessary algebraic manipulation to make a valid conclusion. In part (b), many candidates were not able to provide valid counter-examples. In part (c) It was obvious that quite a few candidates had not seen this type of function before. Those that were able to find the inverse generally did not justify their result, and hence could not earn the final R mark.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f:\mathbb{Z} \times \mathbb{R} \to \mathbb{R},{\text{ }}f(m,{\text{ }}x) = {( - 1)^m}x\). Determine whether <em>f</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) surjective;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) injective.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>P</em> is the set of all polynomials such that \(P = \left\{ {\sum\limits_{i = 0}^n {{a_i}{x^i}|n \in \mathbb{N}} } \right\}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(g:P \to P,{\text{ }}g(p) = xp\). Determine whether <em>g</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) surjective;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) injective.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(h:\mathbb{Z} \to {\mathbb{Z}^ + }\), \(h(x) = \left\{ {\begin{array}{*{20}{c}}<br> {2x,}&{x > 0} \\ <br> {1 - 2x,}&{x \leqslant 0} <br>\end{array}} \right\}\). Determine whether <em>h</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) surjective;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) injective.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) let \(x \in \mathbb{R}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for example, \(f(0,{\text{ }}x) = x\), <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>f</em> is surjective <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) for example, \(f(2,{\text{ }}3) = f(4,{\text{ }}3) = 3,{\text{ but }}(2,{\text{ }}3) \ne (4,{\text{ }}3)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>f</em> is not injective <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) there is no element of <em>P</em> such that \(g(p) = 7\), for example <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>g</em> is not surjective <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(g(p) = g(q) \Rightarrow xp = xq \Rightarrow p = q\), hence <em>g</em> is injective <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) for \(x > 0,{\text{ }}h(x) = 2,{\text{ }}4,{\text{ }}6,{\text{ }}8 \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for \(x \leqslant 0,{\text{ }}h(x) = 1,{\text{ }}3,{\text{ }}5,{\text{ }}7 \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore <em>h</em> is surjective <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) for \(h(x) = h(y)\), since an odd number cannot equal an even number, there are only two possibilities: <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x,{\text{ }}y > 0,{\text{ }}2x = 2y \Rightarrow x = y;\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x,{\text{ }}y \leqslant 0,{\text{ }}1 - 2x = 1 - 2y \Rightarrow x = y\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore <em>h</em> is injective <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> This can be demonstrated in a variety of ways.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was the least successfully answered question on the paper. Candidates often could quote the definitions of surjective and injective, but often could not apply the definitions in the examples. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a) Some candidates failed to show convincingly that the function was surjective, and not injective.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was the least successfully answered question on the paper. Candidates often could quote the definitions of surjective and injective, but often could not apply the definitions in the examples. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">b) Some candidates had trouble interpreting the notation used in the question, hence could not answer the question successfully.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was the least successfully answered question on the paper. Candidates often could quote the definitions of surjective and injective, but often could not apply the definitions in the examples. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">c) Many candidates failed to appreciate that the function is discrete, and hence erroneously attempted to differentiate the function to show that it is monotonic increasing, hence injective. Others who provided a graph again showed a continuous rather than discrete function.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The function \(f:\mathbb{R} \to \mathbb{R}\) is defined as \(f:x \to \left\{ {\begin{array}{*{20}{c}} {1,{\text{ }}x \ge 0} \\ { - 1,{\text{ }}x < 0} \end{array}} \right.\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(f\) is</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>not injective;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>not surjective.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if \(f(a) \times f(b) = 1\)<span class="s1">.</span></p>
<p class="p1">Show that \(R\) is an equivalence relation.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if \(f(a) \times f(b) = 1\)<span class="s1">.</span></p>
<p class="p1">State the equivalence classes of \(R\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(i) <em>eg</em>\(\;\;\;f(2) = f(3)\) <strong><em>M1</em></strong></p>
<p>hence \(f(a) = f(b) \Rightarrow a = b\) <strong><em>R1</em></strong></p>
<p>so not injective <strong><em>AG</em></strong></p>
<p>(ii) <em>eg</em>\(\;\;\;\)Codomain is \(\mathbb{R}\) and range is \(\{ - 1,{\text{ }}1\} \) <strong><em>M1</em></strong></p>
<p>these not the same so not surjective <strong><em>R1AG</em></strong></p>
<p> </p>
<p><strong>Note: </strong>if counter example is given it must be stated it is not in the range to obtain the <strong><em>R1</em></strong>. <em>Eg </em>\(f(x) = 2\) has no solution as \(f(x) \in \{ - 1,{\text{ }}1\} \forall x\).</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>if \(a \ge 0\) then \(f(a) \times f(a) = 1 \times 1 = 1\) <strong><em>A1</em></strong></p>
<p>if \(a < 0\) then \(f(a) \times f(a) = - 1 \times - 1 = 1\) <strong><em>A1</em></strong></p>
<p>in either case \(aRa\) so \(R\) is reflexive <strong><em>R1</em></strong></p>
<p>\(aRb \Rightarrow f(a) \times f(b) = 1 \Rightarrow f(b) \times f(a) = 1 \Rightarrow bRa\) <strong><em>A1</em></strong></p>
<p>so \(R\) is symmetric <strong><em>R1</em></strong></p>
<p>if \(aRb\) then either \(a \ge 0\) and \(b \ge 0\) or \(a < 0\) and \(b < 0\)</p>
<p>if \(a \ge 0\) and \(b \ge 0\) and \(bRc\) then \(c \ge 0\) so \(f(a) \times f(c) = 1 \times 1 = 1\) and \(aRc\) <strong><em>A1</em></strong></p>
<p>if \(a < 0\) and \(b < 0\) and \(bRc\) then \(c < 0\) so \(f(a) \times f(c) = - 1 \times - 1 = 1\) and \(aRc\) <strong><em>A1</em></strong></p>
<p>in either case \(aRb\) and \(bRc \Rightarrow aRc\) so \(R\) is transitive <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept</p>
<p>\(f(a) \times f(b) \times f(b) \times f(c) = 1 \times 1 = 1 \Rightarrow f(a) \times 1 \times {\text{ }}f(c) = 1 \Rightarrow {\text{ }}f(a) \times f(c) = 1\)</p>
<p> </p>
<p><strong>Note: </strong>for each property just award <strong><em>R1 </em></strong>if at least one of the A marks is awarded.</p>
<p> </p>
<p>as \(R\) is reflexive, symmetric and transitive it is an equivalence relation <strong><em>AG</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">equivalence classes are \([0,{\text{ }}\infty [\) and \(] - \infty ,{\text{ }}0[\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A1A0 </em></strong>for both intervals open.</p>
<p class="p1"><em><strong>[2 marks]</strong></em></p>
<p class="p1"><em><strong>Total [14 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f:{\mathbb{R}^ + } \times {\mathbb{R}^ + } \to {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) is defined by \(f(x,{\text{ }}y) = \left( {x{y^2},\frac{x}{y}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>f</em> is a bijection.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for <em>f</em> to be a bijection it must be both an injection and a surjection <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award this </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for stating this anywhere.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">injection:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(f(a{\text{, }}b) = f(c,{\text{ }}d)\) so that <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a{b^2} = c{d^2}\) and \(\frac{a}{b} = \frac{c}{d}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">dividing the equations,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({b^3} = {d^3}\) so \(b = d\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">substituting,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>a</em> = <em>c</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that <em>f</em> is an injection because \(f(a{\text{, }}b) = f(c,{\text{ }}d) \Rightarrow (a{\text{, }}b) = (c,{\text{ }}d)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">surjection:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(f(a{\text{, }}b) = (c,{\text{ }}d)\) where \((c,{\text{ }}d) \in {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then \(c = a{b^2}\) and \(d = \frac{a}{b}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">dividing,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({b^3} = \frac{c}{d}\) so \(b = \sqrt[3]{{\frac{c}{d}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">substituting,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = d \times \sqrt[3]{{\frac{c}{d}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that <em>f</em> is a surjection because</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">given \((c,{\text{ }}d) \in {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) , there exists \((a,{\text{ }}b) \in {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) such that \(f(a{\text{, }}b) = (c,{\text{ }}d)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore <em>f</em> is a bijection <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[11 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates who knew that they were required to give a rigorous demonstration that <em>f</em> was injective and surjective were generally successful, although the formality that is needed in this style of demonstration was often lacking. Some candidates, however, tried unsuccessfully to give a verbal explanation or even a 2-D version of the horizontal line test. In 2-D, the only reliable method for showing that a function <em>f</em> is injective is to show that \(f(a{\text{, }}b) = f(c,{\text{ }}d) \Rightarrow (a{\text{, }}b) = (c,{\text{ }}d)\).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>G</em> be a finite cyclic group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Prove that <em>G</em> is Abelian.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given that <em>a</em> is a generator of <em>G</em>, show that \({a^{ - 1}}\) is also a generator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that if the order of <em>G</em> is five, then all elements of <em>G</em>, apart from the identity, are generators of <em>G</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) let <em>a</em> be a generator and consider the (general) elements \(b = {a^m},{\text{ }}c = {a^n}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(bc = {a^m}{a^n}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {a^n}{a^m}\) (using associativity) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = cb\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore <em>G</em> is Abelian <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) let <em>G</em> be of order <em>p</em> and let \(m \in \{ 1,.......,{\text{ }}p\} \), let <em>a</em> be a generator</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">consider \(a{a^{ - 1}} = e \Rightarrow {a^m}{({a^{ - 1}})^m} = e\) <strong><em>M1R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">this shows that \({({a^{ - 1}})^m}\) is the inverse of \({a^m}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">as <em>m</em> increases from 1 to <em>p</em>, \({a^m}\) takes <em>p</em> different values and it generates <em>G</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows from the uniqueness of the inverse that \({({a^{ - 1}})^m}\) takes <em>p</em> different values and is a generator <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">by Lagrange, the order of any element divides the order of the group, <em>i.e.</em> 5 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the only numbers dividing 5 are 1 and 5 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the identity element is the only element of order 1 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">all the other elements must be of order 5 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so they all generate <em>G</em> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>a</em> be a generator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">successive powers of <em>a</em> and therefore the elements of <em>G</em> are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a,{\text{ }}{a^2},{\text{ }}{a^3},{\text{ }}{a^4}{\text{ and }}{a^5} = e\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">successive powers of \({a^2}\) are \({a^2},{\text{ }}{a^4},{\text{ }}a,{\text{ }}{a^3},{\text{ }}{a^5} = e\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">successive powers of \({a^3}\) are \({a^3},{\text{ }}a,{\text{ }}{a^4},{\text{ }}{a^2},{\text{ }}{a^5} = e\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">successive powers of \({a^4}\) are \({a^4},{\text{ }}{a^3},{\text{ }}{a^2},{\text{ }}a,{\text{ }}{a^5} = e\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">this shows that \({a^2},{\text{ }}{a^3},{\text{ }}{a^4}\) are also generators in addition to <em>a</em> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [13 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to (a) were often disappointing with some solutions even stating that a cyclic group is, by definition, commutative and therefore Abelian. Explanations in (b) were often poor and it was difficult in some cases to distinguish between correct and incorrect solutions. In (c), candidates who realised that Lagrange<span style="letter-spacing: 0.3px;">’</span>s Theorem could be used were generally the most successful. Solutions again confirmed that, in general, candidates find theoretical questions on this topic difficult.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f:\mathbb{R} \to \mathbb{R}\) is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = 2{{\text{e}}^x} - {{\text{e}}^{ - x}}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that <em>f</em> is a bijection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find an expression for \({f^{ - 1}}(x)\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) = 2{{\text{e}}^x} - {{\text{e}}^{ - x}} > 0\) for all <em>x</em> <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so <em>f</em> is an injection <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(2{{\text{e}}^x} - {{\text{e}}^{ - x}} = 2{{\text{e}}^y} - {{\text{e}}^{ - y}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2({{\text{e}}^x} - {{\text{e}}^y}) + {{\text{e}}^{ - y}} - {{\text{e}}^{ - x}} = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2({{\text{e}}^x} - {{\text{e}}^y}) + {{\text{e}}^{ - (x + y)}}({{\text{e}}^x} - {{\text{e}}^y}) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {2 + {{\text{e}}^{ - (x + y)}}} \right)({{\text{e}}^x} - {{\text{e}}^y}) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^x} = {{\text{e}}^y}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = y\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Sufficient working must be shown to gain the above <strong><em>A1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so <em>f</em> is an injection <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept a graphical justification <em>i.e.</em> horizontal line test.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it is also a surjection (accept any justification including graphical) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore it is a bijection <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) let \(y = 2{{\text{e}}^x} - {{\text{e}}^{ - x}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2{{\text{e}}^{2x}} - y{{\text{e}}^x} - 1 = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^x} = \frac{{y \pm \sqrt {{y^2} + 8} }}{4}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \({{\text{e}}^x}\) is never negative, we take the + sign <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({f^{ - 1}}(x) = \ln \left( {\frac{{x + \sqrt {{x^2} + 8} }}{4}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to (a) were often disappointing. Many candidates tried to use the result that, for an injection, \(f(a) = f(b) \Rightarrow a = b\) – although this is the definition, it is often much easier to proceed by showing that the derivative is everywhere positive or everywhere negative or even to use a horizontal line test. Although (b) is based on core material, solutions were often disappointing with some very poor use of algebra seen.</span></p>
</div>
<br><hr><br><div class="specification">
<p>The set of all permutations of the list of the integers 1, 2, 3 4 is a group, <em>S</em><sub>4</sub>, under the operation of function composition.</p>
</div>
<div class="specification">
<p>In the group <em>S</em><sub>4</sub> let \({p_1} = \left( \begin{gathered}<br> \begin{array}{*{20}{c}}<br> 1&2&3&4 <br>\end{array} \hfill \\<br> \begin{array}{*{20}{c}}<br> 2&3&1&4 <br>\end{array} \hfill \\ <br>\end{gathered} \right)\) and \({p_2} = \left( \begin{gathered}<br> \begin{array}{*{20}{c}}<br> 1&2&3&4 <br>\end{array} \hfill \\<br> \begin{array}{*{20}{c}}<br> 2&1&3&4 <br>\end{array} \hfill \\ <br>\end{gathered} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the order of <em>S</em><sub>4</sub>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the proper subgroup <em>H</em> of order 6 containing \({p_1}\), \({p_2}\) and their compositions. Express each element of <em>H</em> in cycle form.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\) for \(p \in {S_4}\).</p>
<p>Using \({p_1}\) and \({p_2}\), explain why \(f\) is not a homomorphism.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>number of possible permutations is 4 × 3 × 2 × 1 <em><strong>(M1)</strong></em></p>
<p>= 24(= 4!) <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find one of \({p_1} \circ {p_1}\), \({p_1} \circ {p_2}\) or \({p_2} \circ {p_1}\) <strong>M1</strong></p>
<p>\({p_1} \circ {p_1} = \left( {132} \right)\) or equivalent (<em>eg</em>, \({p_1}^{ - 1} = \left( {132} \right)\)) <em><strong>A1</strong></em></p>
<p>\( {p_1} \circ {p_2} = \left( {13} \right)\) or equivalent (<em>eg</em>, \({p_2} \circ {p_1} \circ {p_1} = \left( {13} \right)\)) <em><strong>A1</strong></em></p>
<p>\({p_2} \circ {p_1} = \left( {23} \right)\) or equivalent (<em>eg</em>, \({p_1} \circ {p_1} \circ {p_2} = \left( {23} \right)\)) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1A0A0</strong> </em>for one correct permutation in any form; <em><strong>A1A1A0</strong> </em>for two correct permutations in any form.</p>
<p>\(e = \left( 1 \right)\), \({p_1} = \left( {123} \right)\) and \({p_2} = \left( {12} \right)\) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Condone omission of identity in cycle form as long as it is clear it is considered one of the elements of <em>H</em>.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>if \(f\) is a homomorphism \(f\left( {{p_1} \circ {p_2}} \right) = f\left( {{p_1}} \right) \circ f\left( {{p_2}} \right)\)</p>
<p>attempting to express one of \(f\left( {{p_1} \circ {p_2}} \right)\) or \(f\left( {{p_1}} \right) \circ f\left( {{p_2}} \right)\) in terms of \({p_1}\) and \({p_2}\) <em><strong>M1</strong></em></p>
<p>\(f\left( {{p_1} \circ {p_2}} \right) = {p_1} \circ {p_2} \circ {p_1} \circ {p_2}\) <em><strong>A1</strong></em></p>
<p>\(f\left( {{p_1}} \right) \circ f\left( {{p_2}} \right) = {p_1} \circ {p_1} \circ {p_2} \circ {p_2}\) <em><strong>A1</strong></em></p>
<p>\( \Rightarrow {p_2} \circ {p_1} = {p_1} \circ {p_2}\) <em><strong>A1</strong></em></p>
<p>but \({p_1} \circ {p_2} \ne {p_2} \circ {p_1}\) <em><strong>R1</strong></em></p>
<p>so \(f\) is not a homomorphism <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>only if <em><strong>M1</strong> </em>is awarded.</p>
<p><strong>Note:</strong> Award marks only if \({p_1}\) and \({p_2}\) are used; cycle form is not required.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>if \(f\) is a homomorphism \(f\left( {{p_1} \circ {p_2}} \right) = f\left( {{p_1}} \right) \circ f\left( {{p_2}} \right)\)</p>
<p>attempting to find one of \(f\left( {{p_1} \circ {p_2}} \right)\) or \(f\left( {{p_1}} \right) \circ f\left( {{p_2}} \right)\) <em><strong>M1</strong></em></p>
