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</div><h2>SL Paper 1</h2><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \({2^n} \equiv {( - 1)^n}(\bmod 3)\)<span class="s1">, where \(n \in \mathbb{N}\).</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 class="p1"><span class="s1">Hence show that an integer is divisible by 3 if and only if the difference between the </span>sum of its binary (base 2) digits in even-numbered positions and the sum of its binary digits in odd-numbered positions is divisible by 3.</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">Express the hexadecimal (base 16) number \({\text{ABB}}{{\text{A}}_{{\text{16}}}}\) <span class="s1">in binary.</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 class="p1"><strong>METHOD 1</strong></p>
<p class="p2"><span class="Apple-converted-space">\({2^n} = {(3 - 1)^n}\) </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = {3^n} + n{3^{n - 1}}( - 1) + \frac{{n(n - 1)}}{2}{3^{n - 2}}{( - 1)^2} + {\text{ }} \ldots {\text{ }} + {( - 1)^n}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">since all terms apart from the last one are divisible by 3 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>R1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\({2^n} \equiv {( - 1)^n}(\bmod 3)\) </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p2">attempt to reduce the powers of \(2(\bmod 3)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\({2^0} = 1(\bmod 3);{\text{ }}{2^1} = - 1(\bmod 3);{\text{ }}{2^2} = 1(\bmod 3);{\text{ }}{2^3} = - 1(\bmod 3){\text{ }} \ldots \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s2">since \(1(\bmod 3) \times 2 = - 1(\bmod 3)\) </span>and \( - 1(\bmod 3) \times 2 = 1(\bmod 3)\) the result can be generalized <strong><em>R1</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\(2n \equiv {( - 1)^n}(\bmod 3)\) </span><span class="s1"><strong><em>AG</em></strong></span></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 binary number \(N = {({a_n}{a_{n - 1}} \ldots {a_2}{a_1}{a_0})_2}\) has numerical value</p>
<p class="p1"><span class="Apple-converted-space">\({a_0} \times 1 + {a_1} \times 2 + {a_2} \times {2^2} + \ldots + {a_n} \times {2^n}\) </span><strong><em>A1</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\(N = \left( {{a_0} - {a_1} + {a_2} - \ldots {{( - 1)}^n}{a_n}} \right)(\bmod 3)\) </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p1">hence divisibility of \(N\) <span class="s2">by 3 </span>coincides with statement of question <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"><span class="Apple-converted-space">\({\text{ABB}}{{\text{A}}_{16}} = 10 \times {16^3} + 11 \times {16^2} + 11 \times 16 + 10 \times 1\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="s2">\(N = {(1010)_2} \times {2^{12}} + {(1011)_2} \times {2^8} + {(1011)_2} \times {2^4} + {(1010)_2} \times {2^0}\) <span class="Apple-converted-space"> </span></span><strong><em>(M1)(A1)</em></strong></p>
<p class="p3"> </p>
<p class="p2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1 </em></strong><span class="s2">for expressing A and B </span>in binary.</p>
<p class="p3"> </p>
<p class="p2"><span class="Apple-converted-space">\(N = {(1010101110111010)_2}\) </span><strong><em>A1</em></strong></p>
<p class="p2"><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;">
<p class="p1">Many candidates were able to make a beginning to this question and attempted a solution to part (a). Some were let down by being unable to fully explain their reasoning.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b) a number of fully correct answers were seen but some candidates appeared to be completely unaware of what constituted a sensible approach.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Again in part (c) a number of correct answers were seen but a significant number also appeared to have little idea on how to start.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the recurrence relation \({H_{n + 1}} = 2{H_n} + 1,{\text{ }}n \in {\mathbb{Z}^ + }\) where \({H_1} = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({H_2}\), \({H_3}\) and \({H_4}\).</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">Conjecture a formula for \({H_n}\) <span class="s1">in terms of \(n\), for \(n \in {\mathbb{Z}^ + }\).</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">Prove by mathematical induction that your formula is valid for all \(n \in {\mathbb{Z}^ + }\).</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 class="p1"><span class="Apple-converted-space">\({H_2} = 2{H_1} + 1\) </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\( = 3;{\text{ }}{H_3} = 7;{\text{ }}{H_4} = 15\) </span><span class="s1"><strong><em>A1</em></strong></span></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"><span class="Apple-converted-space">\({H_n} = {2^n} - 1\) </span><strong><em>A1</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">let \({\text{P}}(n)\) be the proposition that \({H_n} = {2^n} - 1\) for \(n \in {\mathbb{Z}^ + }\)</p>
<p class="p1">from (a) \({H_1} = 1 = {2^1} - 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so \({\text{P}}(1)\) is true</p>
<p class="p1">assume \({\text{P}}(k)\) is true for some \(k \Rightarrow {H_k} = {2^k} - 1\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">then \({H_{k + 1}} = 2{H_k} + 1\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\( = 2 \times ({2^k} - 1) + 1\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">\( = {2^{k + 1}} - 1\)</p>
<p class="p1"><span class="s2">\({\text{P}}(1)\) </span>is true and \({\text{P}}(k)\) is true \( \Rightarrow {\text{P}}(k + 1)\) is true, hence \({\text{P}}(n)\) is true for all \(n \in {\mathbb{Z}^ + }\) by the principle of mathematical induction <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p3"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Only award the <strong><em>R1 </em></strong>if all earlier marks have been awarded.</p>
<p class="p3"> </p>
<p class="p1"><strong><em>[5 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 was a successful question for many candidates with many wholly correct answers seen. The vast majority of candidates were able to answer parts (a) and (b) and most knew how to start part (c). A few candidates were let down by not realising the need for a degree of formality in the presentation of the inductive proof.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was a successful question for many candidates with many wholly correct answers seen. The vast majority of candidates were able to answer parts (a) and (b) and most knew how to start part (c). A few candidates were let down by not realising the need for a degree of formality in the presentation of the inductive proof.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was a successful question for many candidates with many wholly correct answers seen. The vast majority of candidates were able to answer parts (a) and (b) and most knew how to start part (c). A few candidates were let down by not realising the need for a degree of formality in the presentation of the inductive proof.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the Diophantine equation \(7x - 5y = 1,{\text{ }}x,{\text{ }}y \in \mathbb{Z}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the general solution to this equation.</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">Hence find the solution with minimum positive value of \(xy\).</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">Find the solution satisfying \(xy = 2014\).</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">one solution is \(x = - 2,{\text{ }}y = - 3{\text{ }}\left( {{\text{or }}(3,{\text{ }}4)} \right)\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">the general solution is</p>
<p class="p1"><span class="Apple-converted-space">\(x = - 2 + 5N,{\text{ }}y = - 3 + 7N{\text{ }}({\text{or }}x = 3 + 5M,{\text{ }}y = 4 + 7M)\) </span><strong><em>M1A1</em></strong></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">a listing of small values of the product <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="s1">\( \Rightarrow x = - 2,{\text{ }}y = - 3\) </span>(the least positive value of \(xy\) <span class="s2">is 6</span>) <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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">use of “table” or otherwise to solve</p>
