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</div><h2>SL Paper 1</h2><div class="specification">
<p>Let \(f\,{\text{:}}\,\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) be defined by \(f\left( {x,\,y} \right) = \left( {x + 3y,\,2x - y} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>A</em> is the interval \(\left\{ {x\,{\text{:}}\,0 \leqslant x \leqslant 3} \right\}\) and <em>B</em> is the interval \(\left\{ {y\,{\text{:}}\,0 \leqslant x \leqslant 4} \right\}\) then describe <em>A</em> × <em>B</em> in geometric form.</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>Show that the function \(f\) is a bijection.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the inverse function \({f^{ - 1}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The group \(\left\{ {G,\left. * \right\}} \right.\) is defined on the set \(G = \left\{ {1,2,3,4,5,\left. 6 \right\}} \right.\) where \( * \) denotes </span><span style="font-family: times new roman,times; font-size: medium;">multiplication modulo \(7\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Draw the Cayley table for \(\left\{ {G,\left. * \right\}} \right.\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Determine the order of each element of \(\left\{ {G,\left. * \right\}} \right.\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find all the proper subgroups of \(\left\{ {G,\left. * \right\}} \right.\) .</span></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><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \(x * 6 * x = 3\) where \(x \in G\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(G\) denote the set of \(2 \times 2\) matrices whose elements belong to \(\mathbb{R}\) and whose determinant is equal to 1. Let \( * \) denote matrix multiplication which may be assumed to be associative.</p>
</div>
<div class="specification">
<p>Let \(H\) denote the set of \(2 \times 2\) matrices whose elements belong to \(\mathbb{Z}\) and whose determinant is equal to 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\{ G,{\text{ }} * \} \) is a group.</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 whether or not \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The relations \({\rho _1}\) and \({\rho _2}\) are defined on the Cartesian plane as follows</p>
<p>\(({x_1},{\text{ }}{y_1}){\rho _1}({x_2},{\text{ }}{y_2}) \Leftrightarrow x_1^2 - x_2^2 = y_1^2 - y_2^2\)</p>
<p>\(({x_1},{\text{ }}{y_1}){\rho _2}({x_2},{\text{ }}{y_2}) \Leftrightarrow \sqrt {x_1^2 + x_2^2} \leqslant \sqrt {y_1^2 + y_2^2} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For \({\rho _1}\) and \({\rho _2}\) determine whether or not each is reflexive, symmetric and transitive.</p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For each of \({\rho _1}\) and \({\rho _2}\) which is an equivalence relation, describe the equivalence classes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The permutation \(P\) is given by</p>
<p>\[P = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 3&4&5&6&2&1 \end{array}} \right).\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the order of \(P\), justifying your answer.</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 \({P^2}\).</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>The permutation group \(G\) is generated by \(P\). Determine the element of \(G\) that is of order 2, giving your answer in cycle notation.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The set \(P\) contains all prime numbers less than 2500.</p>
<p class="p1">The set \(Q\) is the set of all subsets of \(P\).</p>
</div>
<div class="specification">
<p class="p1">The set \(S\) contains all positive integers less than <span class="s1">2500</span>.</p>
<p class="p1">The function \(f:{\text{ }}S \to Q\) is defined by \(f(s)\) as the set of primes exactly dividing \(s\), for \(s \in S\).</p>
<p class="p1">For example \(f(4) = \{ 2\} ,{\text{ }}f(45) = \{ 3,{\text{ }}5\} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why only one of the following statements is true</p>
<p class="p2">(i) <span class="Apple-converted-space"> \(17 \subset P\)</span><span class="s1">;</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> \(\{ 7,{\text{ }}17,{\text{ }}37,{\text{ }}47,{\text{ }}57\} \in Q\)</span><span class="s1">;</span></p>
<p class="p2">(iii) <span class="Apple-converted-space"> \(\phi \subset Q\)</span> and \(\phi \in Q\), where \(\phi \) is the empty set.</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"><span class="s1">(i) <span class="Apple-converted-space"> </span>State the value of \(f(1)\), </span>giving a reason for your answer.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Find \(n\left( {f(2310)} \right)\).</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">Determine whether or not \(f\) <span class="s1">is</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>injective;</p>
<p class="p3">(ii) <span class="Apple-converted-space"> </span>surjective.</p>
<div class="marks">[4]</div>
<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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The group \(\{ G,{\text{ }} * \} \) has a subgroup \(\{ H,{\text{ }} * \} \). The relation \(R\) is defined, for \(x,{\text{ }}y \in G\), by \(xRy\) if and only if \({x^{ - 1}} * y \in H\).</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) 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;">(b) Given that \(G = \{ 0,{\text{ }} \pm 1,{\text{ }} \pm 2,{\text{ }} \ldots \} \), \(H = \{ 0,{\text{ }} \pm 4,{\text{ }} \pm 8,{\text{ }} \ldots \} \) and \( * \) denotes addition, find the equivalence class containing the number \(3\).</span></p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(G\) is a group. The elements \(a,b \in G\) , satisfy \({a^3} = {b^2} = e\) and \(ba = {a^2}b\) , where \(e\) is the </span><span style="font-family: times new roman,times; font-size: medium;">identity element of \(G\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \({(ba)^2} = e\) .