<p>\(f\left( {{p_1} \circ {p_2}} \right) = e\) <em><strong>A1</strong></em></p>
<p>\(f\left( {{p_1}} \right) \circ f\left( {{p_2}} \right) = \left( {132} \right)\) <em><strong>(M1)A1</strong></em></p>
<p>so \(f\left( {{p_1} \circ {p_2}} \right) \ne f\left( {{p_1}} \right) \circ f\left( {{p_2}} \right)\) <em><strong>R1</strong></em></p>
<p>so \(f\) is not a homomorphism <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>only if <em><strong>M1</strong> </em>is awarded.</p>
<p><strong>Note:</strong> Award marks only if \({p_1}\) and \({p_2}\) are used; cycle form is not required.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>aRb</em> is defined on {1, 2, 3, 4, 5, 6, 7, 8, 9} if and only if <em>ab</em> is the square of a positive integer. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that <em>R</em> is an equivalence relation. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the equivalence classes of <em>R</em> that contain more than one element.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given the group \((G,{\text{ }} * )\), a subgroup \((H,{\text{ }} * )\) and \(a,{\text{ }}b \in G\), we define \(a \sim b\) if and only if \(a{b^{ - 1}} \in H\). Show that \( \sim \) is an equivalence relation.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(aRa \Rightarrow a \cdot a = {a^2}\) so <em>R </em>is reflexive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(aRb = {m^2} \Rightarrow bRa\) so <em>R </em>is symmetric <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(aRb = ab = {m^2}{\text{ and }}bRc = bc = {n^2}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(a = \frac{{{m^2}}}{b}{\text{ and }}c = \frac{{{n^2}}}{b}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(ac = \frac{{{m^2}{n^2}}}{{{b^2}}} = {\left( {\frac{{mn}}{b}} \right)^2},\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>ac </em>is an integer hence \({\left( {\frac{{mn}}{b}} \right)^2}\) is an integer <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so <em>aRc</em>, hence <em>R </em>is transitive <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>R </em>is therefore an equivalence relation <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) 1<em>R</em>4 and 4<em>R</em>9 <strong>or </strong>2<em>R</em>8 <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so {1, 4, 9} is an equivalence class <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and {2, 8} is an equivalence class <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a \sim a{\text{ since }}a{a^{ - 1}} = e \in H\), the identity must be in <em>H </em>since it is a subgroup. <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence reflexivity. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a \sim b \Leftrightarrow a{b^{ - 1}} \in H\) but <em>H </em>is a subgroup so it must contain \({(a{b^{ - 1}})^{ - 1}} = b{a^{ - 1}}\) <strong><em>M1R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i.e.</em> \(b{a^{ - 1}} \in H{\text{ so }} \sim \) is symmetric <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a \sim b{\text{ and }}b \sim c \Rightarrow a{b^{ - 1}} \in H{\text{ and }}b{c^{ - 1}} \in H\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">But <em>H </em>is closed, so</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((a{b^{ - 1}})(b{c^{ - 1}}) \in H{\text{ or }}a({b^{ - 1}}b){c^{ - 1}} \in H\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a{c^{ - 1}} \in H \Rightarrow a \sim c\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence \( \sim \) is transitive and is thus an equivalence relation <strong><em>R1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Not a difficult question although using the relation definition to fully show transitivity was not well done. It was good to see some students use an operation binary matrix to show transitivity. This was a nice way given that the set was finite. The proof in (b) proved difficult.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Not a difficult question although using the relation definition to fully show transitivity was not well done. It was good to see some students use an operation binary matrix to show transitivity. This was a nice way given that the set was finite. The proof in (b) proved difficult.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Set \(S = \{ {x_0},{\text{ }}{x_1},{\text{ }}{x_2},{\text{ }}{x_3},{\text{ }}{x_4},{\text{ }}{x_5}\} \) and a binary operation \( \circ \) on <em>S</em> is defined as \({x_i} \circ {x_j} = {x_k}\), where \(i + j \equiv k(\bmod 6)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Construct the Cayley table for \(\{ S,{\text{ }} \circ \} \) and hence show that it is a group.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Show that \(\{ S,{\text{ }} \circ \} \) is cyclic.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Let \(\{ G,{\text{ }} * \} \) be an Abelian group of order 6. The element \(a \in {\text{G}}\) has order 2 and the element \(b \in {\text{G}}\) has order 3.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Write down the six elements of \(\{ G,{\text{ }} * \} \).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the order of \({\text{a}} * b\) and hence show that \(\{ G,{\text{ }} * \} \) is isomorphic to \(\{ S,{\text{ }} \circ \} \).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Cayley table for \(\{ S,{\text{ }} \circ \} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\begin{array}{*{20}{c|cccccc}}<br> \circ &{{x_0}}&{{x_1}}&{{x_2}}&{{x_3}}&{{x_4}}&{{x_5}} \\ <br>\hline<br> {{x_0}}&{{x_0}}&{{x_1}}&{{x_2}}&{{x_3}}&{{x_4}}&{{x_5}} \\ <br> {{x_1}}&{{x_1}}&{{x_2}}&{{x_3}}&{{x_4}}&{{x_5}}&{{x_0}} \\ <br> {{x_2}}&{{x_2}}&{{x_3}}&{{x_4}}&{{x_5}}&{{x_0}}&{{x_1}} \\ <br> {{x_3}}&{{x_3}}&{{x_4}}&{{x_5}}&{{x_0}}&{{x_1}}&{{x_2}} \\ <br> {{x_4}}&{{x_4}}&{{x_5}}&{{x_0}}&{{x_1}}&{{x_2}}&{{x_3}} \\ <br> {{x_5}}&{{x_5}}&{{x_0}}&{{x_1}}&{{x_2}}&{{x_3}}&{{x_4}} <br>\end{array}\) <strong><em>A4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A4</em></strong> for no errors, <strong><em>A3</em></strong> for one error, <strong><em>A2</em></strong> for two errors, <strong><em>A1</em></strong> for three errors and <strong><em>A0</em></strong> for four or more errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>S</em> is closed under \( \circ \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({x_0}\) is the identity <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({x_0}\) and \({x_3}\) are self-inverses, <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({x_2}\) and \({x_4}\) are mutual inverses and so are \({x_1}\) and \({x_5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">modular addition is associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence, \(\{ S,{\text{ }} \circ \} \) is a group <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the order of \({x_1}\) (or \({x_5}\)) is 6, hence there exists a generator, and \(\{ S,{\text{ }} \circ \} \) is a cyclic group <strong><em>A1R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[11 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) <em>e</em>, <em>a</em>, <em>b</em>, <em>ab</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and \({b^2},{\text{ }}a{b^2}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \(ba\) and \({b^2}a\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({(ab)^2} = {b^2}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(ab)^3} = a\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(ab)^4} = b\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence order is 6 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">groups <em>G</em> and <em>S</em> have the same orders and both are cyclic <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence isomorphic <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [20 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a) Most candidates had the correct Cayley table and were able to show successfully that the group axioms were satisfied. Some candidates, however, simply stated that an inverse exists for each element without stating the elements and their inverses. Most candidates were able to find a generator and hence show that the group is cyclic.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">b) This part was answered less successfully by many candidates. Some failed to find all the elements. Some stated that the order of <em>ab</em> is 6 without showing any working.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f </em>is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \frac{{1 - {{\text{e}}^{ - x}}}}{{1 + {{\text{e}}^{ - x}}}},{\text{ }}x \in \mathbb{R}{\text{ .}}\]</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the range of <em>f </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Prove that <em>f </em>is an injection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Taking the codomain of <em>f </em>to be equal to the range of <em>f </em>, find an expression for \({f^{ - 1}}(x)\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\left] { - 1,1} \right[\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for the values –1, 1 and </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for the open interval.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\frac{{1 - {{\text{e}}^{ - x}}}}{{1 + {{\text{e}}^{ - x}}}} = \frac{{1 - {{\text{e}}^{ - y}}}}{{1 + {{\text{e}}^{ - y}}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 - {{\text{e}}^{ - x}} + {{\text{e}}^{ - y}} - {{\text{e}}^{ - (x + y)}} = 1 + {{\text{e}}^{ - x}} - {{\text{e}}^{ - y}} - {{\text{e}}^{ - (x + y)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^{ - x}} = {{\text{e}}^{ - y}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = y\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore <em>f </em>is an injection <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) = \frac{{{{\text{e}}^{ - x}}(1 + {{\text{e}}^{ - x}}) + {{\text{e}}^{ - x}}(1 - {{\text{e}}^{ - x}})}}{{{{(1 + {{\text{e}}^{ - x}})}^2}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{2{{\text{e}}^{ - x}}}}{{{{(1 + {{\text{e}}^{ - x}})}^2}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( > 0\) for all <em>x</em>. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore <em>f </em>is an injection. <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A1A0 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for a graphical solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Let \(y = \frac{{1 - {{\text{e}}^{ - x}}}}{{1 + {{\text{e}}^{ - x}}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y(1 + {{\text{e}}^{ - x}}) = 1 - {{\text{e}}^{ - x}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^{ - x}}(1 + y) = 1 - y\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^{ - x}} = \frac{{1 - y}}{{1 + y}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \ln \left( {\frac{{1 + y}}{{1 - y}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({f^{ - 1}}(x) = \ln \left( {\frac{{1 + x}}{{1 - x}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates found the range of <span style="font: 19.0px Times;"><em>f </em></span>correctly. Two algebraic methods were seen for solving (b), either showing that the derivative of <span style="font: 19.0px Times;"><em>f </em></span>is everywhere positive or showing that \(f(a) = f(b) \Rightarrow a = b\) . Candidates who based their ‘proof’ on a graph produced on their graphical calculators were given only partial credit on the grounds that the whole domain could not be shown and, in any case, it was not clear from the graph that <span style="font: 19.0px Times;"><em>f </em></span>was an injection.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R</em> is defined on \(\mathbb{Z} \times \mathbb{Z}\) such that \((a,{\text{ }}b)R(c,{\text{ }}d)\) if and only if <em>a</em> − <em>c</em> is divisible by 3 and <em>b</em> − <em>d</em> is divisible by 2.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Prove that <em>R</em> is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the equivalence class for (2, 1) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Write down the five remaining equivalence classes.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) consider \((x,{\text{ }}y)R(x,{\text{ }}y)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">since <em>x</em> – <em>x</em> = 0 and <em>y</em> – <em>y</em> = 0 , <em>R</em> is reflexive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">assume \((x,{\text{ }}y)R(a,{\text{ }}b)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x - a = 3M\) and \(y - b = 2N\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow a - x = - 3M\) and \(b - y = - 2N\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow (a,{\text{ }}b)R(x,{\text{ }}y)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>R</em> is symmetric</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">assume \((x,{\text{ }}y)R(a,{\text{ }}b)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x - a = 3M\) and \(y - b = 2N\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">assume \((a,{\text{ }}b)R(c,{\text{ }}d)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow a - c = 3P\) and \(b - d = 2Q\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x - c = 3(M + P)\) and \(y - d = 2(N + Q)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \((x,{\text{ }}y)R(c,{\text{ }}d)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>R</em> is transitive</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore <em>R</em> is an equivalence relation <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\left\{ {(x,{\text{ }}y):x = 3m + 2,{\text{ }}y = 2n + 1,{\text{ }}m,{\text{ }}n \in \mathbb{Z}} \right\}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(\{ 3m,{\text{ }}2n\} {\text{ \{ }}3m + 1,{\text{ }}2n\} {\text{ }}\{ 3m + 2,{\text{ }}2n\} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ 3m,{\text{ }}2n + 1\} {\text{ \{ }}3m + 1,{\text{ }}2n + 1\} {\text{ }}m,{\text{ }}n \in \mathbb{Z}\) <strong><em>A1A1A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [14 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Stronger candidates had little problem with part (a) of this question, but proving an equivalence relation is still difficult for many. Equivalence classes still cause major problems and few fully correct answers were seen to this question.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( * \) is defined on the set <em>S</em> = {0, 1, 2, 3} by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[a * b = a + 2b + ab(\bmod 4){\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Construct the Cayley table.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Write down, with a reason, whether or not your table is a Latin square.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Write down, with a reason, whether or not \( * \) is commutative.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Determine whether or not \( * \) is associative, justifying your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find all solutions to the equation \(x * 1 = 2 * x\) , for \(x \in S\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) <br></span></p>
<p style="margin: 0px; font: 28px Helvetica; text-align: justify;"><img src="data:image/png;base64,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" alt><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A3</em></strong> for no errors, <strong><em>A2</em></strong> for one error, <strong><em>A1</em></strong> for two errors and <strong><em>A0</em></strong> for three or more errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) it is not a Latin square because some rows/columns contain the same digit more than once <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it is not commutative because the table is not symmetric about the leading diagonal <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it is not commutative because \(a + 2b + ab \ne 2a + b + ab\) in general <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept a counter example <em>e.g.</em> \(1 * 2 = 3\) whereas \(2 * 1 = 2\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for example \((0 * 1) * 1 = 2 * 1 = 2\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and \(0 * (1 * 1) = 0 * 0 = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \( * \) is not associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">associative if and only if \(a * (b * c) = (a * b) * c\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">which gives</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a + 2b + 4c + 2bc + ab + 2ac + abc = a + 2b + ab + 2c + ac + 2bc + abc\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \( * \) is not associative as \(2ac \ne 2c + ac\) , in general <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) <em>x</em> = 0 is a solution <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>x</em> = 2 is a solution <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [13 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well answered.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the six roots of the equation \({z^6} - 1 = 0\) , giving your answers in the form \(r\,{\text{cis}}\,\theta {\text{, }}r \in {\mathbb{R}^ + }{\text{, }}0 \leqslant \theta < 2\pi \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Show that these six roots form a group <em>G </em>under multiplication of complex numbers.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Show that <em>G </em>is cyclic and find all the generators.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Give an example of another group that is isomorphic to <em>G</em>, stating clearly the corresponding elements in the two groups.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) </span><span style="font-family: 'times new roman', times; font-size: medium;">\({z^6} = 1 = {\text{cis}}\,2n\pi \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The six roots are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{cis}}\,0(1),{\text{ cis}}\frac{\pi }{3},{\text{ cis}}\frac{{2\pi }}{3},{\text{ cis}}\,\pi ( - 1),{\text{ cis}}\frac{{4\pi }}{3},{\text{ cis}}\frac{{5\pi }}{3}\) <strong><em>A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A2 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for 4 or 5 correct roots, </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for 2 or 3 correct roots.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) </span><span style="font-family: 'times new roman', times; font-size: medium;">(i) Closure: Consider any two roots \({\text{cis}}\frac{{m\pi }}{3},{\text{ cis}}\frac{{n\pi }}{3}\). <strong><em>M1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{cis}}\frac{{m\pi }}{3} \times {\text{cis}}\frac{{n\pi }}{3} = {\text{cis}}\,(m + n){\text{(mod6)}}\frac{\pi }{3} \in G\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for a correct Cayley table showing closure.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Identity: The identity is 1. </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Inverse: The inverse of \({\text{cis}}\frac{{m\pi }}{3}{\text{ is cis}}\frac{{(6 - m)\pi }}{3} \in G\) . <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Associative: This follows from the associativity of multiplication. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The 4 group axioms are satisfied. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Successive powers of \({\text{cis}}\frac{\pi }{3}\left( {{\text{or cis}}\frac{{5\pi }}{3}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">generate the group which is therefore cyclic. <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The (only) other generator is \({\text{cis}}\frac{{5\pi }}{3}\left( {{\text{or cis}}\frac{\pi }{3}} \right)\) . <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A0 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for any additional answers.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) The group of the integers 0, 1, 2, 3, 4, 5 under addition modulo 6. </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R2</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The correspondence is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\(m \to {\text{cis}}\frac{{m\pi }}{3}\) <strong><em>R1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Accept any other cyclic group of order 6.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[13 marks]</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>Total [17 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was reasonably well answered by many candidates, although in (b)(iii), some candidates were unable to give another group isomorphic to <em>G</em>.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R</em> is defined on \({\mathbb{Z}^ + }\) by <em>aRb</em> if and only if <em>ab</em> is even. Show that only one of the conditions for <em>R</em> to be an equivalence relation is satisfied.