<p class="p1"><span class="Apple-converted-space">\(35{N^2} - 29N + 6 = 2014{\text{ }}({\text{or }}35{M^2} + 41M + 12 = 2014)\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2">obtain \(N = 8{\text{ }}({\text{or }}M = 7)\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(x = 38,{\text{ }}y = 53\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><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 class="p1">This was also a very successful question with many wholly correct answers seen. A small number of candidates made arithmetic errors in the calculations. Some candidates used unnecessarily long and complex methods for parts (b) and (c) which would have potentially left them short of time elsewhere.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was also a very successful question with many wholly correct answers seen. A small number of candidates made arithmetic errors in the calculations. Some candidates used unnecessarily long and complex methods for parts (b) and (c) which would have potentially left them short of time elsewhere.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was also a very successful question with many wholly correct answers seen. A small number of candidates made arithmetic errors in the calculations. Some candidates used unnecessarily long and complex methods for parts (b) and (c) which would have potentially left them short of time elsewhere.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A number written in base 5 is 4303. Find this as a number written in base 10.</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>1000 is a number written in base 10. Find this as a number written in base 7.</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>4303<sub>5</sub> = 4 × 5<sup>3</sup> + 3 × 5<sup>2</sup> + 0 × 5<sup>1</sup> + 3 × 5<sup>0</sup> <em><strong> (M1)</strong></em><br>= 500 + 75 + 3<br>= 578 <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><strong>METHOD 1</strong></p>
<p>\(1000 = {a_0} + 7{a_1} + 49{a_2} + 343{a_3}\) <em><strong>(M1)</strong></em></p>
<p>(since \(343{a_3} < 1000\))\( \Rightarrow {a_3} = 2\) <em><strong>(A1)</strong></em></p>
<p>\(1000 = {a_0} + 7{a_1} + 49{a_2} + 686\)</p>
<p>\({a_0} + 7{a_1} + 49{a_2} = 314 \Rightarrow {a_2} = 6\) <em><strong>(A1)</strong></em></p>
<p>\({a_0} + 7{a_1} = 20 \Rightarrow {a_1} = 2,\,\,{a_0} = 6\) <em><strong>(A1)</strong></em></p>
<p>\( \Rightarrow {1000_{10}} = {2626_7}\) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>1000 = 7 × 142 + 6 <em><strong>(M1)</strong></em></p>
<p>142 = 7 × 20 + 2 <em><strong>(A1)</strong></em></p>
<p>20 = 7 × 2 + 6 <em><strong>(A1)</strong></em></p>
<p>2 = 7 × 0 + 2 <em><strong>(A1)</strong></em></p>
<p>⇒ (1000)<sub>10</sub> = 2626<sub>7</sub> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<p> </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 simple connected planar graph \(J\) has the following adjacency table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-17_om_13.09.29.png" alt="M17/5/FURMA/HP1/ENG/TZ0/11"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Without attempting to draw \(J\), verify that <em>J </em>satisfies the handshaking lemma;</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>Without attempting to draw \(J\), determine the number of faces in \(J\).</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>The vertices D and G are joined by a single edge to form the graph \(K\). Show that \(K\) is not planar.</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>Explain why a graph containing a cycle of length three cannot be bipartite. </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 by finding a cycle of length three, show that the complement of \(K\) is not bipartite.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>the sum of degrees of the vertices is even (36) or the sum of degrees of the vertices is twice the number of edges <strong><em>A1</em></strong></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>the number of edges \((e)\) is 18 <strong><em>A1</em></strong></p>
<p>using Euler’s relation \(v - e + f = 2\) <strong><em>M1</em></strong></p>
<p>\(f = 2 + 18 - 8 = 12\) <strong><em>A1</em></strong></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>if \(K\) is planar then \(e \leqslant 3v - 6\) <strong><em>M1</em></strong></p>
<p>\(v = 8\) and \(e = 19\) <strong><em>A1</em></strong></p>
<p>the inequality is not satisfied so \(K\) is not planar <strong><em>A1AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let PQRP be a cycle of length 3 in a graph <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept a diagram instead of this statement.</p>
<p> </p>
<p>suppose the graph is bipartite</p>
<p>then P must belong to one of the two disjoint sets of vertices and Q, R must belong to the other disjoint set <strong><em>R1</em></strong></p>
<p>but Q, R cannot belong to the same set because they are linked <strong><em>R1</em></strong></p>
<p>therefore the graph cannot be bipartite <strong><em>AG</em></strong></p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>for example, a suitable cycle of order 3 is AFHA <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for a valid attempt at drawing the complement or constructing its adjacency table.</p>
<p> </p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">c.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.</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>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the Euclidean algorithm to find the greatest common divisor of 74 and 383.</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>Hence find integers <em>s</em> and <em>t</em> such that 74<em>s</em> + 383<em>t</em> = 1.</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>383 = 5 × 74 + 13 <em><strong>M1</strong></em><br>74 = 5 × 13 + 9 <strong><em>A1</em></strong><br>13 = 1 × 9 + 4 <em><strong> (A1)</strong></em><br>9 = 2 × 4 + 1<br>4 = 4 × 1 + 0<br>⇒ gcd (74, 383) = 1 <strong> <em>A1</em></strong></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><strong>EITHER</strong><br>1 = 9 − 2 × 4 <em><strong>(M1)</strong></em><br>= 9 − 2(13 − 1 × 9) = 3 × 9 − 2 × 13 <em><strong>(A1)</strong></em><br>= 3(74 − 5 × 13) − 2 × 13 = 3 × 74 − 17 × 13 <em><strong>(A1)</strong></em><br>= 3 × 74 − 17 (383 − 5 × 74) = 88 × 74 − 17 × 383</p>
<p><strong>OR</strong><br>13 = 383 − 5 × 74 <em><strong>(M1)</strong></em><br>9 = 74 − 5 × 13<br> = 74 − 5(383 − 5 × 74)<br> = 26 × 74 − 5 × 383 <em><strong>(A1)</strong></em><br>4 = 13 − 9<br> = (383 − 5 × 74) − (26 × 74 − 5 × 383)<br> = 6 × 383 − 31 × 74 <em><strong>(A1)</strong></em><br>1 = 9 − 2 × 4<br> = (26 × 74 − 5 × 383) − 2(6 × 383 − 31 × 74)<br> = 88 × 74 − 17 × 383</p>
<p><strong>THEN</strong><br>⇒ <em>s</em> = 88 and <em>t</em> = − 17 <em><strong>A</strong><strong>1A1</strong></em></p>
<p><em><strong>[5 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.</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><span style="font-family: times new roman,times; font-size: medium;">Given that \(a \equiv b(\bmod p)\) , show that \({a^n} \equiv {b^n}(\bmod p)\) for all \(n \in {\mathbb{Z}^ + }\) .</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><span style="font-family: times new roman,times; font-size: medium;">Show that \({29^{13}} + {13^{29}}\) is exactly divisible by \(7\).</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><span style="font-family: times new roman,times; font-size: medium;">\(a \equiv b(\bmod p) \Rightarrow a = b + pN,N \in \mathbb{Z}\) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({a^n} = {(b + pN)^n} = {b^n} + n{b^{n - 1}}pN \ldots \) <strong><em> M1A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = {b^n} + pM\) where \(M \in \mathbb{Z}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">this shows that \({a^n} \equiv {b^n}(\bmod p)\) <strong><em> AG</em></strong></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(29 \equiv 1(\bmod 7) \Rightarrow {29^{13}} \equiv {1^{13}} \equiv 1(\bmod 7)\) <strong><em> M1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(13 \equiv - 1(\bmod 7) \Rightarrow {13^{29}} \equiv {( - 1)^{29}} \equiv - 1(\bmod 7)\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">therefore \({29^{13}} + {13^{29}} \equiv 1 + ( - 1) \equiv 0(\bmod 7)\) <strong><em>M1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">this shows that \({29^{13}} + {13^{29}}\) is exactly divisible by \(7\) <strong><em>AG</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></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" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the general solution to the Diophantine equation \(3x + 5y = 7\).</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">Find the values of \(x\) and \(y\) satisfying the equation for which \(x\) <span class="s1">has the smallest positive integer value greater than \(50\)</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>by any method including trial and error or the Euclidean algorithm, a specific solution is, for example, \(x = 4,{\text{ }}y = - 1\) <strong><em>(A1)(A1)</em></strong></p>