</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><span style="font-family: times new roman,times; font-size: medium;">Express \({(bab)^{ - 1}}\) in its simplest form.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(a \ne e\) ,</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) show that \(b \ne e\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) show that \(G\) is not Abelian.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The set \({{\rm{S}}_1} = \left\{ {2,4,6,8} \right\}\) and \({ \times _{10}}\) denotes multiplication modulo \(10\).</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Write down the Cayley table for \(\left\{ {{{\rm{S}}_1},{ \times _{10}}} \right\}\) .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Show that \(\left\{ {{{\rm{S}}_1},{ \times _{10}}} \right\}\) is a group.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) Show that this group is cyclic.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Now consider the group \(\left\{ {{{\rm{S}}_1},{ \times _{20}}} \right\}\) where \({{\rm{S}}_2} = \left\{ {1,9,11,19} \right\}\) and \({{ \times _{20}}}\) denotes </span><span style="font-family: times new roman,times; font-size: medium;">multiplication modulo \(20\). Giving a reason, state whether or not \(\left\{ {{{\rm{S}}_1},{ \times _{10}}} \right\}\) and </span><span style="font-family: times new roman,times; font-size: medium;">\(\left\{ {{{\rm{S}}_1},{ \times _{20}}} \right\}\) are isomorphic.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em><strong>S</strong></em> be the set of matrices given by</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\left[ \begin{array}{l}<br>a\\<br>c<br>\end{array} \right.\left. \begin{array}{l}<br>b\\<br>d<br>\end{array} \right]\) ; \(a,b,c,d \in \mathbb{R}\), \(ad - bc = 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relation \(R\) is defined on \(S\) as follows. Given \(\boldsymbol{A}\) , </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\boldsymbol{B} \in S\)</span> , \(\boldsymbol{ARB}\) if and only if there </span><span style="font-size: medium;"><span style="font-family: times new roman,times;">exists </span></span><span style="font-size: medium;"><span style="font-family: times new roman,times;"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\boldsymbol{X} \in S\)</span></span> </span></span><span style="font-size: medium;"><span style="font-family: times new roman,times;">such that </span></span><span style="font-family: times new roman,times; font-size: medium;">\(\boldsymbol{A} = \boldsymbol{BX}\)</span><span style="font-size: medium;"><span style="font-family: times new roman,times;"> .<br></span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(R\) 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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relationship between \(a\) , \(b\) , \(c\) and \(d\) is changed to \(ad - bc = n\) . State, with </span><span style="font-family: times new roman,times; font-size: medium;">a reason, whether or not there are any non-zero values of \(n\) , other than \(1\), </span><span style="font-family: times new roman,times; font-size: medium;">for which \(R\) is an equivalence relation.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the set \(S = \{ 0,{\text{ }}1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5\} \) <span class="s1">under the operation of addition modulo \(6\)</span>, denoted by \({ + _6}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Construct the Cayley table for \(\{ S,{\text{ }}{ + _6}\} \).</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">Show that \(\{ S,{\text{ }}{ + _6}\} \) forms 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">State the order of each 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 class="p1">Explain whether or not the group is cyclic.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Prove that the function \(f:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}\) defined by \(f(x,{\text{ }}y) = (2x + y,{\text{ }}x + y)\) is a bijection.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that the set \(S\) of numbers of the form \({2^m} \times {3^n}\) , where \(m,n \in \mathbb{Z}\) , forms a </span><span style="font-family: times new roman,times; font-size: medium;">group \(\left\{ {S, \times } \right\}\) under multiplication.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(\left\{ {S, \times } \right\}\) is isomorphic to the group of complex numbers \(m + n{\rm{i}}\) under </span><span style="font-family: times new roman,times; font-size: medium;">addition, where \(m\), \(n \in \mathbb{Z}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relation \(R\) is defined on the set </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\mathbb{Z}\)</span> by \(aRb\) if and only if \(4a + b = 5n\) , </span><span style="font-family: times new roman,times; font-size: medium;">where \(a,b,n \in \mathbb{Z}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(R\) 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><span style="font-family: times new roman,times; font-size: medium;">State the equivalence classes of \(R\) .</span></p>
<div class="marks">[3]</div>
<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 function \(f:{\mathbb{R}^ + } \times {\mathbb{R}^ + } \to {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) is defined by \(f(x,{\text{ }}y) = \left( {xy,{\text{ }}\frac{x}{y}} \right)\).</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;">Prove that \(f\) is a bijection.</span></p>
</div>
<br><hr><br><div class="specification">
<p>The transformations <em>T</em><sub>1</sub>, <em>T</em><sub>2</sub>, <em>T</em><sub>3</sub>, <em>T</em><sub>4</sub>, in the plane are defined as follows:</p>
<p><em>T</em><sub>1</sub> : A rotation of 360° about the origin<br><em>T</em><sub>2</sub> : An anticlockwise rotation of 270° about the origin<br><em>T</em><sub>3</sub> : A rotation of 180° about the origin<br><em>T</em><sub>4</sub> : An anticlockwise rotation of 90° about the origin.</p>
</div>
<div class="specification">