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>S</em> is defined on \({\mathbb{Z}^ + }\) by <em>aSb</em> if and only if \({a^2} \equiv {b^2}(\bmod 6)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that <em>S</em> is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) For each equivalence class, give the four smallest members.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">reflexive: if <em>a</em> is odd, \(a \times a\) is odd so <em>R</em> is not reflexive <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">symmetric: if <em>ab</em> is even then <em>ba</em> is even so <em>R</em> is symmetric <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">transitive: let <em>aRb</em> and <em>bRc</em>; it is necessary to determine whether or not <em>aRc</em> <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for example <em>5R2</em> and <em>2R3</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(5 \times 3\) is not even, 5 is not related to 3 and <em>R</em> is not transitive <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive: \({a^2} \equiv {a^2}(\bmod 6)\) so <em>S</em> is reflexive <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">symmetric: \({a^2} \equiv {b^2}(\bmod 6) \Rightarrow 6|({a^2} - {b^2}) \Rightarrow 6|({b^2} - {a^2}) \Rightarrow {b^2} \equiv {a^2}(\bmod 6)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so <em>S</em> is symmetric</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">transitive: let <em>aSb</em> and <em>bSc</em> so that \({a^2} = {b^2} + 6M\) and \({b^2} = {c^2} + 6N\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that \({a^2} = {c^2} + 6(M + N)\) so <em>aSc</em> and <em>S</em> is transitive <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>S</em> is an equivalence relation because it satisfies the three conditions <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) by considering the squares of integers (mod 6), the equivalence <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">classes are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{1, 5, 7, 11, \( \ldots \)} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{2, 4, 8, 10, \( \ldots \)} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{3, 9, 15, 21, \( \ldots \)} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{6, 12, 18, 24, \( \ldots \)} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The groups \(\{ K,{\text{ }} * \} \) and \(\{ H,{\text{ }} \odot \} \) are defined by the following Cayley tables.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">G </span><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">H </span><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By considering a suitable function from <em>G</em> to <em>H</em> , show that a surjective homomorphism exists between these two groups. State the kernel of this homomorphism.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">consider the function <em>f</em> given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(E) = e\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(A) = e\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(B) = a\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(C) = a\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then, it has to be shown that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(X * Y) = f(X) \odot f(Y){\text{ for all }}X{\text{ , }}Y \in G\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f\left( {(E{\text{ or }}A) * (E{\text{ or }}A)} \right) = f(E{\text{ or }}A) = e;{\text{ }}f(E{\text{ or }}A) \odot f(E{\text{ or }}A) = e \odot e = e\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f\left( {(E{\text{ or }}A) * (B{\text{ or }}C)} \right) = f(B{\text{ or }}C) = a;{\text{ }}f(E{\text{ or }}A) \odot f(B{\text{ or }}C) = e \odot a = a\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f\left( {(B{\text{ or }}C) * (B{\text{ or }}C)} \right) = f(E{\text{ or }}A) = e;{\text{ }}f(B{\text{ or }}C) \odot f(B{\text{ or }}C) = a \odot a = e\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since the groups are Abelian, there is no need to consider \(f\left( {(B{\text{ or }}C) * (E{\text{ or }}A)} \right)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the required property is satisfied in all cases so the homomorphism exists</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> A comprehensive proof using tables is acceptable.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the kernel is \(\{ E,{\text{ }}A\} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Three functions mapping \(\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}\) are defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{f_1}(m,{\text{ }}n) = m - n + 4;\,\,\,{f_2}(m,{\text{ }}n) = \left| m \right|;\,\,\,{f_3}(m,{\text{ }}n) = {m^2} - {n^2}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two functions mapping \(\mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}\) are defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{g_1}(k) = (2k,{\text{ }}k);\,\,\,{g_2}(k) = \left( {k,{\text{ }}\left| k \right|} \right).\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the range of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({f_1} \circ {g_1}\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({f_3} \circ {g_2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find all the solutions of \({f_1} \circ {g_2}(k) = {f_2} \circ {g_1}(k)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find all the solutions of \({f_3}(m,{\text{ }}n) = p\) in each of the cases <em>p</em> =1 and <em>p</em> = 2 .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) \({f_1} \circ {g_1}(k) = k + 4\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Range \(({f_1} \circ {g_1}) = \mathbb{Z}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({f_3} \circ {g_2}(k) = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Range \(({f_3} \circ {g_2}) = \{ 0\} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the equation to solve is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k - \left| k \right| + 4 = \left| {2k} \right|\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the positive solution is <em>k</em> = 2 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the negative solution is <em>k</em> = –1 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) the equation factorizes: \((m + n)(m - n) = p\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for <em>p</em> = 1 , the possible factors over \(\mathbb{Z}\) are \(m + n = \pm 1,{\text{ }}m - n = \pm 1\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">with solutions (1, 0) and (–1, 0) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for <em>p</em> = 2 , the possible factors over \(\mathbb{Z}\) are \(m + n = \pm 1,{\text{ }} \pm 2;{\text{ }}m - n = \pm 2,{\text{ }} \pm 1\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">there are no solutions over \(\mathbb{Z} \times \mathbb{Z}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [15 marks]</em></strong></span></p>
<div><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of candidates were able to compute the composite functions involved in parts (a) and (b). Part(c) was satisfactorily tackled by a minority of candidates. There were more GDC solutions than the more obvious approach of factorizing a difference of squares. Some candidates seemed to forget that <em>m</em> and <em>n</em> belonged to the set of integers.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">\(\{ G,{\text{ }} * \} \) is a group with identity element \(e\). Let \(a,{\text{ }}b \in G\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State Lagrange’s theorem.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Verify that the inverse of \(a * {b^{ - 1}}\) is equal to \(b * {a^{ - 1}}\).</p>
<p class="p1"> </p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\). Let \(R\) be a relation defined on \(G\) by</p>
<p class="p1">\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]</p>
<p class="p1">Prove that \(R\) is an equivalence relation, indicating clearly whenever you are using one of the four properties required of a group.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be a relation defined on \(G\) by</p>
<p class="p1">\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]</p>
<p class="p1">Show that \(aRb \Leftrightarrow a \in Hb\), where \(Hb\) is the right coset of \(H\) containing \(b\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be a relation defined on \(G\) by</p>
<p class="p1">\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]</p>
<p class="p1">It is given that the number of elements in any right coset of \(H\) is equal to the order of \(H\).</p>
<p class="p1">Explain how this fact together with parts (c) and (d) prove Lagrange’s theorem.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">in a <strong>finite </strong>group the order of any subgroup (exactly) divides the order of the group <strong><em>A1A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">\((a * {b^{ - 1}}) * (b * {a^{ - 1}}) = a * {b^{ - 1}} * b * {a^{ - 1}} = a * e * {a^{ - 1}} = a * {a^{ - 1}} = e\) <span class="Apple-converted-space"> </span><strong><em>M1A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span><em>M1 </em></strong>for multiplying, <strong><em>A1 </em></strong>for at least one of the next 3 expressions,</p>
<p class="p1"><strong><em>A1 </em></strong>for \(e\).</p>
<p class="p1">Allow \((b * {a^{ - 1}}) * (a * {b^{ - 1}}) = b * {a^{ - 1}} * a * {b^{ - 1}} = b * e * {b^{ - 1}} = b * {b^{ - 1}} = e\).</p>
<p class="p2"> </p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">\({(a * {b^{ - 1}})^{ - 1}} = {({b^{ - 1}})^{ - 1}} * {a^{ - 1}}\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">\( = b * {a^{ - 1}}\)<strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(a * {a^{ - 1}} = e \in H\;\;\;\)<span class="s1">(as \(H\) is a subgroup) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></span></p>
<p class="p1">so \(aRa\) and hence \(R\) is reflexive</p>
<p class="p1">\(aRb \Leftrightarrow a * {b^{ - 1}} \in H\). \(H\) is a subgroup so every element has an inverse in \(H\) so</p>
<p class="p1">\({(a * {b^{ - 1}})^{ - 1}} \in H\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">\( \Leftrightarrow b * {a^{ - 1}} \in H \Leftrightarrow bRa\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">so \(R\) is symmetric</p>
<p class="p1">\(aRb,{\text{ }}bRc \Leftrightarrow a * {b^{ - 1}} \in H,{\text{ }}b * {c^{ - 1}} \in H\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">as \(H\) is closed \((a * {b^{ - 1}}) * {\text{(}}b * {c^{ - 1}}) \in H\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">and using associativity <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">\((a * {b^{ - 1}}) * {\text{(}}b * {c^{ - 1}}) = a * ({b^{ - 1}} * b) * {c^{ - 1}} = a * {c^{ - 1}} \in H \Leftrightarrow aRc\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">therefore \(R\) is transitive</p>
<p class="p1">\(R\) is reflexive, symmetric and transitive</p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Can be said separately at the end of each part.</p>
<p class="p2"> </p>
<p class="p1">hence it is an equivalence relation <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(aRb \Leftrightarrow a * {b^{ - 1}} \in H \Leftrightarrow a * {b^{ - 1}} = h \in H\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\( \Leftrightarrow a = h * b \Leftrightarrow a \in Hb\) <span class="Apple-converted-space"> </span><strong><em>M1R1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(d) implies that the right cosets of \(H\) are equal to the equivalence classes of the relation in (c) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">hence the cosets partition \(G\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">all the cosets are of the same size as the subgroup \(H\) so the order of \(G\) must be a multiple of \(\left| H \right|\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [19 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many students obtained just half marks in (a) for not stating the requirement of the order to be finite.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) should have been more straightforward than many found.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (c) it was evident that most candidates knew what to do, but being a more difficult question fell down on a lack of rigour. Nonetheless, many candidates obtained full or partial marks on this question part.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (d) enabled many candidates to obtain, at least partial marks, but there were few students with the insight to be able to answer part (e) satisfactorily.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (d) enabled many candidates to obtain, at least partial marks, but there were few students with the insight to be able to answer part (e) satisfactorily.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given a set \(U\), and two of its subsets \(A\) and \(B\), prove that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B),{\text{ where }}A\backslash B = A \cap B'.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Let \(S = \{ A,{\text{ }}B,{\text{ }}C,{\text{ }}D\} \) where \(A = \emptyset ,{\text{ }}B = \{ 0\} ,{\text{ }}C = \{ 0,{\text{ }}1\} \) and \(D = \{ {\text{0, 1, 2}}\} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">State, with reasons, whether or not each of the following statements is true.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) The operation \ is closed in \(S\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) The operation \( \cap \) has an identity element in \(S\) but not all elements have an inverse.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Given \(Y \in S\), the equation \(X \cup Y = Y\) always has a unique solution for \(X\) in \(S\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \((A\backslash B) \cup (B\backslash A) = (A \cap B') \cup (B \cap A')\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {(A \cap B') \cup B} \right) \cap \left( {(A \cap B') \cup A'} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {(A \cup B) \cap \underbrace {(B' \cup B)}_U} \right) \cap \left( {\underbrace {(A \cup A')}_U \cap (B' \cup A')} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (A \cup B') \cap (B' \cup A') = (A \cup B) \cap (A \cap B)'\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (A \cup B)\backslash (A \cap B)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) false <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">counterexample</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(D\backslash C = \{ 2\} \notin S\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) true <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">as \(A \cap D = A,{\text{ }}B \cap D = B,{\text{ }}C \cap D = C\) and \(D \cap D = D\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(D\) is the identity <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(A\) (or \(B\) or \(C\)) has no inverse as \(A \cap X = D\) is impossible <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) false <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">when \(Y = D\) the equation has more than one solution (four solutions) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">For part (a), candidates who chose to prove the given statement using the properties of Sets were often successful with the proof. Some candidates chose to use the definition of equality of sets, but made little to no progress. In a few cases candidates attempted to use Venn diagrams as a proof. Part (b) was challenging for most candidates, and few correct answers were seen.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The relation \(R\) is defined on \(\mathbb{Z}\) by \(xRy\) if and only if \({x^2}y \equiv y\bmod 6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the product of three consecutive integers is divisible by \(6\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence prove that \(R\) is reflexive.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the set of all \(y\) for which \(5Ry\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the set of all \(y\) for which \(3Ry\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Using your answers for (c) and (d) show that \(R\) is not symmetric.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">in a product of three consecutive integers either one or two are even <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">and one is a multiple of \(3\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">so the product is divisible by \(6\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">to test reflexivity, put \(y = x\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">then \({x^2}x - x = (x - 1)x(x + 1) \equiv 0\bmod 6\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">so \(xRx\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">if \(5Ry\) then \(25y \equiv y\bmod 6\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(24y \equiv 0\bmod 6\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">the set of solutions is \(\mathbb{Z}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Only one of the method marks may be implied.</p>
<p class="p1"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">if \(3Ry\) then \(9y \equiv y\bmod 6\)</p>
<p class="p1">\(8y \equiv 0\bmod 6 \Rightarrow 4y \equiv 0\bmod 3\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">the set of solutions is \(3\mathbb{Z}\) (<em>ie</em> multiples of \(3\)) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">from part (c) \(5R3\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">from part (d) \(3R5\) is false <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(R\) is not symmetric <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Accept other counterexamples.</p>
<p class="p1"><em><strong>[2 marks] </strong></em></p>
<p class="p1"><em><strong>Total [12 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">A surprising number of candidates thought that an example was sufficient evidence to answer this part.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Again, a lack of confidence with modular arithmetic undermined many candidates' attempts at this part.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(c) and (d) Most candidates started these parts, but some found solutions as fractions rather than integers or omitted zero and/or negative integers.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(c) and (d) Most candidates started these parts, but some found solutions as fractions rather than integers or omitted zero and/or negative integers.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some candidates regarded \(R\) as an operation, rather than a relation, so returned answers of the form \(aRb \ne bRa\).</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine, giving reasons, which of the following sets form groups under the operations given below. Where appropriate you may assume that multiplication is associative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\mathbb{Z}\) under subtraction.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The set of complex numbers of modulus 1 under multiplication.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) The set {1, 2, 4, 6, 8} under multiplication modulo 10.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) The set of rational numbers of the form</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{3m + 1}}{{3n + 1}},{\text{ where }}m,{\text{ }}n \in \mathbb{Z}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">under multiplication.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) not a group <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">subtraction is not associative on \(\mathbb{Z}\) (or give counter-example) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">there is a right-identity, 0, but it is not a left-identity <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the set forms a group <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the closure is a consequence of the following relation (and the closure of \(\mathbb{C}\) itself):</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{z_1}{z_2}} \right| = \left| {{z_1}} \right|\left| {{z_2}} \right|\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the set contains the identity 1 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">that inverses exist follows from the relation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{z^{ - 1}}} \right| = {\left| z \right|^{ - 1}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for non-zero complex numbers <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) not a group <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for example, only the identity element 1 has an inverse <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) the set forms a group <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{2m + 1}}{{3n + 1}} \times \frac{{3s + 1}}{{3t + 1}} = \frac{{9ms + 3s + 3m + 1}}{{9nt + 3n + 3t + 1}} = \frac{{3(3ms + s + m) + 1}}{{3(3nt + n + t) + 1}}\) <strong><em>M1R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">shows closure</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the identity 1 corresponds to <em>m</em> = <em>n</em> = 0 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">an inverse corresponds to interchanging the parameters <em>m</em> and <em>n</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [13 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">There was a mixed response to this question. Some candidates were completely out of their depth. Stronger candidates provided satisfactory answers to parts (a) and (c). For the other parts there was a general lack of appreciation that, for example, closure and the existence of inverses, requires that products and inverses have to be shown to be members of the set.