<p>\(3(4) + 5( - 1) = 7\;\;\;\)(equation i)</p>
<p>\(3x + 5y = 7\;\;\;\)(equation ii)</p>
<p>equation ii – equation i: \(3(x - 4) + 5(y + 1) = 0\)</p>
<p>\(\frac{{4 - x}}{5} = \frac{{y + 1}}{3} = N\) <strong><em>(M1)</em></strong></p>
<p>\(x = 4 - 5N\) <strong><em>A1</em></strong></p>
<p>\(y = 3N - 1\) <strong><em>A1</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>smallest positive integer \( > 50\) occurs when \(N = - 10\) <strong><em>(M1)</em></strong></p>
<p>\(x = 54,{\text{ }}y = - 31\) <strong><em>A1</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">This question was well answered in general although a minority appeared not to realise that Diophantine means a solution in integers. It was expected that candidates would find a particular solution by inspection but some took a little longer by going via the Euclidean Algorithm.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well answered in general although a minority appeared not to realise that Diophantine means a solution in integers. It was expected that candidates would find a particular solution by inspection but some took a little longer by going via the Euclidean Algorithm.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the linear congruence \(ax \equiv b(\bmod p)\) where \(a,{\text{ }}b \in {\mathbb{Z}^ + },{\text{ }}p\) is a prime and \(\gcd (a,{\text{ }}p) = 1\). Using Fermat’s little theorem, show that \(x \equiv {a^{p - 2}}b(\bmod p)\).</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><strong>Hence </strong>find the smallest value of \(x\) greater than 100 satisfying the linear congruence \(3x \equiv 13(\bmod 19)\).</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>multiplying both sides by \({a^{p - 2}}\), <strong><em>M1</em></strong></p>
<p>\({a^{p - 1}}x \equiv {a^{p - 2}}b(\bmod p)\) <strong><em>A1</em></strong></p>
<p>using \({a^{p - 1}} \equiv 1(\bmod p)\) <strong><em>R1</em></strong></p>
<p>therefore, \(x \equiv {a^{p - 2}}b(\bmod p)\) <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using the above result,</p>
<p>\(x \equiv {3^{17}} \times 13(\bmod 19){\text{ }}\left( { \equiv 16\,7882\,2119(\bmod 19)} \right)\) <strong><em>A1</em></strong></p>
<p>\( \equiv 17(\bmod 19)\) <strong><em>(M1)A1</em></strong></p>
<p>\(x = 112\) <strong><em>A1</em></strong></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="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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(5982 = 162 \times 36 + 150\) </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\(162 = 150 \times 1 + 12\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\(150 = 12 \times 12 + 6\)</p>
<p class="p1">\(12 = 6 \times 2 + 0 \Rightarrow \gcd \) is 6 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><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>for example, \(\gcd (4,{\text{ }}4) = 4\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\(4 \ne 2\) </span><span class="s1"><strong><em>R1</em></strong></span></p>
<p class="p2">so \(R\) is not reflexive <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1">for example</p>
<p class="p1">\(\gcd (4,{\text{ }}2) = 2\) and \(\gcd (2,{\text{ }}8) = 2\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p1">but \(\gcd (4,{\text{ }}8) = 4{\text{ }}( \ne 2)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>R1</em></strong></span></p>
<p class="p2">so \(R\) is not transitive <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span><strong>EITHER</strong></p>
<p class="p2">even numbers <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">not divisible by 6 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong>OR</strong></p>
<p class="p3"><span class="Apple-converted-space">\(\{ 2 + 6n:n \in \mathbb{N}\} {\text{ }} \cup \{ 4 + 6n:n \in \mathbb{N}\} \) </span><span class="s1"><strong><em>A1A1</em></strong></span></p>
<p class="p2"><strong>OR</strong></p>
<p class="p1"><span class="Apple-converted-space">\(2,{\text{ }}4,{\text{ }}8,{\text{ }}10,{\text{ }} \ldots \) </span><span class="s1"><strong><em>A2</em></strong></span></p>
<p class="p2"><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;">
<p class="p1">This was a successful question for many students with many wholly correct answers seen. Part (a) was successfully answered by most candidates and those candidates usually had a reasonable understanding of how to complete part (b). A number were not fully successful in knowing how to explain their results.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was a successful question for many students with many wholly correct answers seen. Part (a) was successfully answered by most candidates and those candidates usually had a reasonable understanding of how to complete part (b). A number were not fully successful in knowing how to explain their results.</p>
<div class="question_part_label">b.</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;">Express the number \(47502\) as a product of its prime factors.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The positive integers \(M\) , \(N\) are such that \(\gcd (M,N) = 63\) and </span><span style="font-family: times new roman,times; font-size: medium;">\(lcm(M,N) = 47502\) . Given that \(M\) is even and \(M < N\) , find the two possible </span><span style="font-family: times new roman,times; font-size: medium;">pairs of values for \(M\) , \(N\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/merry.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> (M1)</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">therefore \(47502 = 2 \times {3^2} \times 7 \times 13 \times 29\) <strong><em>A1</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">noting that \(MN = \gcd \times {\rm{lcm}} = 2 \times {3^4} \times {7^2} \times 13 \times 29\) <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">the possibilities are</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\((M,N) = (126,23751)\) <strong><em>A1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\((M,N) = (1638,1827)\) <strong><em>A1A1</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></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;">As expected, the factorization in (a) was successfully completed by most candidates. </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;">Part (b) caused problems for some candidates. The most common mistake was that only one pair of values for \(M\), \(N\) was given. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the recurrence relation \({u_n} = 4{u_{n - 1}} - 4{u_{n - 2}}\) given that \({u_0} = {u_1} = 1\).</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>Consider \({v_n}\) which satisfies the recurrence relation \(2{v_n} = 7{v_{n - 1}} - 3{v_{n - 2}}\) subject to the initial conditions \({v_0} = {v_1} = 1\).</p>
<p>Prove by using strong induction that \({v_n} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^n} + \frac{1}{5}{\left( 3 \right)^n}\) for \(n \in \mathbb{N}\).</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>auxiliary equation is \({m^2} - 4m + 4 = 0\) <em><strong>M1A1</strong></em></p>
<p>hence \(m\) has a repeated root of 2 <em><strong>(A1)</strong></em></p>
<p>solution is of the form \({u_n} = a{\left( 2 \right)^n} + bn{\left( 2 \right)^n}\) <em><strong>M1</strong></em></p>
<p>using the initial conditions <em><strong>M1</strong></em></p>
<p>\( \Rightarrow a = 1\) and \(b = - \frac{1}{2}\)</p>
<p>\( \Rightarrow {u_n} = {2^n} - \frac{n}{2}{\left( 2 \right)^n}\) <em><strong>A1</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>\({v_n} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^n} + \frac{1}{5}{\left( 3 \right)^n}\)</p>
<p>let \(n = 0\) \({v_0} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^0} + \frac{1}{5}{\left( 3 \right)^0} = \frac{4}{5} + \frac{1}{5} = 1\)</p>
<p>let \(n = 1\) \({v_1} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^1} + \frac{1}{5}{\left( 3 \right)^1} = \frac{2}{5} + \frac{3}{5} = 1\)</p>
<p>hence true for \(n = 0\) and \(n = 1\) <em><strong>M1A1</strong></em></p>
<p>assume that \({v_j} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^j} + \frac{1}{5}{\left( 3 \right)^j}\) is true for all \(j > k + 1\) <em><strong>M1</strong></em></p>
<p>hence \({v_k} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^k} + \frac{1}{5}{\left( 3 \right)^k}\) and \({v_{k - 1}} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^{k - 1}} + \frac{1}{5}{\left( 3 \right)^{k - 1}}\)</p>
<p>\({v_{k + 1}} = \frac{{7{v_k} - 3{v_{k - 1}}}}{2}\)</p>
<p>\({v_{k + 1}} = \frac{{7\left[ {\frac{4}{5}{{\left( {\frac{1}{2}} \right)}^k} + \frac{1}{5}{{\left( 3 \right)}^k}} \right] - 3\left[ {\frac{4}{5}{{\left( {\frac{1}{2}} \right)}^{k - 1}} + \frac{1}{5}{{\left( 3 \right)}^{k - 1}}} \right]}}{2}\) <em><strong>M1A1</strong></em></p>