<p>The transformation <em>T</em><sub>5</sub> is defined as a reflection in the \(x\)-axis.</p>
</div>
<div class="specification">
<p>The transformation <em>T</em> is defined as the composition of <em>T</em><sub>3</sub> followed by <em>T</em><sub>5</sub> followed by <em>T</em><sub>4</sub>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Copy and complete the following Cayley table for the transformations of <em>T</em><sub>1</sub>, <em>T</em><sub>2</sub>, <em>T</em><sub>3</sub>, <em>T</em><sub>4</sub>, under the operation of composition of transformations.</p>
<p><img src="data:image/png;base64,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"></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><em>Show that T</em><sub>1</sub>, <em>T</em><sub>2</sub>, <em>T</em><sub>3</sub>, <em>T</em><sub>4 </sub>under the operation of composition of transformations form a group. Associativity may be assumed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that this group is cyclic.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the 2 × 2 matrices representing <em>T</em><sub>3</sub>, <em>T</em><sub>4</sub> and <em>T</em><sub>5</sub>.</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>Find the 2 × 2 matrix representing <em>T</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give a geometric description of the transformation <em>T</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
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<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;">\(\{ G,{\text{ }} * \} \) is a group of order \(N\) and \(\{ H,{\text{ }} * \} \) is a proper subgroup of \(\{ G,{\text{ }} * \} \) of order \(n\).</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) Define the right coset of \(\{ H,{\text{ }} * \} \) containing the element \(a \in G\).</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) Show that each right coset of \(\{ H,{\text{ }} * \} \) contains \(n\) elements.</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) Show that the union of the right cosets of \(\{ H,{\text{ }} * \} \) is equal to \(G\).</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;">(d) Show that any two right cosets of \(\{ H,{\text{ }} * \} \) are either equal or disjoint.</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) Give a reason why the above results can be used to prove that \(N\) is a multiple of \(n\).</span></p>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The set \(S\) contains the eight matrices of the form\[\left( {\begin{array}{*{20}{c}}<br>a&0&0\\<br>0&b&0\\<br>0&0&c<br>\end{array}} \right)\]where \(a\), \(b\), \(c\) can each take one of the values \( + 1\) or \( - 1\) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that any matrix of this form is its own inverse.</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><span style="font-family: times new roman,times; font-size: medium;">Show that \(S\) forms an Abelian group under matrix multiplication.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Giving a reason, state whether or not this group is cyclic.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Prove that the number \(14 641\) is the fourth power of an integer in any base greater </span><span style="font-family: times new roman,times; font-size: medium;">than \(6\).</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For \(a,b \in \mathbb{Z}\)</span><span style="font-family: times new roman,times; font-size: medium;"> the relation \(aRb\) is defined if and only if \(\frac{a}{b} = {2^k}\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(k \in \mathbb{Z}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Prove that \(R\) is an equivalence relation.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) List the equivalence classes of \(R\) on the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The group \(\left\{ {G, + } \right\}\) is defined by the operation of addition on the set \(G = \left\{ {2n|n \in \mathbb{Z}} \right\}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The group \(\left\{ {H, + } \right\}\) is defined by the operation of addition on the set \(H = \left\{ {4n|n \in \mathbb{Z}} \right\}\) </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Prove that \(\left\{ {G, + } \right\}\) and \(\left\{ {H, + } \right\}\) are isomorphic.</span></p>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the Euclidean algorithm to find \(\gcd (162,{\text{ }}5982)\).</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"><span class="s1">The relation \(R\) </span>is defined on \({\mathbb{Z}^ + }\) by \(nRm\) if and only if \(\gcd (n,{\text{ }}m) = 2\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>By finding counterexamples show that \(R\) is neither reflexive nor transitive.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the set of solutions of \(nR6\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A sample of size 100 is taken from a normal population with unknown mean <em>μ</em> and known variance 36.</p>
</div>
<div class="specification">
<p>Another investigator decides to use the same data to test the hypotheses <em>H</em><sub>0</sub> : <em>μ</em> = 65 , <em>H<span style="font-size: 11.6667px;">1</span></em> : <em>μ</em> = 67.9.</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>An investigator wishes to test the hypotheses <em>H</em><sub>0</sub> : <em>μ</em> = 65, <em>H</em><sub>1</sub> : <em>μ</em> > 65.</p>
<p>He decides on the following acceptance criteria:</p>
<p>Accept <em>H</em><sub>0</sub> if the sample mean \(\bar x\) ≤ 66.5</p>
<p>Accept <em>H</em><sub>1</sub> if \(\bar x\) > 66.5</p>
<p>Find the probability of a Type I error.</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>She decides to use the same acceptance criteria as the previous investigator. Find the probability of a Type II error.</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>Find the critical value for \({\bar x}\) if she wants the probabilities of a Type I error and a Type II error to be equal.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br>