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the set <em>S </em>defined by \(S = \{ s \in \mathbb{Q}:2s \in \mathbb{Z}\} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">You may assume that \( + \) (addition) and \( \times \) (multiplication) are associative binary operations</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">on \(\mathbb{Q}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the six smallest non-negative elements of \(S\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that \(\{ S,{\text{ }} + \} \) is a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Give a reason why \(\{ S,{\text{ }} \times \} \) is not a group. Justify your answer.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The relation \(R\) is defined on \(S\) by \({s_1}R{s_2}\) if \(3{s_1} + 5{s_2} \in \mathbb{Z}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(R\) is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine the equivalence classes.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{0, }}\frac{{\text{1}}}{{\text{2}}}{\text{, 1, }}\frac{{\text{3}}}{{\text{2}}}{\text{, 2, }}\frac{{\text{5}}}{{\text{2}}}\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes: <em>A2 </em></strong>for all correct, <strong><em>A1 </em></strong>for three to five correct.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">closure: if \({s_1},{\text{ }}{s_2} \in S\), then \({s_1} = \frac{m}{2}\) and \({s_2} = \frac{n}{2}\) for some \(m,{\text{ }}n \in {\text{¢}}\). <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept two distinct examples (<em>eg</em>, \(\frac{1}{2} + \frac{1}{2} = 1;{\text{ }}\frac{1}{2} + 1 = \frac{3}{2}\)) for the <strong><em>M1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({s_1} + {s_2} = \frac{{m + n}}{2} \in S\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">the sum of two half-integers <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">is a half-integer <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">identity: 0 is the (additive) identity <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">inverse: \(s + ( - s) = 0\), where \( - s \in S\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">it is associative (since \(S \subset \S\)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">the group axioms are satisfied <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">the set is not closed under multiplication, <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">for example, \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\), but \(\frac{1}{4} \notin S\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">not every element has an inverse, <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">for example, 3 does not have an inverse <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive: consider \(3s + 5s\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 8s \in {\text{¢}} \Rightarrow \) reflexive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">symmetric: if \({s_1}R{s_2}\), consider \(3{s_2} + 5{s_1}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for example, \( = 3{s_1} + 5{s_2} + (2{s_1} - 2{s_2}) \in {\text{¢}} \Rightarrow \)symmetric <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">transitive: if \({s_1}R{s_2}\) and \({s_2}R{s_3}\), consider <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(3{s_1} + 5{s_3} = (3{s_1} + 5{s_2}) + (3{s_2} + 5{s_3}) - 8{s_2}\) <em><strong>M1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \in {\text{¢}} \Rightarrow \)transitive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so <em>R </em>is an equivalence relation <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) </span><span style="font-family: 'times new roman', times; font-size: medium;">\({C_1} = {\text{¢}}\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({C_2} = \left\{ { \pm \frac{1}{2},{\text{ }} \pm \frac{3}{2},{\text{ }} \pm \frac{5}{2},{\text{ }} \ldots } \right\}\) <em><strong>A1A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong> </strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>A1 </em></strong>for half odd integers and <strong><em>A1 </em></strong>for ±.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \(\Delta\) is defined on the set \(S =\) {1, 2, 3, 4, 5} by the following Cayley table.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-10_om_13.21.35.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State whether <em>S </em>is closed under the operation Δ and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) State whether Δ is commutative and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) State whether there is an identity element and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Determine whether Δ is associative and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Find the solutions of the equation \(a\Delta b = 4\Delta b\), for \(a \ne 4\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) yes <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">because the Cayley table only contains elements of <em>S </em><strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) yes <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">because the Cayley table is symmetric <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) no <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">because there is no row (and column) with 1, 2, 3, 4, 5 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) attempt to calculate \((a\Delta b)\Delta c\) and \(a\Delta (b\Delta c)\) for some \(a,{\text{ }}b,{\text{ }}c \in S\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">counterexample: for example, \((1\Delta 2)\Delta 3 = 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1\Delta (2\Delta 3) = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Δ is not associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept a correct evaluation of \((a\Delta b)\Delta c\) and \(a\Delta (b\Delta c)\) for some \(a,{\text{ }}b,{\text{ }}c \in S\) for the <strong><em>M1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; color: #3f3f3f;"><em><strong><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span></strong></em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; color: #3f3f3f;"><em><strong><span style="font-family: 'times new roman', times; font-size: medium;"> </span></strong></em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) for example, attempt to enumerate \(4\Delta b\) for <em>b</em> = 1, 2, 3, 4, 5 and obtain (3, 2, 1, 4, 1) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">find \((a,{\text{ }}b) \in \left\{ {{\text{(2, 2), (2, 3)}}} \right\}\) for \(a \ne 4\) (or equivalent) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A1A0 </em></strong>if extra ‘solutions’ are listed.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [12 marks]</em></strong></span></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( * \) is defined for \(a{\text{, }}b \in {\mathbb{Z}^ + }\) by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[a * b = a + b - 2.\]</span></p>
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<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine whether or not \( * \) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) closed,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) commutative,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) associative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Find the identity element.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the set of positive integers having an inverse under \( * \).</span></p>
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<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) It is not closed because</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 * 1 = 0 \notin {\mathbb{Z}^ + }\) . <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(a * b = a + b - 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(b * a = b + a - 2 = a * b\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">It is commutative. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) It is not associative. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider \((1 * 1) * 5\) and \(1 * (1 * 5)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The first is undefined because \(1 * 1 \notin {\mathbb{Z}^ + }\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The second equals 3. <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;">Notes: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1R2 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for stating that non-closure implies non-associative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1R1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">to candidates who show that \(a * (b * c) = (a * b) * c = a + b + c - 4\) and therefore conclude that it is associative, ignoring the non-closure.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) The identity <em>e </em>satisfies</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a * e = a + e - 2 = a\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(e = 2\,\,\,\,\,({\text{and }}2 \in {\mathbb{Z}^ + })\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(a * {a^{ - 1}} = a + {a^{ - 1}} - 2 = 2\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a + {a^{ - 1}} = 4\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">So the only elements having an inverse are 1, 2 and 3. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Due to commutativity there is no need to check two sidedness of identity and inverse.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[5 marks]</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>Total [12 marks]</em></strong></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Almost all the candidates thought that the binary operation was associative, not realising that the non-closure prevented this from being the case. In the circumstances, however, partial credit was given to candidates who ‘proved’ associativity. Part (b) was well done by many candidates.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A\), \(B\), \(C\) and \(D\) are subsets of \(\mathbb{Z}\) .</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A = \{ \left. m \right|m{\text{ is a prime number less than 15}}\}\)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(B = \{ \left. m \right|{m^4} = 8m\} \)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(C = \{ \left. m \right|(m + 1)(m - 2) < 0\} \)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(D = \{ \left. m \right|{m^2} < 2m + 4\} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) List the elements of each of these sets.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine, giving reasons, which of the following statements are true and which are false.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) \(n(D) = n(B) + n(B \cup C)\)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \(D\backslash B \subset A\)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) \(B \cap A' = \emptyset \)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iv) \(n(B\Delta C) = 2\)</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) by inspection, or otherwise,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>A</em> = {2, 3, 5, 7, 11, 13} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>B</em> = {0, 2} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>C</em> = {0, 1} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>D</em> = {–1, 0, 1, 2, 3} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) true <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n(B) + n(B \cup C) = 2 + 3 = 5 = n(D)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) false <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(D\backslash B = \{ - 1,{\text{ }}1,{\text{ }}3\} \not\subset A\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) false <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(B \cap A' = \{ 0\} \ne \emptyset \) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) true <em><strong>A1</strong></em> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n(B\Delta C) = n\{ 1,{\text{ }}2\} = 2\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [12 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">It was surprising and disappointing that many candidates regarded 1 as a prime number. One of the consequences of this error was that it simplified some of the set-theoretic calculations in part(b), with a loss of follow-through marks. Generally speaking, it was clear that the majority of candidates were familiar with the set operations in part(b).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Consider the set <em>A</em> = {1, 3, 5, 7} under the binary operation \( * \), where \( * \) denotes multiplication modulo 8.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Write down the Cayley table for \(\{ A,{\text{ }} * \} \).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Show that \(\{ A,{\text{ }} * \} \) is a group.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Find all solutions to the equation \(3 * x * 7 = y\). Give your answers in the form \((x,{\text{ }}y)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Now consider the set <em>B</em> = {1, 3, 5, 7, 9} under the binary operation \( \otimes \), where \( \otimes \) denotes multiplication modulo 10. Show that \(\{ B,{\text{ }} \otimes \} \) is not a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Another set <em>C</em> can be formed by removing an element from <em>B</em> so that \(\{ C,{\text{ }} \otimes \} \) is a group.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) State which element has to be removed.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Determine whether or not \(\{ A,{\text{ }} * \} \) and \(\{ C,{\text{ }} \otimes \} \) are isomorphic.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) </span><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A2</em></strong> for 15 correct, <strong><em>A1</em></strong> for 14 correct and <strong><em>A0</em></strong> otherwise.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) it is a group because:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the table shows closure <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">multiplication is associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it possesses an identity 1 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">justifying that every element has an inverse <em>e.g.</em> all self-inverse <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) (since \( * \) is commutative, \(5 * x = y\))</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so solutions are (1, 5), (3, 7), (5, 1), (7, 3) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> Award <strong><em>A1</em></strong> for 3 correct and <strong><em>A0</em></strong> otherwise.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Do not penalize extra incorrect solutions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) </span><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> It is not necessary to see the Cayley table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a valid reason <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>e.g.</em> from the Cayley table the 5 row does not give a Latin square, or 5 does not have an inverse, so it cannot be a group</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) remove the 5 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) they are not isomorphic because all elements in <em>A</em> are self-inverse this is not the case in <em>C</em>, (e.g. \(3 \otimes 3 = 9 \ne 1\)) <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept any valid reason.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>Total [14 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates are generally confident when dealing with a specific group and that was the situation again this year. Some candidates lost marks in (a)(ii) by not giving an adequate explanation for the truth of some of the group axioms, eg some wrote ‘every element has an inverse’. Since the question told the candidates that \(\{ A,{\text{ }} * )\) was a group, this had to be the case and the candidates were expected to justify their statement by noting that every element was self-inverse. Solutions to (c)(ii) were reasonably good in general, certainly better than solutions to questions involving isomorphisms set in previous years.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let {<em>G</em> , \( * \)} be a finite group of order <em>n</em> and let <em>H</em> be a non-empty subset of <em>G</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that any element \(h \in H\) has order smaller than or equal to <em>n</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) If <em>H</em> is closed under \( * \), show that {<em>H</em> , \( * \)} is a subgroup of {<em>G</em> , \( * \)}.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) if \(h \in H\) then \(h \in G\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence, (by Lagrange) the order of <em>h</em> exactly divides <em>n</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and so the order of <em>h</em> is smaller than or equal to <em>n</em> <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the associativity in <em>G</em> ensures associativity in <em>H</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(closure within <em>H</em> is given)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">as <em>H</em> is non-empty there exists an \(h \in H\) , let the order of <em>h</em> be <em>m</em> then \({h^m} = e\) and as <em>H</em> is closed \(e \in H\) <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows from the earlier result that \(h * {h^{m - 1}} = {h^{m - 1}} * h = e\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">thus, the inverse of <em>h</em> is \({h^{m - 1}}\) which \( \in H\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the four axioms are satisfied showing that \(\{ H{\text{ , }} * \} \) is a subgroup <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [9 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to this question were extremely disappointing. This property of subgroups is mentioned specifically in the Guide and yet most candidates were unable to make much progress in (b) and even solutions to (a) were often unconvincing.</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1"><span class="s1">The group \(\{ G,{\text{ }} * \} \) </span>is Abelian and the bijection \(f:{\text{ }}G \to G\) is defined by \(f(x) = {x^{ - 1}},{\text{ }}x \in G\).</p>
<p class="p1">Show that \(f\) is an isomorphism.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">we need to show that \(f(a * b) = f(a) * f(b)\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>This <strong><em>R1 </em></strong>may be awarded at any stage.</p>