<p>\({v_{k + 1}} = \frac{{7\left[ {\frac{8}{5}{{\left( {\frac{1}{2}} \right)}^{k + 1}} + \frac{1}{{15}}{{\left( 3 \right)}^{k + 1}}} \right] - 3\left[ {\frac{{16}}{5}{{\left( {\frac{1}{2}} \right)}^{k + 1}} + \frac{1}{{45}}{{\left( 3 \right)}^{k + 1}}} \right]}}{2}\) <em><strong>(A1)</strong></em></p>
<p>\({v_{k + 1}} = \frac{{\frac{{56}}{5}{{\left( {\frac{1}{2}} \right)}^{k + 1}} + \frac{7}{{15}}{{\left( 3 \right)}^{k + 1}} - \frac{{48}}{5}{{\left( {\frac{1}{2}} \right)}^{k + 1}} - \frac{1}{{15}}{{\left( 3 \right)}^{k + 1}}}}{2}\) <em><strong>(A1)</strong></em></p>
<p>\({v_{k + 1}} = \frac{{\frac{8}{5}{{\left( {\frac{1}{2}} \right)}^{k + 1}} + \frac{6}{{15}}{{\left( 3 \right)}^{k + 1}}}}{2}\) <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Only one of the above <em><strong>(A1)</strong></em> can be implied.</p>
<p>\({v_{k + 1}} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^{k + 1}} + \frac{1}{{15}}{\left( 3 \right)^{k + 1}}\)</p>
<p>since the basis step and the inductive step have been verified, the Principle of Mathematical Induction tells us that \({v_n} = \frac{4}{5}{\left( {\frac{1}{2}} \right)^n} + \frac{1}{5}{\left( 3 \right)^n}\) is<br>the general solution <em><strong>R1</strong></em></p>
<p><em><strong>[9 marks]</strong></em></p>
<p><strong>Note:</strong> Only award final <em><strong>R1</strong></em> if at least 5 previous marks have been awarded.</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><span style="font-family: times new roman,times; font-size: medium;">Prove that \(3k + 2\) and \(5k + 3\) , \(k \in \mathbb{Z}\) are relatively prime.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>EITHER</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(3k + 2\) and \(5k + 3\) , </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(k \in \mathbb{Z}\)</span> are relatively prime </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">if, for all \(k\) , there exist \(m,n \in \mathbb{Z}\) such that </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(m(3k + 2) + n(5k + 3) = 1\) <strong><em> R1M1A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow 3m + 5n = 0\) <strong><em>A1</em></strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2m + 3n = 1\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(m = 5\) , \(n = - 3\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">hence they are relatively prime <strong><em>AG</em> </strong></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">OR </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(5k + 3 = 1 \times (3k + 2) + 2k + 1\) <strong><em>M1A1 </em></strong><br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(3k + 2 = 1 \times (2k + 1) + k + 1\) <strong><em>A1</em></strong><br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2k + 1 = 1 \times (k + 1) + k\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k + 1 = 1 \times (k) + 1\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">GDC = 1 <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so \(5k + 3\) and \(3k + 2\) are relatively prime <strong><em>AG </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This was often done using Euclid's algorithm rather than writing \(m(3k + 2) + n(5k + 3) = 1\) for the relatively prime numbers. </span></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;">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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(14641\) (base \(a > 6\) ) \( = {a^4} + 4{a^3} + 6{a^2} + 4a + 1\) , <em><strong>M1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = {(a + 1)^4}\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">this is the fourth power of an integer <strong><em>AG</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(aRa\) since \(\frac{a}{a} = 1 = {2^0}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , hence \(R\) is reflexive <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(aRb \Rightarrow \frac{a}{b} = {2^k} \Rightarrow \frac{b}{a} = {2^{ - k}} \Rightarrow bRa\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">so R is symmetric <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(aRb\) and \(bRc \Rightarrow \frac{a}{b} = {2^m}\), \(m \in \mathbb{Z}\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \(bRc \Rightarrow \frac{b}{c} = {2^n}\) , \(n \in \mathbb{Z}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow \frac{a}{b} \times \frac{b}{c} = \frac{a}{c} = {2^{m + n}}\) , \(m + n \in \mathbb{Z}\) </span><span style="font-family: times new roman,times; font-size: medium;"> <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow aRc\) so transitive <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">hence \(R\) is an equivalence relation <em><strong>AG</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) equivalence classes are {1, 2, 4, 8} , {3, 6} , {5, 10} , {7} , {9} <strong><em>A3</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A2</strong></em> if one class missing, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> if two classes missing, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A0</strong></em> if three or more classes missing.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></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><span style="font-family: times new roman,times; font-size: medium;">This was not difficult but a surprising number of candidates were unable to do it. Care with notation and logic were lacking. </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 question was at first straightforward but some candidates mixed up the properties of an equivalence relation with those of a group. The idea of an equivalence class is still not clearly understood by many candidates so that some were missing. </span></p>
<div class="question_part_label">b.</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;">Using mathematical induction or otherwise, prove that the number \({(1020)_n}\) , </span><span style="font-family: times new roman,times; font-size: medium;">that is the number \(1020\) written with base \(n\) , is divisible by \(3\) for all values of \(n\) </span><span style="font-family: times new roman,times; font-size: medium;">greater than \(2\).</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;">Explain briefly why the case \(n = 2\) has to be excluded.</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({(1020)_n} = {n^3} + 2n\) <em><strong>(R1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">so we are required to prove that \({n^3} + 2n\) is divisible by \(3\) for \(n \ge 3\) <strong><em>(R1)</em></strong></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">when \(n = 3\) , \({n^3} + 2n = 33\) which is divisible by \(3\) so the result is true for \(n = 3\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">assume the result is true for \(n = k\) , i.e. \({k^3} + 2k\) is divisible by \(3\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">for \(n = k + 1\) ,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({(k + 1)^3} + 2(k + 1) = {k^3} + 3{k^2} + 3k + 1 + 2k + 2\) <em><strong>M1</strong></em></span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> \( = ({k^3} + 2k) + 3({k^2} + k + 1)\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">the second term is clearly divisible by \(3\) and the first term is divisible by \(3\) </span><span style="font-family: times new roman,times; font-size: medium;">by hypothesis <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">therefore true for \(n = k \Rightarrow \) true for \(n = k + 1\) and since shown true for \(n = 3\) ,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">the result is proved by induction <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award the final <em><strong>R1</strong></em> only if the two <em><strong>M1</strong></em> marks have been awarded.