<p class="p1">let \(a,{\text{ }}b \in G\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">consider \(f(a) * f(b)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p3"><span class="Apple-converted-space">\( = {a^{ - 1}} * {b^{ - 1}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">consider \(f(a * b) = {(a * b)^{ - 1}}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p3"><span class="Apple-converted-space">\( = {b^{ - 1}} * {a^{ - 1}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s2">\( = {a^{ - 1}} * {b^{ - 1}}\) </span>since \(G\) is Abelian <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">hence \(f\) is an isomorphism <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">A surprising number of candidates wasted time and unrewarded effort showing that the mapping \(f\), stated to be a bijection in the question, actually was a bijection. Many candidates failed to get full marks by not properly using the fact that the group was stated to be Abelian. There were also candidates who drew the graph of \(y = \frac{1}{x}\) or otherwise assumed that the inverse of \(x\) was its reciprocal - this is unacceptable in the context of an abstract group question.</p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The group <em>G</em> has a subgroup <em>H</em>. The relation <em>R</em> is defined on <em>G</em> by <em>xRy</em> if and only if \(x{y^{ - 1}} \in H\), for \(x,{\text{ }}y \in G\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>R</em> is an equivalence relation.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The Cayley table for <em>G</em> is shown below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The subgroup <em>H</em> is given as \(H = \{ e,{\text{ }}{a^2}b\} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the equivalence class with respect to <em>R</em> which contains <em>ab</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Another equivalence relation \(\rho \) is defined on <em>G</em> by \(x\rho y\) if and only if \({x^{ - 1}}y \in H\), for \(x,{\text{ }}y \in G\). Find the equivalence class with respect to \(\rho \) which contains <em>ab</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x{x^{ - 1}} = e \in H\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow xRx\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>R</em> is reflexive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">if <em>xRy</em> then \(x{y^{ - 1}} \in H\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {(x{y^{ - 1}})^{ - 1}} \in H\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">now \((x{y^{ - 1}}){(x{y^{ - 1}})^{ - 1}} = e\) and \(x{y^{ - 1}}y{x^{ - 1}} = e\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {(x{y^{ - 1}})^{ - 1}} = y{x^{ - 1}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \(y{x^{ - 1}} \in H \Rightarrow yRx\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>R</em> is symmetric <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">if <em>xRy</em>, <em>yRz</em> then \(x{y^{ - 1}} \in H,{\text{ }}y{z^{ - 1}} \in H\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow (x{y^{ - 1}})(y{z^{ - 1}}) \in H\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x({y^{ - 1}}y){z^{ - 1}} \in H\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {x^{ - 1}}z \in H\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>R</em> is transitive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence <em>R</em> is an equivalence relation <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) for the equivalence class, solving:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x{(ab)^{ - 1}} = e{\text{ or }}x{(ab)^{ - 1}} = {a^2}b\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ ab,{\text{ }}a\} \) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(ab{(x)^{ - 1}} = e{\text{ or }}ab{(x)^{ - 1}} = {a^2}b\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ ab,{\text{ }}a\} \) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) for the equivalence class, solving:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({x^{ - 1}}(ab) = e{\text{ or }}{x^{ - 1}}(ab) = {a^2}b\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ ab,{\text{ }}{a^2}\} \) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(ab)^{ - 1}}x = e{\text{ or }}{(ab)^{ - 1}}x = {a^2}b\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ ab,{\text{ }}{a^2}\} \) <strong><em>A2</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Stronger candidates made a reasonable start to (a), and many were able to demonstrate that the relation was reflexive and transitive. However, the majority of candidates struggled to make a meaningful attempt to show the relation was symmetric, with many making unfounded assumptions. Equivalence classes still cause major problems and few fully correct answers were seen to (b).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Stronger candidates made a reasonable start to (a), and many were able to demonstrate that the relation was reflexive and transitive. However, the majority of candidates struggled to make a meaningful attempt to show the relation was symmetric, with many making unfounded assumptions. Equivalence classes still cause major problems and few fully correct answers were seen to (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R</em> is defined on {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} by <em>aRb</em> if and only if \(a(a + 1) \equiv b(b + 1)(\bmod 5)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>R</em> is an equivalence relation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the equivalence defining <em>R</em> can be written in the form</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(a - b)(a + b + 1) \equiv 0(\bmod 5).\]</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, determine the equivalence classes.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">reflexive: \(a(a + 1) \equiv a(a + 1)(\bmod 5)\), therefore <em>aRa</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">symmetric: \(aRb \Rightarrow a(a + 1) = b(b + 1) + 5N\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow b(b + 1) = a(a + 1) - 5N \Rightarrow bRa\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">transitive:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(aRb{\text{ and }}bRc \Rightarrow a(a + 1) = b(b + 1) + 5M{\text{ and }}b(b + 1) = c(c + 1) + 5N\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that \(a(a + 1) = c(c + 1) + 5(M + N) \Rightarrow aRc\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(aRb{\text{ and }}bRc \Rightarrow a(a + 1) \equiv b(b + 1)(\bmod 5){\text{ and}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(b(b + 1) \equiv c(c + 1)(\bmod 5)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a(a + 1) - b(b + 1) \equiv 0(\bmod 5);{\text{ }}b(b + 1) - c(c + 1) \equiv 0(\bmod 5)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a(a + 1) - c(c + 1) \equiv 0\bmod 5 \Rightarrow a(a + 1) \equiv c(c + 1)\bmod 5 \Rightarrow aRc\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the equivalence can be written as</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({a^2} + a - {b^2} - b \equiv 0(\bmod 5)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((a - b)(a + b) + a - b \equiv 0(\bmod 5)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((a - b)(a + b + 1) \equiv 0(\bmod 5)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the equivalence classes are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{1, 3, 6, 8, 11}</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{2, 7, 12}</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{4, 5, 9, 10} <strong><em>A4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A3</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for 2 correct classes, </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A2</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for 1 correct class.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates knew the properties of equivalence relations but did not show sufficient working out in the transitive case. Others did not do the modular arithmetic correctly, still others omitted the \(\bmod(5)\) in part or throughout.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates knew the properties of equivalence relations but did not show sufficient working out in the transitive case. Others did not do the modular arithmetic correctly, still others omitted the \(\bmod (5)\) in part or throughout.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates knew the properties of equivalence relations but did not show sufficient working out in the transitive case. Others did not do the modular arithmetic correctly, still others omitted the \(\bmod(5)\) in part or throughout.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the set \({S_3} = \{ {\text{ }}p,{\text{ }}q,{\text{ }}r,{\text{ }}s,{\text{ }}t,{\text{ }}u\} \) of permutations of the elements of the set \(\{ 1,{\text{ }}2,{\text{ }}3\} \), defined by</p>
<p class="p1"><span class="Apple-converted-space"> </span>\(p = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 1&2&3 \end{array}} \right),{\text{ }}q = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 1&3&2 \end{array}} \right),{\text{ }}r = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 3&2&1 \end{array}} \right),{\text{ }}s = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 2&1&3 \end{array}} \right),{\text{ }}t = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 2&3&1 \end{array}} \right),{\text{ }}u = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 3&1&2 \end{array}} \right).\)</p>
<p class="p1">Let \( \circ \) denote composition of permutations, so \(a \circ b\) means \(b\) followed by \(a\). You may assume that \(({S_3},{\text{ }} \circ )\) forms a group.</p>
<p class="p1"> </p>
<p class="p1"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Complete the following Cayley table</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-08_om_09.28.14.png" alt></p>
<p class="p1" style="text-align: left;"><em><strong>[5 marks]</strong></em></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) State the inverse of each element.</p>
<p class="p1">(ii) Determine the order of each element.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the subgroups containing</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(r\),</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(u\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space"><img src="images/Schermafbeelding_2016-01-08_om_09.33.26.png" alt> </span><span class="s1"><strong>(M1)A4</strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Award <strong><em>M1</em></strong> for use of Latin square property and/or attempted multiplication, <strong><em>A1</em></strong> for the first row or column, <strong><em>A1</em></strong> for the squares of \(q\), \(r\) and \(s\), then <strong><em>A2</em></strong> for all correct.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\({p^{ - 1}} = p,{\text{ }}{q^{ - 1}} = q,{\text{ }}{r^{ - 1}} = r,{\text{ }}{s^{ - 1}} = s\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\({t^{ - 1}} = u,{\text{ }}{u^{ - 1}} = t\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Allow FT from part (a) unless the working becomes simpler.</p>
<p class="p3"> </p>
<p class="p4"><span class="s2">(ii) <span class="Apple-converted-space"> </span></span>using the table or direct multiplication <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">the orders of \(\{ p,{\text{ }}q,{\text{ }}r,{\text{ }}s,{\text{ }}t,{\text{ }}u\} \) are \(\{ 1,{\text{ }}2,{\text{ }}2,{\text{ }}2,{\text{ }}3,{\text{ }}3\} \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A3</em></strong></span></p>
<p class="p2"> </p>
<p class="p4"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Award <strong><em>A1</em></strong> for two, three or four correct, <strong><em>A2</em></strong> for five correct.</p>
<p class="p4"><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(\{ p,{\text{ }}r\} {\text{ }}\left( {{\text{and }}({S_3},{\text{ }} \circ )} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(\{ p,{\text{ }}u,{\text{ }}t\} {\text{ }}\left( {{\text{and }}({S_3},{\text{ }} \circ )} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Award <strong><em>A0A1</em></strong> if the identity has been omitted.</p>
<p class="p3">Award <strong><em>A0</em></strong> in (i) or (ii) if an extra incorrect “subgroup” has been included.</p>
<p class="p3"><em><strong>[2 marks]</strong></em></p>
<p class="p3"><em><strong>Total [13 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The majority of candidates were able to complete the Cayley table correctly. Unfortunately, many wasted time and space, laboriously working out the missing entries in the table - the identity is \(p\) and the elements \(q\), \(r\) and \(s\) are clearly of order two, so 14 entries can be filled in without any calculation. A few candidates thought \(t\) and \(u\) had order two.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Generally well done. A few candidates were unaware of the definition of the order of an element.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Often well done. A few candidates stated extra, and therefore incorrect subgroups.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The permutation \({p_1}\) of the set {1, 2, 3, 4} is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{p_1} = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&4&1&3 <br>\end{array}} \right)\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) State the inverse of \({p_1}\).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the order of \({p_1}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Another permutation \({p_2}\) is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{p_2} = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 3&2&4&1 <br>\end{array}} \right)\]</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Determine whether or not the composition of \({p_1}\) and \({p_2}\) is commutative.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the permutation \({p_3}\) which satisfies</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{p_1}{p_3}{p_2} = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 1&2&3&4 <br>\end{array}} \right){\text{.}}\]</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) the inverse is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 3&1&4&2 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 \to 2 \to 4 \to 3 \to 1\) (is a cycle of length 4) <strong><em>R3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \({p_1}\) is of order 4 <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p_1^2 = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 4&3&1&2 <br>\end{array}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it is now clear that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p_1^4 = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 1&2&3&4 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \({p_1}\) is of order 4 <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({p_1}{p_2} = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&4&1&3 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 3&2&4&1 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 1&4&3&2 <br>\end{array}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({p_2}{p_1} = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 3&2&4&1 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&4&1&3 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&1&3&4 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">composition is not commutative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> In this part do not penalize candidates who incorrectly reverse the order both times.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">pre and postmultiply by \(p_1^{ - 1}\), \(p_2^{ - 1}\)to give</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({p_3} = p_1^{ - 1}p_2^{ - 1}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 3&1&4&2 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 4&2&1&3 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&1&3&4 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">starting from</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&4&1&3 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> {}&{}&{}&{} <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 3&2&4&1 <br>\end{array}} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">successively deducing each missing number, to get</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&4&1&3 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&1&3&4 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 3&2&4&1 <br>\end{array}} \right)\) <strong><em>A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [13 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates scored well on this question although some gave the impression of not having studied this topic. The most common error in (b) was to believe incorrectly that \({p_1}{p_2}\) means \({p_1}\) followed by \({p_2}\). This was condoned in (i) but penalised in (ii). The Guide makes it quite clear that this is the notation to be used.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>R</em> be a relation on the set \(\mathbb{Z}\) such that \(aRb \Leftrightarrow ab \geqslant 0\), for <em>a</em>, <em>b</em> \( \in \mathbb{Z}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine whether <em>R</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Write down with a reason whether or not <em>R</em> is an equivalence relation.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) \({a^2} \geqslant 0\) for all \(a \in \mathbb{Z}\), hence <em>R</em> is reflexive <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(aRb \Rightarrow ab \geqslant 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow ba \geqslant 0\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow bRa\), hence <em>R</em> is symmetric <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(aRb{\text{ and }}bRc \Rightarrow ab \geqslant 0{\text{ and }}bc \geqslant 0,{\text{ is }}aRc?\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">no, for example, \( - 3R0\) and \(0R5\), but \( - 3R5\) is not true <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>aRc</em> is not generally true, hence <em>R</em> is not transitive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <em>R</em> does not satisfy all three properties, hence <em>R</em> is not an equivalence relation <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Although the properties of an equivalence relation were well known, few candidates provided a counter-example to show that the relation is not transitive. Some candidates interchanged the definitions of the reflexive and symmetric properties.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R</em> is defined for <em>a</em> , \(b \in {\mathbb{Z}^ + }\) such that <em>aRb</em> if and only if \({a^2} - {b^2}\) is divisible by 5.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>R</em> is an equivalence relation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Identify the three equivalence classes.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">reflexive: <em>aRa</em> because \({a^2} - {a^2} = 0\) (which is divisible by 5) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">symmetric: let <em>aRb</em> so that \({a^2} - {b^2} = 5M\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that \({a^2} - {b^2} = - 5M\) which is divisible by 5 so <em>bRa</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">transitive: let <em>aRb</em> and <em>bRc</em> so that \({a^2} - {b^2} = 5M\) and \({b^2} - {c^2} = 5N\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({a^2} - {b^2} + {b^2} - {c^2} = 5M + 5N\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({a^2} - {c^2} = 5M + 5N\) which is divisible by 5 so <em>aRc</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \) <em>R</em> is an equivalence relation <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the equivalence classes are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{1, 4, 6, 9, …} <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{2, 3, 7, 8, …} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{5, 10, …} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Do not award any marks for classes containing fewer elements than shown above.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates solved (a) correctly but solutions to (b) were generally poor. Most candidates seemed to have a weak understanding of the concept of equivalence classes and were unaware of any systematic method for finding the equivalence classes. If all else fails, a trial and error approach can be used. Here, starting with 1, it is easily seen that 4, 6,… belong to the same class and the pattern can be established. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates solved (a) correctly but solutions to (b) were generally poor. Most candidates seemed to have a weak understanding of the concept of equivalence classes and were unaware of any systematic method for finding the equivalence classes. If all else fails, a trial and error approach can be used. Here, starting with 1, it is easily seen that 4, 6, … belong to the same class and the pattern can be established. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(G\) be a group of order 12 with identity element <em>e</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(a \in G\) such that \({a^6} \ne e\) and \({a^4} \ne e\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Prove that \(G\) is cyclic and state two of its generators.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Let \(H\) be the subgroup generated by \({a^4}\). Construct a Cayley table for \(H\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">State, with a reason, whether or not it is necessary that a group is cyclic given that all its proper subgroups are cyclic.