</span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">there are three cases to consider, let \(N\) be a positive integer</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">case 1: \(n = 3N\) , in this case</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n({n^2} + 2) = 3N(9{N^2} + 2)\) which is divisible by \(3\) <strong><em>M1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">case 2: \(n = 3N + 1\) , in this case,</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n({n^2} + 2) = (3N + 1)(9{N^2} + 6N + 3)\) which is divisible by \(3\) <strong><em>M1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">case 3: \(n = 3N + 2\) , in this case,</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n({n^2} + 2) = (3N + 2)(9{N^2} + 12N + 6)\) which is divisible by \(3\) <strong><em>M1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">this proves the required result for all \(n > 2\)</span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">numbers to base \(2\) do not use the digit \(2\) or equivalent <strong><em>R1</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></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;">A significant number of candidates struggled with this question, but a number of wholly correct answers were seen. The majority of candidates tried to use a method of proof by induction although a number got lost in trying to establish the result for \(k + 1\) . Again the presentation and explanation of some of the solutions was poor. Some excellent solutions were seen using other methods including one which noted simply that all positive integers \(n\) are either 0 or \( \pm 1\) modulo \(3\) so that \(n({n^2} + |2) \equiv 0\) modulo \(3\) for all \(n\). </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;">A significant number of candidates struggled with this question, but a number of wholly correct answers were seen. The majority of candidates tried to use a method of proof by induction although a number got lost in trying to establish the result for \(k + 1\) . Again the presentation and explanation of some of the solutions was poor. Some excellent solutions were seen using other methods including one which noted simply that all positive integers \(n\) are either 0 or \( \pm 1\) modulo \(3\) so that \(n({n^2} + |2) \equiv 0\) modulo \(3\) for all \(n\). </span></p>
<div class="question_part_label">b.</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;">Prove that if \({\rm{gcd}}(a,b) = 1\) and \({\rm{gcd}}(a,c) = 1\) , then \({\rm{gcd}}(a,bc) = 1\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) A simple graph has <strong><em>e</em></strong> edges and <strong><em>v</em></strong> vertices, where \(v > 2\) . Prove that if all </span><span style="font-family: times new roman,times; font-size: medium;">the vertices have degree at least <strong><em>k</em></strong> , then \(2e \ge kv\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong> prove that every planar graph has at least one vertex of degree less </span><span style="font-family: times new roman,times; font-size: medium;">than 6.</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 align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">since \({\rm{gcd}}(a,b) = 1\) and \({\rm{gcd}}(a,c) = 1\) then</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(ax + by = 1\) and \(ap + cq = 1\) for \(x\), \(y\) , \(p\), \(q \in \mathbb{Z}\)</span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>M1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">hence</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\((ax + by)(ap + cq) = 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a(xap + xcq + byp) + bc(yq) = 1\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">since \((xap + xcq + byp)\) and \((yq)\) are integers <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">then \({\rm{gcd}}(a,bc) = 1\) <em><strong> AG</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">if \({\rm{gcd}}(a,bc) \ne 1\) , some prime <strong><em>p</em></strong> divides <strong><em>a</em></strong> and <strong><em>bc</em></strong> <em><strong>M1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p\) divides \(b\) or \(c\) <em><strong> M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">either \({\rm{gcd}}(a,b)\) or \({\rm{gcd}}(a,c) \ne 1\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">contradiction \( \Rightarrow {\rm{gcd}}(a,bc) = 1\) <strong><em>R1</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) each edge contributes 2 to the degree sum <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">so \(2e = \) degree sum <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">so \(2e \ge kv\) <strong><em>AG</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong><em> </em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(k = 6\) so \(2e \ge 6v\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">for a planar graph with \(v\) vertices and <strong><em>e</em></strong> edges, \(e \le 3v - 6\) <strong><em> M1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">so \(2e \le 6v - 12\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">this is a contradiction so at least one vertex must have degree \( < 6\) <strong><em>R1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Alternative proofs are possible.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></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><span style="font-family: times new roman,times; font-size: medium;">A "verbal" argument rather than a symbolic approach was often taken, leading to some doubt about the validity of the "proofs". This discursive approach in trying to prove propositions often leads down blind alleys to confusion. </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;">This was reasonably well done although clear proofs were not abundant and the same comments mentioned in (a) apply. In (ii) the word "Hence" was sometimes ignored. Candidates should realize that such "signpost" words are there to help them. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="images/thor.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The above diagram shows the weighted graph \(G\).</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;">Determine whether or not \(G\) is bipartite.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the adjacency matrix for \(G\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the number of distinct walks of length \(4\) beginning and ending at A.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(G\) is bipartite <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">because it contains a triangle. <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>R1</strong></em> for a valid attempt at showing that the vertices cannot be </span><span style="font-family: times new roman,times; font-size: medium;">divided into two disjoint sets.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i)<br></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\boldsymbol{M}} = \left( {\begin{array}{*{20}{c}}<br> 0&1&0&0&0&1 \\ <br> 1&0&1&1&1&0 \\ <br> 0&1&0&1&0&0 \\ <br> 0&1&1&0&1&1 \\ <br> 0&1&0&1&0&1 \\ <br> 1&0&0&1&1&0 <br>\end{array}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) We require the (A, A) element of \({\boldsymbol{M}^4}\) which is \(13\). <strong><em>M1A2</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></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" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the Euclidean Algorithm to show that \(275\) and \(378\) are relatively prime.</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><span style="font-family: times new roman,times; font-size: medium;">Find the general solution to the diophantine equation \(275x + 378y = 1\) .</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(378 = 1 \times 275 + 103\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(275 = 2 \times 103 + 69\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(103 = 1 \times 69 + 34\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(69 = 2 \times 34 + 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">showing that gcd \( = 1\) , <em>i.e.</em> relatively prime. <strong><em>R1</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Working backwards,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(1 = 69 - 2 \times (103 - 69)\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = 3 \times 69 - 2 \times 103\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = 3 \times (275 - 2 \times 103) - 2 \times 103\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = 3 \times 275 - 8 \times 103\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = 3 \times 275 - 8 \times (378 - 275)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = 11 \times 275 - 8 \times 378\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A solution is \(x = 11\) , \(y = - 8\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The general solution is \(x = 11 + 378n\) , \(y = - 8 - 275n\) where \(n \in \mathbb{Z}\) <em><strong>M1A1 N6</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></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.</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>An integer \(N\) given in base \(r\), can be expressed in base \(s\) in the form</p>