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) the order of \(a\) is a divisor of the order of \(G\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">since the order of \(G\) is 12, the order of \(a\)<em> </em>must be 1, 2, 3, 4, 6 or 12 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">the order cannot be 1, 2, 3 or 6, since \({a^6} \ne e\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">the order cannot be 4, since \({a^4} \ne e\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">so the order of \(a\)<em> </em>must be 12</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">therefore, \(a\)<em> </em>is a generator of \(G\), which must therefore be cyclic <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">another generator is <em>eg</em> \({a^{ - 1}},{\text{ }}{a^5},{\text{ }} \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(H = \{ e,{\text{ }}{a^4},{\text{ }}{a^8}\} \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-17_om_18.40.15.png" alt> <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">no <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>the group of symmetries of a triangle \({S_3}\)<span style="font: 17.0px 'Times New Roman';"><em> </em></span>is not cyclic but all its (proper) subgroups are cyclic</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>the Klein four-group is not cyclic but all its (proper) subgroups are cyclic <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (a), many candidates could not provide a logical sequence of steps to show that \(G\) is cyclic. In particular, although they correctly quoted Lagrange’s theorem, they did not always consider all the orders of a, i.e., all the factors of 12, omitting in particular 1 as a factor. Some candidates did not state the second generator, in particular \({a^{ - 1}}\). Very few candidates were successful in finding the required subgroup, although they were obviously familiar with setting up a Cayley table.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that {1, −1, i, −i} forms a group of complex numbers <em>G</em> under multiplication.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Consider \(S = \{ e,{\text{ }}a,{\text{ }}b,{\text{ }}a * b\} \) under an associative operation \( * \) where <em>e</em> is the identity element. If \(a * a = b * b = e\) and \(a * b = b * a\) , show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(a * b * a = b\) ,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(a * b * a * b = e\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Write down the Cayley table for \(H = \{ S{\text{ , }} * \} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that <em>H</em> is a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Show that <em>H</em> is an Abelian group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) For the above groups, <em>G</em> and <em>H</em> , show that one is cyclic and write down why the other is not. Write down all the generators of the cyclic group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Give a reason why </span><em style="font-family: 'times new roman', times; font-size: medium;">G</em><span style="font-family: 'times new roman', times; font-size: medium;"> and </span><em style="font-family: 'times new roman', times; font-size: medium;">H</em><span style="font-family: 'times new roman', times; font-size: medium;"> are not isomorphic.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">see the Cayley table, (since there are no new elements) the set is closed <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">1 is the identity element <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">1 and –1 are self inverses and i and -i form an inverse pair, hence every element has an inverse <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">multiplication is associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence {1, –1, i, –i} form a group <em>G</em> under the operation of multiplication <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) <em>aba</em> = <em>aab</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= <em>eb</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= <em>b</em> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <em>abab</em> = <em>aabb</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= <em>ee</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= <em>e</em> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><img src="data:image/png;base64,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" alt><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for 1 or 2 errors, <strong><em>A0</em></strong> for more than 2.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) see the Cayley table, (since there are no new elements) the set is closed <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>H</em> has an identity element <em>e</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">all elements are self inverses, hence every element has an inverse <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the operation is associative as stated in the question</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence {<em>e </em>, <em>a </em>, <em>b </em>, <em>ab</em>} forms a group <em>G</em> under the operation \( * \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) since there is symmetry across the leading diagonal of the group table, the group is Abelian <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) consider the element i from the group <em>G</em> <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{i}}^2} = - 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{i}}^3} = - {\text{i}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{i}}^4} = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">thus i is a generator for <em>G</em> and hence <em>G</em> is a cyclic group <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">–i is the other generator for <em>G</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for the group <em>H</em> there is no generator as all the elements are self inverses <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) since one group is cyclic and the other group is not, they are not isomorphic <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [17 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were aware of the group axioms and the properties of a group, but they were not always explained clearly. A number of candidates did not understand the term “Abelian”. Many candidates understood the conditions for a group to be cyclic. Many candidates did not realise that the answer to part (e) was actually found in part (d), hence the reason for this part only being worth 1 mark. Overall, a number of fully correct solutions to this question were seen. </span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relations <em>R</em> and <em>S</em> are defined on quadratic polynomials <em>P</em> of the form</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[P(z) = {z^2} + az + b{\text{ , where }}a{\text{ , }}b \in \mathbb{R}{\text{ , }}z \in \mathbb{C}{\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) The relation <em>R</em> is defined by \({P_1}R{P_2}\) if and only if the sum of the two zeros of \({P_1}\) is equal to the sum of the two zeros of \({P_2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that <em>R</em> is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine the equivalence class containing \({z^2} - 4z + 5\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The relation <em>S</em> is defined by \({P_1}S{P_2}\) if and only if \({P_1}\) and \({P_2}\) have at least one zero in common. Determine whether or not <em>S</em> is transitive.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) <em>R</em> is reflexive, <em>i.e.</em> <em>PRP</em> because the sum of the zeroes of <em>P</em> is equal to the sum of the zeros of <em>P</em> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>R</em> is symmetric, <em>i.e.</em> \({P_1}R{P_2} \Rightarrow {P_2}R{P_1}\) because the sums of the zeros of \({P_1}\) and \({P_2}\) are equal implies that the sums of the zeros of \({P_2}\) and \({P_1}\) are equal <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">suppose that \({P_1}R{P_2}\) and \({P_2}R{P_3}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it follows that \({P_1}R{P_3}\) so <em>R</em> is transitive, because the sum of the zeros of \({P_1}\) is equal to the sum of the zeros of \({P_2}\) which in turn is equal to the sum of the zeros of \({P_3}\) , which implies that the sum of the zeros of \({P_1}\) is equal to the sum of the zeros of \({P_3}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the three requirements for an equivalence relation are therefore satisfied <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the zeros of \({z^2} - 4z + 5\) are \(2 \pm {\text{i}}\) , for which the sum is 4 <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{z}}^2} + az + b\) has zeros of \(\frac{{ - a \pm \sqrt {{a^2} - 4b} }}{2}\) , so the sum is –<em>a</em> <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept use of the result (although not in the syllabus) that the sum of roots is minus the coefficient of <em>z</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence – <em>a</em> = 4 and so <em>a</em> = – 4 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the equivalence class is \({z^2} - 4z + k{\text{ , }}(k \in \mathbb{R})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) for example, \((z - 1)(z - 2)S(z - 1)(z - 3)\) and</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((z - 1)(z - 3)S(z - 3)(z - 4)\) but \((z - 1)(z - 2)S(z - 3)(z - 4)\) is not true <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so <em>S</em> is not transitive <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [12 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to show, in (a), that <em>R</em> is an equivalence relation although few were able to identify the required equivalence class. In (b), the explanation that <em>S</em> is not transitive was often unconvincing.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R </em>is defined on ordered pairs by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(a,{\text{ }}b)R(c,{\text{ }}d){\text{ if and only if }}ad = bc{\text{ where }}a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in {\mathbb{R}^ + }.\]</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that <em>R </em>is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Describe, geometrically, the equivalence classes.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Reflexive: \((a,{\text{ }}b)R(a,{\text{ }}b)\) because \(ab = ba\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Symmetric: \((a,{\text{ }}b)R(c,{\text{ }}d) \Rightarrow ad = bc \Rightarrow cb = da \Rightarrow (c,{\text{ }}d)R(a,{\text{ }}b)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Transitive: \((a,{\text{ }}b)R(c,{\text{ }}d) \Rightarrow ad = bc\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\((c,{\text{ }}d)R(e,{\text{ }}f) \Rightarrow cf = de\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{ad}}{{de}} = \frac{{bc}}{{cf}}{\text{ so }}af = be\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">It follows that \((a,{\text{ }}b)R(e,{\text{ }}f)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \((a,{\text{ }}b)R(c,{\text{ }}d) \Rightarrow \frac{a}{b} = \frac{c}{d}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Equivalence classes are therefore points lying, in the first quadrant, on straight lines through the origin. <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;">Notes: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Accept a correct sketch.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">if “in the first quadrant” is omitted.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Do not penalise candidates who fail to exclude the origin.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [9 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was well answered by many candidates although some misunderstandings of the terminology were seen. Some candidates appeared to believe, incorrectly, that reflexivity was something to do with \((a,{\text{ }}a)R(a,{\text{ }}a)\) and some candidates confuse the terms ‘reflexive’ and ‘symmetric’. Many candidates were unable to describe the equivalence classes geometrically.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the set <em>S</em> = {1, 3, 5, 7, 9, 11, 13} under the binary operation multiplication modulo 14 denoted by \({ \times _{14}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Copy and complete the following Cayley table for this binary operation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Give one reason why \(\{ S,{\text{ }}{ \times _{14}}\} \) is not a group.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that a new set <em>G</em> can be formed by removing one of the elements of <em>S</em> such that \(\{ G,{\text{ }}{ \times _{14}}\} \) is a group.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the order of each element of \(\{ G,{\text{ }}{ \times _{14}}\} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the proper subgroups of \(\{ G,{\text{ }}{ \times _{14}}\} \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>A4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A3</em></strong> for one error, <strong><em>A2</em></strong> for two errors, <strong><em>A1</em></strong> for three errors, <strong><em>A0</em></strong> for four or more errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">any valid reason, for example <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">not a Latin square</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">7 has no inverse</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">delete 7 (so that <em>G</em> = {1, 3, 5, 9, 11, 13}) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">closure – evident from the table <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">associative because multiplication is associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the identity is 1 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">13 is self-inverse, 3 and 5 form an inverse</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">pair and 9 and 11 form an inverse pair <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the four conditions are satisfied so that \(\{ G,{\text{ }}{ \times _{14}}\} \) is a group <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>A4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A3</em></strong> for one error, <strong><em>A2</em></strong> for two errors, <strong><em>A1</em></strong> for three errors, <strong><em>A0</em></strong> for four or more errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{1}</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{1, 13}\(\,\,\,\,\,\){1, 9, 11} <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were no problems with parts (a), (b) and (d).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were no problems with parts (a), (b) and (d).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were no problems with parts (a), (b) and (d) but in part (c) candidates often failed to state that the set was associative under the operation because multiplication is associative. Likewise they often failed to list the inverses of each element simply stating that the identity was present in each row and column of the Cayley table.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates did not answer part (d) correctly and often simply listed all subsets of order 2 and 3 as subgroups.<br></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operator multiplication modulo 14, denoted by \( * \), is defined on the set <em>S</em> = {2, 4, 6, 8, 10, 12}.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Copy and complete the following operation table.</span><br><img src="data:image/png;base64,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" alt></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that {<em>S</em> , \( * \)} is a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the order of each element of {<em>S</em> , \( * \)}.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence show that {<em>S</em> , \( * \)} is cyclic and find all the generators.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The set <em>T</em> is defined by \(\{ x * x:x \in S\} \). Show that {<em>T</em> , \( * \)} is a subgroup of {<em>S</em> , \( * \)}.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img 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" alt> <strong><em>A4 <br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A4</em></strong> for all correct, <strong><em>A3</em></strong> for one error, <strong><em>A2</em></strong> for two errors, <strong><em>A1</em></strong> for three errors and <strong><em>A0</em></strong> for four or more errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) closure: there are no new elements in the table <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">identity: 8 is the identity element <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">inverse: every element has an inverse because there is an 8 in every row and column <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">associativity: (modulo) multiplication is associative <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore {<em>S </em>, \( * \)} is a group <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the orders of the elements are as follows</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>A4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A4</em></strong> for all correct, <strong><em>A3</em></strong> for one error, <strong><em>A2</em></strong> for two errors, <strong><em>A1</em></strong> for three errors and <strong><em>A0</em></strong> for four or more errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) </span><strong style="font-family: 'times new roman', times; font-size: medium;">EITHER</strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the group is cyclic because there are elements of order 6 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the group is cyclic because there are generators <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">10 and 12 are the generators <strong><em>A1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[11 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">looking at the Cayley table, we see that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>T</em> = {2, 4, 8} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">this is a subgroup because it contains the identity element 8, no new elements are formed and 2 and 4 form an inverse pair <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for any two conditions</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well done in general. Some candidates, however, when considering closure and associativity simply wrote ‘closed’ and ‘associativity’ without justification. Here, candidates were expected to make reference to their Cayley table to justify closure and to state that multiplication is associative to justify associativity. In (c), some candidates tried to show the required result without actually identifying the elements of <em>T</em>. This approach was invariably unsuccessful.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well done in general. Some candidates, however, when considering closure and associativity simply wrote ‘closed’ and ‘associativity’ without justification. Here, candidates were expected to make reference to their Cayley table to justify closure and to state that multiplication is associative to justify associativity. In (c), some candidates tried to show the required result without actually identifying the elements of <em>T</em>. This approach was invariably unsuccessful.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well done in general. Some candidates, however, when considering closure and associativity simply wrote ‘closed’ and ‘associativity’ without justification. Here, candidates were expected to make reference to their Cayley table to justify closure and to state that multiplication is associative to justify associativity. In (c), some candidates tried to show the required result without actually identifying the elements of <em>T</em>. This approach was invariably unsuccessful.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The group \(\{ G,{\text{ }} * \} \) <span class="s1">is defined on the set \(G\) </span>with binary operation \( * \)<span class="s1">. \(H\) is a subset of \(G\) defined by \(H = \{ x:{\text{ }}x \in G,{\text{ }}a * x * {a^{ - 1}} = x{\text{ for all }}a \in G\} \)</span>. Prove that \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">associativity: This follows from associativity in \(\{ G,{\text{ }} * \} \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>R1</em></strong></span></p>
<p class="p2">the identity \(e \in H\) since \(a * e * {a^{ - 1}} = a * {a^{ - 1}} = e\) (for all \(a \in G\)) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Condone the use of the commutativity of <em>e </em>if that is involved in an alternative simplification of the LHS.</p>