<p>\(N = {a_0} + {a_1}s + {a_2}{s^2} + {a_3}{s^3} + \ldots \) where \({a_0},{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }} \ldots \in \{ 0,{\text{ }}1,{\text{ }}2,{\text{ }} \ldots ,{\text{ }}s - 1\} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">In base \(5\)<span class="s1"> </span>an integer is written \(1031\)<span class="s1">. </span>Express this integer in base \(10\).</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">Given that \(N = 365,{\text{ }}r = 10\) and \(s = 7\) find the values of \({a_0},{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }} \ldots \).</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">(i) <span class="Apple-converted-space"> </span>Given that \(N = 899,{\text{ }}r = 10\) and \(s = 12\) find the values of \({a_0},{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }} \ldots \)</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence write down the integer in base \(12\), which is equivalent to \(899\) in base \(10\)<span class="s1">.</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 class="p1">Show that \(121\)<span class="s1"> </span>is always a square number in any base greater than \(2\).</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">\(1031 = 1 \times {5^3} + 0 \times {5^2} + 3 \times 5 + 1\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( = 125 + 0 + 15 + 1\)</p>
<p class="p1">\( = 141\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(365 = 1 \times {7^3} + 0 \times {7^2} + 3 \times 7 + 1\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( \Rightarrow {a_0} = 1,{\text{ }}{a_1} = 3,{\text{ }}{a_2} = 0,{\text{ }}{a_3} = 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></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>\(899 = 6 \times {12^2} + 2 \times 12 + 11\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( \Rightarrow {a_0} = 11,{\text{ }}{a_1} = 2,{\text{ }}{a_2} = 6\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"> </p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({(899)_{10}} = {(62B)_{12}}\)</p>
<p class="p1">(where \(B\) represents the digit in base \(12\) given by \({a_0} = 11\)) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>Note: </strong>Accept any letter in place of \(B\) provided it is defined</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(121\) in base \(r\) is \(1 + 2r + {r^2}\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">\( = {(r + 1)^2}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">which is a square for all \(r\) <span class="Apple-converted-space"> </span><strong><em>AG</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">This question was very well answered in general, although some candidates failed to see that \(121 = {11^2}\) in all number bases greater than \(2\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general, although some candidates failed to see that \(121 = {11^2}\) in all number bases greater than 2.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general, although some candidates failed to see that \(121 = {11^2}\) in all number bases greater than 2.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general, although some candidates failed to see that \(121 = {11^2}\) in all number bases greater than 2.</p>
<div class="question_part_label">d.</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;">(i) Use the Euclidean algorithm to find gcd(\(6750\), \(144\)) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Express your answer in the form \(6750r + 144s\) where <strong><em>r</em></strong> , \(s \in \mathbb{Z}\) .</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;">Consider the base \(15\) number CBA, where A, B, C represent respectively the </span><span style="font-family: times new roman,times; font-size: medium;">digits ten, eleven, twelve.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Write this number in base \(10\).</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Hence express this number as a product of prime factors, writing the factors </span><span style="font-family: times new roman,times; font-size: medium;">in base 4.</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><span style="font-family: times new roman,times; font-size: medium;">(i) \(6750 = 46 \times 144 + 126\) <strong><em>M1A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(144 = 1 \times 126 + 18\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(126 = 7 \times 18\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\gcd(6750,144) = 18\) <strong><em>A1 N0</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(18 = 144 - 1 \times 126\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 144 - (6750 - 46 \times 144)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 47 \times 144 + ( - 1) \times 6750\) <em><strong>A1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></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) \(n = 10 + 11 \times 15 + 12 \times {15^2}\) <strong><em>(M1)(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 2875\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(2875 = {5^3} \times 23\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 11 \times 11 \times 11 \times 113\) in base \(4\) <em><strong> A1A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: <em><strong>A1</strong></em> for \(11 \times 11 \times 11\) , <em><strong>A1</strong></em> for \(113\) . </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></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><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to use the Euclidean algorithm to find a gcd and to express the answer in</span> <span style="font-family: times new roman,times; font-size: medium;">a different form. All but the weakest candidates were able to make a meaningful start to this question.<br></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;">In part (b) many correct answers were seen and a majority of students were able to use the work they had studied on number bases correctly. All but the weakest candidates were able to make a meaningful start to this question.</span></p>
<div class="question_part_label">b.</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;">(i) Sum the series \(\sum\limits_{r = 0}^\infty {{x^r}} \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong>, using sigma notation, deduce a series for</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (a) \(\frac{1}{{1 + {x^2}}}\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (b) \(\arctan x\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (c) \(\frac{\pi }{6}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Show that \(\sum\limits_{n = 1}^{100} {n! \equiv 3(\bmod 15)} \) .</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><span style="font-family: times new roman,times; font-size: medium;">(i) \(\sum\limits_{r = 0}^\infty {{x^r}} = 1 + x + {x^2} + {x^3} + {x^4} + \ldots = \frac{1}{{1 - x}}\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) (a) replacing <strong><em>x</em></strong> by \( - {x^2}\) gives <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{1}{{1 - ( - {x^2})}} = 1 + ( - {x^2}) + {( - {x^2})^2} + {( - {x^2})^3} + {( - {x^2})^4} + \ldots \) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{1}{{1 + {x^2}}} = 1 - {x^2} + {x^4} - {x^6} + {x^8} - \ldots \) <strong><em>(A1)</em> </strong></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = \sum\limits_{r = 0}^\infty {{{( - 1)}^r}{x^{2r}}} \) <strong><em>A1 N2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) \(\arctan x = \int {\frac{{{\rm{d}}x}}{{1 + {x^2}}}} = x - \frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} - \frac{{{x^7}}}{7} + \ldots + c\) <em><strong>M1A1</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 0 \Rightarrow c = 0\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\arctan x = \sum\limits_{r = 0}^\infty {{{( - 1)}^r}\frac{{{x^{2r + 1}}}}{{2r + 1}}} \) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) by taking \(x = \frac{1}{{\sqrt 3 }}\) <strong><em>M1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\arctan \frac{1}{{\sqrt 3 }} = \frac{\pi }{6} = \sum\limits_{r = 0}^\infty {\frac{{{{( - 1)}^r}{{\left( {\frac{1}{{\sqrt 3 }}} \right)}^{2r + 1}}}}{{2r + 1}}} \) <em><strong>A1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[11 marks] </span></strong></em></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;">\(\sum\limits_{n = 1}^{100} {n! = 1! + 2! + 3! + 4! + 5! + \ldots } \) <strong><em>M1</em> </strong> </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 1 + 2 + 6 + 24 + 120 + \ldots \)</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( \equiv 1 + 2 + 6 + 24 + 0 + 0 + 0 + \ldots (\bmod 15)\) <strong><em>M1A1 </em></strong> </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( \equiv 33(\bmod 15)\) <strong><em>A1 </em></strong></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( \equiv 3(\bmod 15)\) <strong><em>AG </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (b) many did not recognize the sum of a simple geometric series to infinity and got involved in some heavy Maclaurin work thus wasting time. The clear instruction "Hence" was ignored by many candidates so that the question became more difficult and time consuming than it should have been. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Part (c) proved not to be as difficult as expected. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The positive integer \(N\) is represented by \(4064\) in base \(b\) and \(2612\) in base \(b + 1\) .</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;">Determine the value of \(b\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the representation of \(N\)</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) in base \(10\);</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) in base \(12\).</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">the equation satisfied by \(b\) is</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(4{b^3} + 6b + 4 = 2{(b + 1)^3} + 6{(b + 1)^2} + (b + 1) + 2\) <strong><em>M1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(2{b^3} - 12{b^2} - 13b - 7 = 0\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(b = 7\) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(N = 4 \times {7^3} + 6 \times 7 + 4 = 1418\)</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">or \(N = 2 \times {8^3} + 6 \times {8^2} + 1 \times 8 + 2 = 1418\) <em><strong>(M1)A1</strong></em></span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(12\left| \!{\underline {\,<br> {1418} \,}} \right. \) <em><strong>(M1)</strong></em></span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(12\left| \!{\underline {\,<br> {118} \,}} \right. \) remainder \(2\) <em><strong>(A1)</strong></em></span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(12\left| \!{\underline {\,<br> 9 \,}} \right. \) remainder A (where A \( = 10\) ) <em><strong>(A1)</strong></em></span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(1418 = {(9{\rm{A}}2)_{12}}\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept alternative correct methods.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></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.</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;">Find the positive square root of the base 7 number \({(551662)_7}\), giving your answer as a base 7 number.</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;">converting to base 10</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;">\({(551662)_7} = 2 + 6 \times 7 + 6 \times {7^2} + 1 \times {7^3} + 5 \times {7^4} + 5 \times {7^5}\) <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;">\( = 96721\) <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;">\(\sqrt {96721} = 311\) <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;">converting back to base 7</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;">\(7)\underline {311} \) <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;">\()\underline {44} (3\)</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;">\()\underline 6 (2\) <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;">it follows that \(\sqrt {{{(551662)}_7}} = {(623)_7}\) <strong><em>A1</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: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept \(623\).</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></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>
<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) Show that the solution to the linear congruence \(ax \equiv b(\bmod p)\), where \(a,{\text{ }}x,{\text{ }}b,{\text{ }}p \in {\mathbb{Z}^ + },{\text{ }}p\) is prime and \(a\), \(p\) are relatively prime, is given by \(x \equiv {a^{p - 2}}b(\bmod p)\).</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) Consider the congruences</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;"> \(7x \equiv 13(\bmod 19)\)</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;"> \(2x \equiv 1(\bmod 7)\).</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) Use the result in (a) to solve the first congruence, giving your answer in the form \(x \equiv k(\bmod 19)\) where \(1 \leqslant k \leqslant 18\).</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) Find the set of integers which satisfy both congruences simultaneously.</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) using Fermat’s little theorem, <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;">\({a^{p - 1}} \equiv 1(\bmod p)\) <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;">multiplying both sides of the congruence by \({a^{p - 2}}\), <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;">\({a^{p - 1}}x \equiv {a^{p - 2}}b(\bmod p)\) <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;">\(x \equiv {a^{p - 2}}b(\bmod p)\) <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>[4 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;"><strong><em> </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) (i) the solution is</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;">\(x \equiv {7^{17}} \times 13(\bmod 19)\) <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;">consider</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;">\({7^3} = 343 \equiv 1(\bmod 19)\) <strong><em>(A1)</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: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Other powers are possible.</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: 21.0px Helvetica;"><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: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x \equiv {\left( {{7^3}} \right)^5} \times {7^2} \times 13(\bmod 19)\) <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;">\( \equiv {7^2} \times 13(\bmod 19)\) <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;">\( \equiv 10(\bmod 19)\) <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;">(ii) using any method, including trial and error, the solution to the second congruence is given by \(x \equiv 32(\bmod 7)\) (or equivalent) <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;">a simultaneous solution is \(x = 67\) (or equivalent, <em>eg</em> \(-66\)) <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;">the full solution is \(x = 67 + 133N\) (where \(N \in \mathbb{Z}\)) (or equivalent) <strong><em>A1</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: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not <strong><em>FT </em></strong>an incorrect answer from (i).</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: 21.0px 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="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \({n^2} + 2n + 3 \equiv N(\bmod 8)\) , where \(n \in {\mathbb{Z}^ + }\) and \(0 \le N \le 7\) , prove that \(N\) can </span><span style="font-family: times new roman,times; font-size: medium;">take one of only three possible values.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">consider the following</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/tiny.png" alt> </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1A1 </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">we see that the only possible values so far are \(2\), \(3\) and \(6\) <strong><em>R1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">also, the table suggests that these values repeat themselves but we have to prove this </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">let \(f(n) = {n^2} + 2n + 3\) , consider </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(n + 4) - f(n) = {(n + 4)^2} + 2(n + 4) + 3 - {n^2} - 2n - 3\) <strong><em>M1</em> </strong></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 8n + 24\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">since \(8n + 24\) is divisible by \(8\), <strong><em>M1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(n + 4) \equiv (n)(\bmod 8)\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">this confirms that the values do repeat every \(4\) values of \(n\) so that \(2\), \(3\) and \(6\) are the only values taken for all values of \(n\) <strong><em>R1 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[9 marks] </span></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solutions to this question were extremely disappointing in general. Many candidates spotted that there was a periodicity in the values of the polynomial as far as \(n = 4\) and even \(n = 8\) but then the majority simply assumed that this would continue without any need for proof. This is equivalent to noting that \({2^{2n + 1}} - 1\) is prime for \(n = 1\), \(2\) and \(3\) and simply assuming that this will be true for larger values of \(n\). </span></p>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A sequence \(\left\{ {{u_n}} \right\}\) satisfies the recurrence relation \({u_{n + 2}} = 2{u_{n + 1}} - 5{u_n}\) , \(n \in \mathbb{N}\) . Obtain an </span><span style="font-family: times new roman,times; font-size: medium;">expression for \({u_n}\) in terms of <strong><em>n</em></strong> given that \({u_0} = 0\) and \({u_1} = 1\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">the auxiliary equation is \({m^2} - 2m + 5 = 0\) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the roots are \(1 \pm 2{\rm{i}}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the general solution is \({u_n} = A{\left(1 + 2{\rm{i}}\right)^n} + B{\left(1 - 2{\rm{i}}\right)^n}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \({u_0} = 0\) and \({u_1} = 1\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(A + B = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(A(1 + 2{\rm{i}}) + B(1 - 2{\rm{i}}) = 1\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Only award above <em><strong>A1</strong></em> if both equations are correct.