<p class="p2">closure: Let \(x,{\text{ }}y \in H\) so that \(a * x * {a^{ - 1}} = x\) and \(a * y * {a^{ - 1}} = y\) for all \(a \in G\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2">multiplying, \(x * y = a * x * {a^{ - 1}} * a * y * {a^{ - 1}}\) (for all \(a \in G\)) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = a * x * y * {a^{ - 1}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">therefore \(x * y \in H\) (proving closure) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2">inverse: Let \(x \in H\) so that \(a * x * {a^{ - 1}} = x\) (for all \(a \in G\))</p>
<p class="p2"><span class="Apple-converted-space">\({x^{ - 1}} = {(a * x * {a^{ - 1}})^{ - 1}}\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = a * {x^{ - 1}} * {a^{ - 1}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">therefore \({x^{ - 1}} \in H\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">hence \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accuracy marks cannot be awarded if commutativity is assumed for general elements of \(G\).</p>
<p class="p2"><strong><em>[9 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">This is an abstract question, clearly defined on a subset. Far too many candidates almost immediately deduced, erroneously, that the full group was Abelian. Almost no marks were then available.</p>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">The following Cayley table for the binary operation multiplication modulo 9, denoted by \( * \)</span>, is defined on the set \(S = \{ 1,{\text{ }}2,{\text{ }}4,{\text{ }}5,{\text{ }}7,{\text{ }}8\} \).</p>
<p class="p1"><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Copy and complete the table.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Show that \(\{ S,{\text{ }} * \} \) </span>is an Abelian group.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the two proper subgroups of \(\{ S,{\text{ }} * \} \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the coset of each of these subgroups with respect to the element 5.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve the equation \(2 * x * 4 * x * 4 = 2\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-02-06_om_17.09.58.png" alt="M16/5/MATHL/HP3/ENG/TZ0/SG/M/01.a"> <strong><em>A3</em></strong></p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A3 </em></strong>for correct table, <strong><em>A2 </em></strong>for one or two errors, <strong><em>A1 </em></strong>for three or four errors and <strong><em>A0 </em></strong>otherwise.</p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">the table contains only elements of \(S\), showing closure <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2">the identity is 1 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s2">every element has an inverse since 1 </span>appears in every row and column, or a complete list of elements and their correct inverses <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">multiplication of numbers is associative <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2">the four axioms are satisfied therefore \(\{ S,{\text{ }} * \} \) <span class="s1">is a group</span></p>
<p class="p1">the group is Abelian because the table is symmetric (about the leading diagonal) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-02-06_om_17.17.38.png" alt="M16/5/MATHL/HP3/ENG/TZ0/SG/M/01.c"> <strong><em>A3</em></strong></p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A3 </em></strong>for all correct values, <strong><em>A2 </em></strong>for 5 correct, <strong><em>A1 </em></strong>for 4 correct and <strong><em>A0 </em></strong>otherwise.</p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>the subgroups are \(\{ 1,{\text{ }}8\} \); \(\{ 1,{\text{ }}4,{\text{ }}7\} \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1A1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the cosets are \(\{ 4,{\text{ }}5\} \); \(\{ 2,{\text{ }}5,{\text{ }}8\} \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1A1</em></strong></span></p>
<p class="p2"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">use of algebraic manipulations <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">and at least one result from the table, used correctly <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\(x = 2\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(x = 7\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">testing at least one value in the equation <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">obtain \(x = 2\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">obtain \(x = 7\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">explicit rejection of all other values <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The majority of candidates were able to complete the Cayley table correctly.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Generally well done. However, it is not good enough for a candidate to say something along the lines of 'the operation is closed or that inverses exist by looking at the Cayley table'. A few candidates thought they only had to prove commutativity.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Often well done. A few candidates stated extra, and therefore incorrect subgroups.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">The majority found only one solution, usually the obvious \(x = 2\), but sometimes only the less obvious \(x = 7\).</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The binary operation multiplication modulo 10, denoted by ×<sub>10</sub>, is defined on the set <em>T</em> = {2 , 4 , 6 , 8} and represented in the following Cayley table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that {<em>T</em>, ×<sub>10</sub>} is a group. (You may assume associativity.)</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By making reference to the Cayley table, explain why<em> T</em> is Abelian.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the order of each element of {<em>T</em>, ×<sub>10</sub>}.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that {<em>T</em>, ×<sub>10</sub>} is cyclic and write down all its generators.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The binary operation multiplication modulo 10, denoted by ×<sub>10</sub> , is defined on the set <em>V</em> = {1, 3 ,5 ,7 ,9}.</p>
<p>Show that {<em>V</em>, ×<sub>10</sub>} is not a group.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>closure: there are no new elements in the table <em><strong>A1</strong></em></p>
<p>identity: 6 is the identity element <em><strong>A1</strong></em></p>
<p>inverse: every element has an inverse because there is a 6 in every row and column (2<sup>−1</sup> = 8, 4<sup>−1</sup> = 4, 6<sup>−1</sup> = 6, 8<sup>−1</sup> = 2) <em><strong>A1</strong></em></p>
<p>we are given that (modulo) multiplication is associative <em><strong>R1</strong></em></p>
<p>so {<em>T</em>, ×<sub>10</sub>} is a group <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the Cayley table is symmetric (about the main diagonal) <em><strong>R1</strong></em></p>
<p>so<em> T</em> is Abelian <em><strong> AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>considering powers of elements <em><strong>(M1)</strong></em></p>
<p><img src="data:image/png;base64,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"> <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A2</strong> </em>for all correct and <em><strong>A1</strong> </em>for one error.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>{<em>T</em>, ×<sub>10</sub>} is cyclic because there is an element of order 4 <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Accept “there are elements of order 4”.</p>
<p><strong>OR</strong></p>
<p>{<em>T</em>, ×<sub>10</sub>} is cyclic because there is generator <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Accept “because there are generators”.</p>
<p><strong>THEN</strong></p>
<p>2 and 8 are generators <em><strong>A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>considering singular elements <em><strong>(M1)</strong></em></p>
<p>5 has no inverse (5 ×<sub>10</sub> a = 1, a∈<em>V</em> has no solution) <em><strong>R1</strong></em></p>
<p><strong>OR</strong></p>
<p>considering Cayley table for {<em>V</em>, ×<sub>10</sub>}</p>
<p><img 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NOv7cLRH0I48JIcvQRbqyuKtAIMW59RLLSvelsg0j/DObIS2/MFf5hKjbAK5cbngdbP4eTubQOQrv8BI0+XIF+YB2AehPytMG4HWm2XuHGoktuK2oalMJavgiBrbupyvHRgL9a3H8DhTo4cqbcHJ44swrERnz8WhDz9ntnKkK3e2hzF0D3YLk12GT775AjeeHPzg90oIa+WMOq8goeODcAnl/3k9ybsphIKKOUvU82fQ68PjXE8jhv9/wxd1V8hn5vQlVParVm2Gvp02XlMfKRb6O9ZgqotKzlZxlteuvs8OnKykBGyZLVGeAKrczwcOVIJo91taOj4CPv2N8LqcPPXjRmsA9z+/oTbGRtgKkwPiwEvuc/C0pOvaEs0Md5ZvG44mv4L9ve9hONV+YrF+1Wnfsp9vw401byDPvN7qMp9WJ3ZnPVceTHQ+930u2rSkLlCC7GnHzcU6vedngkF90hunLI0Q4SIzrqXsbFIj+d5HxtUEL86HjUPQvbSCDsov2B1IaOiSLHuK5mFyh2IhFFHLbQLdCgqfwtHv/wCX/bxOQISUFx54CwfC3RFKK87iy9tF9HHTSzwVGToHgU6zuNC3/TBYmFFJharXJtnxfhoHkOZzQPm88DZ9gFMeqCzrhIV7zupJTIrgBP0Jv7Qvul4YW1or03sZVF5ldNgYeGBwKCx8yhMaMZGfS2sg+OxJ6PKJ6Sh0OIMGI/m7UDdRo4GkOcje9NOmIST2L3/U1zzO85xiK522LqHkaPTRryRqbIAZy9TGgG5G3bAYv8K9pocdDa0oZvDscHZA5rYd4pXaF+VO5Bgoc6DkLsdbx15Ewbx/+JiL8+tELndKCDXuBdHGjdDbHeil5dWyEI99nSeRxXq8fiCOdAaD6Pr2yvo7lyp6MBhUCtV8atJR+HrZpjQAZtzSBVZokwoTSB+oX0TxIEECkST9Qy2Gh5SunRU+rz5yCpYB4NKcxebbGmQml2M6pbL/pknnpZKPIXv0WvaiU3Z82PzyES4a6oWuhwddBmpiZBbyuNsE/B3Xz2C4rxFs33ne94voRwIvB70Xv8LPK1beE/Bkv8ECd6BPlxfnwddyKyk5Jc7KOE4RIcFFa2rcG6PnpOZaEHZI369HvTllvPtRCOQ8LQZr+4rmbFqHYg0aEW5thjmlm6I8uwaaRCOt9/HjdqdMIROZ+VCU8YxaK2Edo0ZLS4RfhyiHW9bhlC7ay3SVVuKsSicwD+QT9XvwLMtabAcfwW5HDlQSXThzBlXoE74FyfowqH9X2PtqwV8O9FYqFpC3DN+3Vd+PGziA8i9Air6jF1mjaXLmZwv+SuUHmSnnR7mUyiL6uLhY2NXm1mpEGABLGelB08zp+dOTGmojcGIvYYJsj7oTazR3svGYip94ObqYsCYz9PBavRCoF4IpezgyS9Y71hsa4XaGPhLZsTOTJP1Qa4XAjM0Xo2ZfVAlAxmE7yprLHuXOWOsA/KjojEIi0hIAaWmXjkogAwF0ZG1gfSAGJAeBOxitLrAVefHlHugFBEgAkSACDwoAXIgD0qQricCRIAIcEqAHAinBU9iEwEiQAQelEDYGMiD3oyuJwJEgAgQgeQlEDlOPjdU1MiDocd4S0cbMCIGvBGgAWS5xKkuEIOgHkRaAOrCiiRC20SACBABIjAjAuRAZoSJTiICRIAIEIFIAuRAIonQNhEgAkSACMyIADmQGWGik4gAESACRCCSQAI4EDmIUp5/IE8ezEtJ0aK46Zp/PahIYbjYHnXArJU5BL9b0OSeHmApqVkQA4AYEAO5ksdZD1TvQKTB/41PWl0h9rCA39gPGMfg51a0iiE4DOsUjYEc8uQ4JYkBSA9ADOTqF/+6EDaNN04W4S6PHcY3J84g7dhNsMK0u5zHySFvD04cWYRjIz4ULlS9749NoRADgBgQA7l2qUAP1O1ARr/GPzQcRZ1uMXTD61Fg+Etkc7R0d7gFljDa3YaGjmbo9i/BcPFzMBRm8xXGFcQAxEBexJ7qgloYBBatjr5Ub/BYfH7/yHobN08u5y4vJQx9DTvdO6JIdqItXazIg3/uIfKyzYaJZbz9S9wLTG86HdOlvImBCusF6UFgCXOqC6qwB5NBQFRnLCYN6R3mcZ5jjSZDwJnoG+K29v1kluKZ8HmYs+0DZvLHhBCY/uDFmMXFUK1OEAM5MAjpATGIux4kgAMJWus7zGP/e6ZHLjPZbwZ3xuxXtcYzKLFvgNlrDAxCDbOPxCagEDFQYQskWP7BX9IDxohB3Bgk0EjsPAiFr2CPCWi1XcJo+AABf1uadBS+boYJHbA5h/iTX5aYGBAD0oNA3Y9TXVD3IPo0s5iKDJ0OOdByNng8DURgR6oWuhwdkJH6MydwsJsYAMSAGMhVPQ56kEAtEJmQFwN9OTBvykaCZTw2ltzrQV9uOTZlz4/N/RPhrsQAIAbEwG8elbcH6rXDkgjXmf8JlzgeMGOSiO5DDehcux16Dv8DIYkunDnjgigFcXTh0P6vsfbVAixMBEM/C3kkBgAxIAZyVVKNHgTH4lQ3YBocGPNPWV3OSg/+N2ZXaAqvzERtPHyeDlbjn3kFBqGUHTz5RUyn8BKDQM0gPaC6QHXh5+tCWERCCig19ZpMQXQoiI6sDaQHxID0IGAXo9UF9XZhTdlyShEBIkAEiIAKCZADUWGhUJaIABEgAolAgBxIIpQS5ZEIEAEioEICYWMgKswfZYkIEAEiQARUQiBynDzsj4SRB1WS57hkI9qAUVwyEseHEgMaQJbVj/SAGAT1INIcURdWJBHaJgJEgAgQgRkRIAcyI0x0EhEgAkSACEQSIAcSSYS2iQARIAJEYEYEVOxAJIw6aqFNSfH3wcr9sFPfLWhy/zgjAZPnpFtwmPNCGITwKG6Ce2KJk+SR9+ckkeB1d8J6qh7GrFo4RrkRPBzIqANmbYgOpHBWJ+SljlrMWOO3Czkw1tvg9nKmC/7lnYwTNrIYZus1eMO1JOZbKnYgQ3DaOiBGQ2BYh4IszhYQHL0EW6srCg0Bhq3PIEvFJRkl0/e9S3I3Y9vej3H6tV04+sN93ybBLxzH4OdWtIZWDq7qxDBcDdU4PFSC4z4GNnYCz12uhv63bRjkxocM45u2LmDbh/CwO/BcLMGNyhexz3FLUd1Wr9kZvYKvH3oLHllBWPDrw4h9D37DkcEMaIOEUecVPHRsAL5JFjKTm7CbSrC1IJOb1Yk12WX47JMjeOPNzYpWFFU9zNuDE0cW4diIb6pu2MqQrd7aPKv4JLcVtQ1LYSxfBUGWOXU5XjqwF+vbD+Bwp7IGdFYF+wU3k67/ASNPlyBfmAdgHoT8rTBuVz5WkmpVTrq9BH9tKgooSBCs5MYpy3X8DUcGMyD6T7idsQGmwvQwRyG5z8LSk89fayyoD1z+ShjtbkNDx0fYt78RVodb8W6L+GL/EX0XzqMjJwsZqVPmSyM8gdU5Hm6CzWmWrYY+XXYeEx/pFvp7lqBqy0pFV+eeKoFgRlTyqxGWYVmIgsjZkvp+D2vGRhTz1n0lv2FkL40IoiVXpC5kVBRx032lEtWMbzb8L1HNECGis+5lbCzS43mzlaP+fy8Ger+bXgaaNGSu0ELs6ccNbrqx/FYRXrcDTTXvoM/8HqpyH57OJoZ7VOtApsssG8xuPPnCXyI9gXI9XY5Z2iP144I1HS+sXRLWKpmlu9Nt1EpA8xjKbB4wnwfOtg9g0gOddZWoeN/JSUtEjkr6KNBxHhf6pk+kEVZkYjE39kGeWJOPBboilNedxZe2i+hTeCJB4qD2G8xHUJy3SK1VW9F8+VtjT+qRx2FwLUVBq/VhGgG5G3bAYv8K9pocdDa0oZuLGWnzkb1pJ0zCSeze/ymu+Q3mOERXO2zdw8jR8RTuOg2FFmfgZaJ5O1C3UfGJBAnjQMhghlqyidZY8VOK9neG5oDSKiGgSUfh62aY0AGbc0glmYpxNhbqsafzPKpQj8cXzIHWeBhd315Bd+dKriaUTFKWXyaMe3GkcTPEdid6FWyFJIgDIYM5qSxyglpjYTi430jVQpejgy4jlRMUGqRmF6O65bJ/FpqnpRJP4Xv0mnZiUzZn0/snS3w+sgrWwTC5rUwiMRwIGcwwbaDWWBgO2vB60JdbzqnxHIfosKCidRXO7dFz3CKX4B3ow/X1edBFTD6KZQVJCAdCBjNUBag1FkqDt7QkunDmjAvixEwjSezCof1fY+2rBZwZz8CKBKfqd+DZljRYjr+CXAUNZ3z1bhyD1kpo15jR4hIhq4Ik2vG2ZQi1u9YqOskoARyIbDB7oVd4fnN8FeQuT5dbY11PYEs+p5MJ/Et4/Cvoyk8C4lso+lUGipuu+SvRXagl0aEhXHzneWjnpCBFa0RD17/gN3uqUOj/Q1kSifmzogSXOJqDBRXtGF35d+hpeWXiD3U/e1GSHZiLXz3xb/Dr3jq8lKfFnJQc/O0nY9jYfBhljy1UVNawgFIUD2SKPcVAoBgIsjaQHhAD0oOAXYxWFxKgBTJl1ClFBIgAESAC6iFADkQ9ZUE5IQJEgAgkFAFyIAlVXJRZIkAEiIB6CISNgagnW5QTIkAEiAARUBuByHHyuaEZjDwYeoy3dLQBI2LAGwEaQJZLnOoCMQjqQaQFoC6sSCK0TQSIABEgAjMiQA5kRpjoJCJABIgAEYgkQA4kkghtEwEiQASIwIwIkAOZESY6iQgQASJABCIJqNyByOv8t8K8RusfyNMaD+Ef3aORMvCzLYlwtZixJiUFKSk5MNbbOIpEJxezHEAnz68L8sBu2Le4CW6eItH5l3QJZbAFTe7pAZaStnJIIroPGaH160ExzNZrnATUCi3RwHpg1lP1MGbVwhGHeDAqdiASvK7f4aXDo9h4vB+MjcDx3BUY9bWwDo6HUuQkPQxXQzUOD5XguI+BjZ3Ac5erFQ8gE1fYo5dga3VFyYIAw9ZnOArtO47Bz61oFUNQGNahgJtQz8P4pq0L2PYhPOwOPBdLcKPyRexz3AoBkvxJyd2MbXs/xunXduHoD3GSl018AHlpfRV9fFdZo+EZZrLfnMqUr5+dLlvOBJOdjUztjUlKbTx8vY3MINQw+4hvUl7fwGlWJuSGM5o8+uAJdTHwsRH7u2yPfYBNEZBlvMnspgrW2PvHBxc4yh3UxWAig2MX2UHDnjBdiJL1WdulNga+vgvsi4E7IfLJOpAbU7ugNgZTwv+R9TZuZoiwDVPHZy8VjYFqWyDyEu6fdqSHB8nR/GssX/04xNbP4YxDcy1OPh6AvCLxeXTkZCEjZMlqjfAEVud40Gq7hOTv2PsJtzM2wFSYHhYDXnKfhaUnn6O3bwmj3W1o6PgI+/Y3wupwc9d1o1m2Gvr0eVPVUbqF/p4lqKIVu6eYKJRSqQMJBEe5PA3CPCzOzIIg9qH/Bk/dWF4M9H43jQY0achcoYXY048bSd//Pw9C9lKEx9yTHWsXMiqK+Om+ktw4ZWmGCBGddS9jY5Eez5utnI2FBauCPAbgQFPNO+gzv4eq3IeDB+hXIQIqdSAapGZkIQcX8OmF/umxHoQsZC4OeQNRCFb8HpOKDN2jQMd5XOibPlAqrMjEYpWWZEyZ+SNVpuOFtUvCWiUxfWa8b655DGU2D5jPA2fbBzDpgc66SlS87+SsJSJPqMjHAl0RyuvO4kvbRfQpGAs83mqgluer1uxosktgNmnRsfsgmq8FOmgk0Yn/bnNBjOjKUQvM2OVjPrI37YRJOInd+z/FNX9FkWeotcPWPYwcnTbizTx2OVHTnbmOVKkRkLthByz2r2CvyUFnQxu6uerWTUOhxRlwpM3bgbqNfE0oUUlFVK0DAdJQuOc4Oqp+wu7HfzURfe1b/FP3dc5m3ExoykI99nSeRxXq8fiCOdAaD6Pr2yvo7lyJrQWZ/LyBT1YcCu3rR6FJR+HrZpjQAZtzaJIONwnZkRr34kjjZojtTvRSK0TRolexAwGQmo1fV7fAwxiYpwVVT81BT+8WmDdlc2gwNUjNLkZ1y2V5unYP4WAAAAIMSURBVBw8LZV4Ct+j17QTm7LnK6o0qniYv/vqERTncRraN7QQUrXQ5ejCJ5yEHk/69HxkFayDIenlVJ+A6nYgIbwk8R+xu+Icfn3OhMKFCZPtEAlmMzkO0WFBResqnNujh7JRkGdTjvu/F9fdV5HYvB705Zbz+SLhZxGYdHN9fR50IbMUIzHR9uwTUL8l9rrhOFWPv332GP7M8l85n2kR+OfpqfodeLYlDZbjryCXywrDb/eVJLpw5owL4sSsO0nswqH9X2PtqwWcvEiMY9BaCe0aM1pcon+CjSTa8bZlCLW71iJd/RZt9q14PO8Y/JtJtD+JBI/F5XfEzkwCGGBgpkY76x0L//tYrPOkLh7yn+hqmAAw6E2s0d7LxmINIBAcRoGn3Mcj5D+Zlr3LnArohLr0gDGfp4PV6AUm5wtCKTt48ouY1w11MfCxsavNrNRvG2T7sJyVHjzNnJ7QPxbeh07d4xJ1MZjI7KSNlDnIX4EZGq9G/NH2HoL9gsPRGIRFJKSAUlOunILoUBAdWRtID4gB6UHALkarC9Tgm/IZlCICRIAIEIFfQIAcyC+ARacSASJABIjAFAFyIFMsKEUEiAARIAK/gEDYGMgvuI5OJQJEgAgQAc4IRI6Tzw2VP/Jg6DHe0tEGjIgBbwRoAFkucaoLxCCoB5EWYLIFEnmAtokAESACRIAI3I3A/we2Y8SGgYXsqwAAAABJRU5ErkJggg=="> <em><strong>M1</strong></em></p>
<p>the Cayley table is not a Latin square (or equivalent) <em><strong>R1</strong></em></p>
<p><strong>OR</strong></p>
<p>considering cancellation law</p>
<p><em>eg</em>, 5 ×<sub>10</sub><sub> </sub>9 = 5 ×<sub>10</sub> 1 = 5 <em><strong>M1</strong></em></p>
<p>if {<em>V</em>, ×<sub>10</sub>} is a group the cancellation law gives 9 = 1 <em><strong>R1</strong></em></p>
<p><strong>OR</strong></p>
<p>considering order of subgroups</p>
<p><em>eg</em>, {1, 9} is a subgroup <em><strong>M1</strong></em></p>
<p>it is not possible to have a subgroup of order 2 for a group of order 5 (Lagrange’s theorem) <em><strong>R1</strong></em></p>
<p><strong>THEN</strong></p>
<p>so {<em>V</em>, ×<sub>10</sub>} is not a group <em><strong> AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the sets <em>A</em> = {1, 3, 5, 7, 9} , <em>B</em> = {2, 3, 5, 7, 11} and <em>C</em> = {1, 3, 7, 15, 31} .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\left( {A \cup B} \right) \cap \left( {A \cup C} \right)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <em>A</em> \ <em>C</em> ≠ <em>C </em>\ <em>A</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <em>S</em> be a set containing \(n\) elements where \(n \in \mathbb{N}\).</p>
<p>Show that S has \({2^n}\) subsets.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>\(\left( {A \cup B} \right) \cap \left( {A \cup C} \right) = \left\{ {1,\,2,\,3,\,5,\,7,\,9,\,11} \right\} \cap \left\{ {1,\,3,\,5,\,7,\,9,\,15,\,31} \right\}\) <em><strong>M1A1</strong></em></p>
<p><strong>OR</strong></p>
<p>\(A \cup \left( {B \cap C} \right) = \left\{ {1,\,3,\,5,\,7,\,9,\,11} \right\} \cup \left\{ {3,\,7} \right\}\) <em><strong>M1A1</strong></em></p>
<p><strong>OR</strong></p>
<p>\({B \cap C}\) is contained within <em>A</em> <em><strong>(M1)A1</strong></em></p>
<p><strong>THEN</strong></p>
<p>= {1, 3, 5, 7, 9} (= <em>A</em>) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept a Venn diagram representation.</p>
<p><img 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"></p>
<p><em><strong> [3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>A</em> \ <em>C = </em>{5, 9} <em><strong>A1</strong></em></p>
<p><em>C </em>\ <em>A </em>= {15, 31} <em><strong>A1</strong></em></p>
<p>so <em>A</em> \ <em>C</em> ≠ <em>C </em>\ <em>A <strong>AG</strong></em></p>
<p><strong>Note:</strong> Accept a Venn diagram representation.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>if \(S = \emptyset \) then \(n = 0\) and the number of subsets of <em>S</em> is given by 2<sup>0</sup> = 1 <em><strong>A1</strong></em></p>
<p>if \(n > 0\)</p>
<p>for every subset of <em>S</em>, there are 2 possibilities for each element \(x \in S\) either \(x\) will be in the subset or it will not <em><strong>R1</strong></em></p>
<p>so for all \(n\) elements there are \(\left( {2 \times 2 \times \ldots \times 2} \right){2^n}\) different choices in forming a subset of <em>S</em> <em><strong>R1</strong></em></p>
<p>so <em>S</em> has \({2^n}\) subsets <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> If candidates attempt induction, award <em><strong>A1</strong> </em>for case \(n = 0\), <em><strong>R1</strong> </em>for setting up the induction method (assume \(P\left( k \right)\) and consider \(P\left( {k + 1} \right)\) and <em><strong>R1</strong> </em>for showing how the \(P\left( k \right)\) true implies \(P\left( {k + 1} \right)\) true).</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>\(\sum\limits_{k = 0}^n {\left( \begin{gathered}<br> n \hfill \\<br> k \hfill \\ <br>\end{gathered} \right)} \) is the number of subsets of <em>S</em> (of all possible sizes from 0 to \(n\)) <em><strong>R1</strong></em></p>
<p>\({\left( {1 + 1} \right)^n} = \sum\limits_{k = 0}^n {\left( \begin{gathered}<br> n \hfill \\<br> k \hfill \\ <br>\end{gathered} \right)} \left( {{1^k}} \right)\left( {{1^{n - k}}} \right)\) <em><strong>M1</strong></em></p>
<p>\({2^n} = \sum\limits_{k = 0}^n {\left( \begin{gathered}<br> n \hfill \\<br> k \hfill \\ <br>\end{gathered} \right)} \) (= number of subsets of <em>S</em>) <em><strong>A1</strong></em></p>
<p>so <em>S</em> has \({2^n}\) subsets <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following Cayley table for the set <em>G</em> = {1, 3, 5, 7, 9, 11, 13, 15} under the operation \({ \times _{16}}\), where \({ \times _{16}}\) denotes multiplication modulo 16.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the values of <em>a</em>, <em>b</em>, <em>c</em>, <em>d</em>, <em>e</em>, <em>f</em>, <em>g</em>, <em>h</em>, <em>i</em> and <em>j</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that \({ \times _{16}}\) is associative, show that the set <em>G</em>, together with the operation \({ \times _{16}}\), forms a group.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The Cayley table for the set \(H = \{ e,{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }}{a_3},{\text{ }}{b_1},{\text{ }}{b_2},{\text{ }}{b_3},{\text{ }}{b_4}\} \) under the operation \( * \), is shown below.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Given that \( * \) is associative, show that <em>H</em> together with the operation \( * \) forms a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find two subgroups of order 4.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\{ G,{\text{ }}{ \times _{16}}\} \) and \(\{ H,{\text{ }} * \} \) are not isomorphic.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\{ H,{\text{ }} * \} \) is not cyclic.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(a = 9,{\text{ }}b = 1,{\text{ }}c = 13,{\text{ }}d = 5,{\text{ }}e = 15,{\text{ }}f = 11,{\text{ }}g = 15,{\text{ }}h = 1,{\text{ }}i = 15,{\text{ }}j = 15\) <strong><em>A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A2</em></strong> for one or two errors,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>A1</em></strong> for three or four errors,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>A0</em></strong> for five or more errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) since the Cayley table only contains elements of the set </span><em style="font-family: 'times new roman', times; font-size: medium;">G</em><span style="font-family: 'times new roman', times; font-size: medium;">, then it is closed </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">there is an identity element which is 1 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">{3, 11} and {5, 13} are inverse pairs and all other elements are self inverse <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence every element has an inverse <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A0R0</em></strong> if no justification given for every element having an inverse.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since the set is closed, has an identity element, every element has an inverse and it is associative, it is a group <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) since the Cayley table only contains elements of the set <em>H</em>, then it is closed <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">there is an identity element which is <em>e</em> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\{ {a_1},{\text{ }}{a_3}\} \) form an inverse pair and all other elements are self inverse <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence every element has an inverse <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A0R0</em></strong> if no justification given for every element having an inverse.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since the set is closed, has an identity element, every element has an inverse and it is associative, it is a group <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) any 2 of \(\{ e,{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }}{a_3}\} ,{\text{ }}\{ e,{\text{ }}{a_2},{\text{ }}{b_1},{\text{ }}{b_2}\} ,{\text{ }}\{ e,{\text{ }}{a_2},{\text{ }}{b_3},{\text{ }}{b_4}\} \) <strong><em>A2A2</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the groups are not isomorphic because \(\{ H,{\text{ }} * \} \) has one inverse pair whereas \(\{ G,{\text{ }}{ \times _{16}}\} \) has two inverse pairs <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any other valid reason:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>e.g.</em> the fact that \(\{ G,{\text{ }}{ \times _{16}}\} \) is commutative and \(\{ H,{\text{ }} * \} \) is not.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a group is not cyclic if it has no generators <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for the group to have a generator there must be an element in the group of order eight <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><img src="data:image/png;base64,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alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since there is no element of order eight in the group, it is not cyclic <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a group is not cyclic if it has no generators <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">only possibilities are \({a_1}\), \({a_3}\) since all other elements are self inverse <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">this is not possible since it is not possible to generate any of the “<em>b</em>” elements from the “<em>a</em>” elements – the elements \({a_1},{\text{ }}{a_2},{\text{ }}{a_3},{\text{ }}{a_4}\) form a closed set <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were aware of the group axioms and the properties of a group, but they were not always explained clearly. Surprisingly, a number of candidates tried to show the non-isomorphic nature of the two groups by stating that elements of different groups were not in the same position rather than considering general group properties. Many candidates understood the conditions for a group to be cyclic, but again explanations were sometimes incomplete. Overall, a good number of substantially correct solutions to this question were seen.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were aware of the group axioms and the properties of a group, but they were not always explained clearly. Surprisingly, a number of candidates tried to show the non-isomorphic nature of the two groups by stating that elements of different groups were not in the same position rather than considering general group properties. Many candidates understood the conditions for a group to be cyclic, but again explanations were sometimes incomplete. Overall, a good number of substantially correct solutions to this question were seen.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were aware of the group axioms and the properties of a group, but they were not always explained clearly. Surprisingly, a number of candidates tried to show the non-isomorphic nature of the two groups by stating that elements of different groups were not in the same position rather than considering general group properties. Many candidates understood the conditions for a group to be cyclic, but again explanations were sometimes incomplete. Overall, a good number of substantially correct solutions to this question were seen.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were aware of the group axioms and the properties of a group, but they were not always explained clearly. Surprisingly, a number of candidates tried to show the non-isomorphic nature of the two groups by stating that elements of different groups were not in the same position rather than considering general group properties. Many candidates understood the conditions for a group to be cyclic, but again explanations were sometimes incomplete. Overall, a good number of substantially correct solutions to this question were seen.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(g:\mathbb{Z} \to \mathbb{Z}\) is defined by \(g(n) = \left| n \right| - 1{\text{ for }}n \in \mathbb{Z}\) . Show that <em>g </em>is neither surjective nor injective.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The set <em>S </em>is finite. If the function \(f:S \to S\) is injective, show that <em>f </em>is surjective.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the set \({\mathbb{Z}^ + }\) as both domain and codomain, give an example of an injective function that is not surjective.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">non-S: for example –2 does not belong to the range of <em>g </em><strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">non-I: for example \(g(1) = g( - 1) = 0\) <strong><em>R1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note</strong>: Graphical arguments have to recognize that we are dealing with sets of integers and not all real numbers</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">as <em>f </em>is injective \(n\left( {f(S)} \right) = n(S)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em><img 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" alt> R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative explanations.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>f </em>is surjective <strong><em>AG<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">for example, \(h(n) = n + 1\) <strong><em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Only award the <strong><em>A1 </em></strong>if the function works.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">I: \(n + 1 = m + 1 \Rightarrow n = m\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">non-S: 1 has no pre-image as \(0 \notin {\mathbb{Z}^ + }\) <strong><em>R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Nearly all candidates were aware of the conditions for an injection and a surjection in part (a). However, many missed the fact that the function in question was mapping from the set of integers to the set of integers. This led some to lose marks by applying graphical tests that were relevant for functions on the real numbers but not appropriate in this case. However, many candidates were able to give two integer counter examples to prove that the function was neither injective or surjective. In part (b) candidates seemed to lack the communication skills to adequately demonstrate what they intuitively understood to be true. It was usually not stated that the number of elements in the sets of the image and pre – image was equal. Part (c) was well done by many candidates although a significant minority used functions that mapped the positive integers to non – integer values and thus not appropriate for the conditions required of the function.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Nearly all candidates were aware of the conditions for an injection and a surjection in part (a). However, many missed the fact that the function in question was mapping from the set of integers to the set of integers. This led some to lose marks by applying graphical tests that were relevant for functions on the real numbers but not appropriate in this case. However, many candidates were able to give two integer counter examples to prove that the function was neither injective or surjective. In part (b) candidates seemed to lack the communication skills to adequately demonstrate what they intuitively understood to be true. It was usually not stated that the number of elements in the sets of the image and pre – image was equal. Part (c) was well done by many candidates although a significant minority used functions that mapped the positive integers to non – integer values and thus not appropriate for the conditions required of the function.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Nearly all candidates were aware of the conditions for an injection and a surjection in part (a). However, many missed the fact that the function in question was mapping from the set of integers to the set of integers. This led some to lose marks by applying graphical tests that were relevant for functions on the real numbers but not appropriate in this case. However, many candidates were able to give two integer counter examples to prove that the function was neither injective or surjective. In part (b) candidates seemed to lack the communication skills to adequately demonstrate what they intuitively understood to be true. It was usually not stated that the number of elements in the sets of the image and pre – image was equal. Part (c) was well done by many candidates although a significant minority used functions that mapped the positive integers to non – integer values and thus not appropriate for the conditions required of the function.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the functions \(f:A \to B\) and \(g:B \to C\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that if both <em>f</em> and <em>g</em> are injective, then \(g \circ f\) is also injective.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that if both <em>f</em> and <em>g</em> are surjective, then \(g \circ f\) is also surjective.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show, using a single counter example, that both of the converses to the results in part (a) and part (b) are false.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>s</em> and <em>t</em> be in <em>A</em> and \(s \ne t\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since <em>f</em> is injective \(f(s) \ne f(t)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since <em>g</em> is injective \(g \circ f(s) \ne g \circ f(t)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \(g \circ f\) is injective <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>z</em> be an element of <em>C</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we must find <em>x</em> in <em>A</em> such that \(g \circ f(x) = z\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since <em>g</em> is surjective, there is an element <em>y</em> in <em>B</em> such that \(g(y) = z\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since <em>f</em> is surjective, there is an element <em>x</em> in <em>A</em> such that \(f(x) = y\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">thus \(g \circ f(x) = g(y) = z\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \(g \circ f\) is surjective <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">converses: if \(g \circ f\) is injective then <em>g</em> and <em>f</em> are injective</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">if \(g \circ f\) is surjective then <em>g</em> and <em>f</em> are surjective <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><img 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" alt><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> There will be many alternative counter-examples.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was found difficult by a large number of candidates and no fully correct solutions were seen. A number of students made thought-through attempts to show it was surjective, but found more difficulty in showing it was injective. Very few were able to find a single counter example to show that the converses of the earlier results were false. Candidates struggled with the abstract nature of the question.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was found difficult by a large number of candidates and no fully correct solutions were seen. A number of students made thought-through attempts to show it was surjective, but found more difficulty in showing it was injective. Very few were able to find a single counter example to show that the converses of the earlier results were false. Candidates struggled with the abstract nature of the question.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was found difficult by a large number of candidates and no fully correct solutions were seen. A number of students made thought-through attempts to show it was surjective, but found more difficulty in showing it was injective. Very few were able to find a single counter example to show that the converses of the earlier results were false. Candidates struggled with the abstract nature of the question.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f:\mathbb{R} \to \mathbb{R}\) is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {2x + 1}&{{\text{for }}x \leqslant 2} \\ <br> {{x^2} - 2x + 5}&{{\text{for }}x > 2.} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Sketch the graph of <em>f</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) By referring to your graph, show that <em>f</em> is a bijection.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({f^{ - 1}}(x)\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: times new roman,times;"><img 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" alt><span style="font-size: medium;"> <strong><em>A1A1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for each part of the piecewise function. Award <strong><em>A1A0</em></strong> if the two parts of the graph are of the correct shape but <em>f</em> is not continuous at <em>x</em> = 2. Do not penalise a discontinuity in the derivative at <em>x</em> = 2.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) demonstrating the need to show that <em>f</em> is both an injection and a surjection (seen anywhere) <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>f</em> is an injection by any valid reason <em>eg</em> horizontal line test, strictly increasing function <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the range of <em>f</em> is \(\mathbb{R}\) so that <em>f</em> is a surjection <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>f</em> is therefore a bijection <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">considering the linear section, put</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = 2x + 1\) or \(x = 2y + 1\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \frac{{y - 1}}{2}\) or \(y = \frac{{x - 1}}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \({f^{ - 1}}(x) = \frac{{x - 1}}{2},{\text{ }}x \leqslant 5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = {(x - 1)^2} + 4\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(x - 1)^2} = y - 4\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 1 \pm \sqrt {y - 4} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 1 + \sqrt {y - 4} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">taking the + sign to give the right hand half of the parabola <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \({f^{ - 1}}(x) = 1 + \sqrt {x - 4} ,{\text{ }}x > 5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">considering the quadratic section, put</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = {x^2} - 2x + 5\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({x^2} - 2x + 5 - y = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \frac{{2 \pm \sqrt {4 - 4(5 - y)} }}{2}{\text{ }}( = 1 \pm \sqrt {y - 4} )\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">taking the + sign to give the right hand half of the parabola <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \({f^{ - 1}}(x) = \frac{{2 + \sqrt {4 - 4(5 - x)} }}{2},{\text{ }}x > 5{\text{ }}({f^{ - 1}}(x) = 1 + \sqrt {x - 4} ,{\text{ }}x > 5)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for omission of \({f^{ - 1}}(x)\) or omission of the domain. Penalise the omission of the notation \({f^{ - 1}}(x)\) only once. The domain must be seen in both cases.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For the most part the piecewise function was correctly graphed. Even though the majority of candidates knew that it is required to establish that the function is an injection and a surjection in order to prove it is a bijection, many just quoted the definition of injection or surjection and did not relate their reason to the graph.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates found the inverse of the first part of the piecewise function but some struggled with the algebra of the second part. In finding the inverse of the quadratic part of the function some candidates omitted the plus or minus sign in front of the square root. Others who had it often forgot to eliminate the negative sign and so did not gain the reasoning mark. Most did not state the correct domain for either part of the inverse function.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine, using Venn diagrams, whether the following statements are true.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(A' \cup B' = (A \cup B)'\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \((A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B)\)</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove, without using a Venn diagram, that \(A\backslash B\) and \(B\backslash A\) are disjoint sets.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><img 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" alt><span class="Apple-style-span" style="font-family: times new roman,times; font-size: medium; line-height: 20px; display: inline; float: none;"><span class="Apple-style-span" style="line-height: 20px; display: inline; float: none;"> <strong><em>A1</em></strong></span></span><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since the shaded regions are different, \(A' \cup B' \ne (A \cup B)'\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \) not true</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img 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" alt></span><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px; display: inline; float: none;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px; display: inline; float: none;"><strong><em> <span style="font-family: times new roman,times; font-size: medium;">A1</span></em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;"><img 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" alt> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since the shaded regions are the same \((A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \) true</span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A\backslash B = A \cup B'\) and \(B\backslash A = B \cap A'\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">consider \(A \cap B' \cap B \cap A'\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">now \(A \cap B' \cap B \cap A' = \emptyset \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since this is the empty set, they are disjoint <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept alternative valid proofs.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was accessible to most candidates, but a number drew incorrect Venn diagrams. In some cases the clarity of the diagram made it difficult to follow what the candidate intended. Candidates found (b) harder, although the majority made a reasonable start to the proof. Once again a number of candidates were let down by poor explanation.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was accessible to most candidates, but a number drew incorrect Venn diagrams. In some cases the clarity of the diagram made it difficult to follow what the candidate intended. Candidates found (b) harder, although the majority made a reasonable start to the proof. Once again a number of candidates were let down by poor explanation.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br>