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">solving</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(A\left(1 + 2{\rm{i}} - 1 + 2{\rm{i}}\right) = 1\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(A = \frac{1}{{4{\rm{i}}}}\) or \( - \frac{{\rm{i}}}{4}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1</span></em></strong></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(B = - \frac{1}{{4{\rm{i}}}}\) or </span><span style="font-family: Times New Roman; font-size: medium;">\(\frac{{\rm{i}}}{4}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">therefore</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({u_n} = \frac{1}{{4{\rm{i}}}}{\left(1 + 2{\rm{i}}\right)^n} - \frac{1}{{4{\rm{i}}}}{\left(1 - 2{\rm{i}}\right)^n}\) or \(\frac{{\rm{i}}}{4}{\left(1 - 2{\rm{i}}\right)^n} - \frac{{\rm{i}}}{4}{\left(1 + 2{\rm{i}}\right)^n}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [9 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The figure below shows the graph <strong><em>G</em></strong> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chick.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<address><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the adjacency matrix for \(G\) .</span></address>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the number of walks of length \(4\) beginning and ending at B.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Draw \(G'\), the complement of \(G\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the degrees of all the vertices of \(G\) and all the vertices of \(G'\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Hence, or otherwise, determine whether or not \(G\) and \(G'\) 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><span style="font-family: times new roman,times; font-size: medium;">(i) the adjacency matrix is </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br>0&1&0&0&0\\<br>1&0&1&0&1\\<br>0&1&0&1&0\\<br>0&0&1&0&1\\<br>0&1&0&1&0<br>\end{array}} \right)\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A2</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A2</strong></em> for correct matrix, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for one or two errors and </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A0</em></strong> for more than two errors. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the required number of walks is the (B, B) element in the fourth power of the adjacency matrix <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">using a GDC gives the answer as 13 <em><strong>A2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></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)<br></span></p>
<p><br><img src="data:image/png;base64,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" alt><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/door2.png" alt> </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1 </span></strong></em></p>
<p> </p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) In \(G\), the vertex of degree 1 is adjacent to the vertex of degree 3, whereas in \(G'\) the vertex of degree \(1\) is adjacent to a vertex of degree \(2\). They are not therefore isomorphic. <em><strong>A1R1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept alternative correct solutions. </span></p>
<p> </p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></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><span style="font-family: times new roman,times; font-size: medium;">Many candidates wrote down the adjacency matrix correctly and then proceeded to raise it to the fourth power, although at least one candidate actually enumerated, correctly, the \(13\) walks. </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;">Part (b) was reasonably well done with many candidates realising that, although the degrees of the vertices were the same, their relative positions meant that the two graphs were not isomorphic. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A simple graph \(G\) is represented by the following adjacency table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-15_om_07.29.58.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Draw the simple graph \(G\).</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 class="p1">Explain why \(G\) does not contain an Eulerian circuit.</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">Show that \(G\) has a Hamiltonian cycle.</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">State whether or not \(G\) is planar, giving a reason for your answer.</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 whether or not the simple graph \(G\) is bipartite, giving a reason for your answer.</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">Draw the complement \(G'\) of \(G\).</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"><img 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MQd5WIZQVpmHPHJq+yuL7/3c/sEWIuTpz8RnJW/Xn3qoK7H2pS2I6A43Y69KUugKQIIUi6MvDJ2tCwAjlmWeIo41UUrCfXzediVr2O6bt1yDjoGdV3mprYuAcXpurxNTQLNEEj3fMaO8n8ZIkDL7vryez/3SUDHtI56HTqoPIGLngqDBHogoDjtoRYtgwRmIoAQjTuarvsyagRpKUb537AfAjqmddW1Dmpd9WFu5iOgOJ2PpTFJoDkC6Z5n3CgXunKJJwqD+MSNiSBVjDZXxbNkmOMCx5TjhOAY01mwzhKJDuosGI2kMgKK08oqxOxIYGkC6aaPMD3WXT8cO7p0noy/XgIIUxzTTH568+bNHS/HE9dTZzqo9dSFOZmHgOJ0Ho7GIoEqCSAsMnaUC9gxdxRnNN31iFJFR5VVuUmmho4pE58Qpob6COig1lcn5uh6AorT69n5SwlUSyBCFHeUFyKjDMOuervrSzp+hoDCtL3jgPPeWfzt1Zs5fk1AcfqaiVsk0BSB0h3N2NGyADihw7GjuqMlIT8PCXBMIXIQOxwr6cof7uf/9RHQQa2vTszRdAKK0+nM/IUENieAeCjdUbru2VaGTGLCJeWlIC3p+PkUAY6jcvKTXfmnSNW7XQe13roxZ+MIKE7HcXIvCWxKIMs60UUfUVpmaOiOMnbUIIGpBHRMpxKrd38d1HrrxpxdJqA4vczIPSSwCYG4o3k86NAdRZDiiCJEeafrXnd0k6rqIlEd0y6q8UUhhg7qH/7wh0Nb8WIn/5FAhQQUpxVWilnaJ4G4o3FGcUnLEDGKEE2Xffm9nyVwLYGhY0pXPmuZGtonoIPafh3usQSK0z3WumWuhgCCNGKUz8cEaYSo7mg11dZVRnRMu6rOo4XRQT2KxY0VE1CcVlw5Zq0/AhGgiNB015elpGueVylIy+/9LIE5CeiYzkmz7rhoc1iBgTaINubjx4928dddZbvOneJ019Vv4dcgwMUgSzzxmVcZ6K7PE5nijpbf+1kCSxDQMV2Cat1x6qDWXT/m7gsBxekXFn6SwCwEEJ+4FGV3fRkxrgUilBeilP8NEliTAMcoLhrHKTdHjjFdk/62aemgbsvf1McRUJyO4+ReEjhLYChGh+5ouuojRhWkZ3H65YIEODZZx5SbJ4QpM7g5Lg37IaCDup+6brWkitNWa858b0oAMcpFPt31dJGWIe5oxo4qRks6ft6KwNAxZdwhx6hhfwRKB5VeHI4F26n9HQe1llhxWmvNmK+qCCA+I0Zp1HkNBSkNfClGbeirqsLdZ2YoTHVMd39IHNxzhnfQltF2cUzYbnlc1EBAcVpDLZiH6ghEjCJCM3b0mBgtx47SRWqQQI0EOI7pyued41THtMZa2iZPwy5+HdRt6sFUXxJQnL7k4X87JoD45OJNVz3vOE1DQRpnFFGKw6DLsOMDppGi65g2UlEbZpM2DweVoIO6YUWY9DMBxekzCj/sjUDEKEIU94D3UoziMCE+y+563dG9HSVtl5djWse07TpcK/cIVI4V2kDaPB3UtcibzjECitNjVNzWLQEaXoQoDTGO0tAdRXyWYhRxqiDt9nDoumA6pl1X7yKF00FdBKuRXkFAcXoFNH/SDoG4o3FGeS9DxGgEKe+K0ZKQn1skoGPaYq3VkWcd1DrqYe+5UJzu/QjosPwIUhrYCFL+55WA+ByOHVWQho7vrRMYOqZOfmq9RtfPfylQHYO6Pn9TvLtTnHoUNE+AizFOEa8I0rJQQ3eUxtYggR4J6Jj2WKvblKkUqI5B3aYO9pyq4nTPtd9w2RGkCNGIUf4vQ9xRnnyTrnrd0ZKQn3sjoDDtrUa3L08pUHVQt6+PPeVAcbqn2m64rHFHI0a5EJchyzqV3fXl936WQM8EOC+ymDrnAoup20PQc42vV7ZSoOqgrsd97ykpTvd+BFRafsaIlu4on4fuKBfhUozijOqOVlqhZmsxAghTlgDi/OCcYIwpIsIggbkIlDc/OqhzUTWecwQUp+fo+N2qBHBD86IxPCZGuejyooH0Arxq9ZhYhQRK0cD5gGPqeVFhRXWQJR3UDiqxoSIoThuqrN6yWrqjNHz5P+XEBS3dUT7zMkhAAneH8dY6ph4JaxIY3gy5UP+a9PeVluJ0X/W9eWlLZ5TPQ3cU12coSDfPtBmQQGUEFAmVVciOsuOxt6PK3rCoitMN4e8hadxQRCgNGi/EKNsS4o4yq56u+vyf732XgAReEuA80jF9ycT/1iWgQF2X9x5TU5zusdYXLHO65mm8IkpLMUrSuKN5IUjtql+wQoy6KwKKgq6qs+nCeCw2XX3VZ15xWn0V1Z9BxCcNVQTpMXcUMYoQjRh1Vn399WoO6yLA+RXHlPPJyU911c8eczMUqB6TezwKlimz4nQZrl3Hihgdjh0duqMRoRGkitGuDwkLtzCBoQhwIsrCwI1+NIHy2KQX7PHx0d6w0fTc8RQBxekpMm5/JoDwjCBlVj3CNNuyE+ITNydPZKKRUpCGju8SuJ6Ajun17PzlOgRKgaqrvw7z3lNRnPZew1eWL2KURidjR8uoIkbTXc+7YrQk5GcJ3E7AtSVvZ2gM6xAob6J0UNdh3nMqitOea3dC2eKE0sBEkGYb0UR40k0fd5Rt2T4hKXeVgARGEECYvn379rDCBTd/duWPgOYumxLg2pHH6OqgbloVzSeuOG2+Cq8vABOXhmNHy9gQnjQwvDJ2tPzezxKQwDIEPnz4cJj8ROwK02UYG+syBHRQl+G6t1gVpzuqcZxQBGncUT7zSogTGiHKRTHbso/vEpDAsgQQpjimnK+cizim9lAsy9zY5yWggzovzz3GpjjtvNbjjtJYxCUti8zYoLijcUjL7/0sAQmsR6B0TBWm63E3pfkJ6KDOz3RPMSpOO6vtoTuamfUpZpzQPJEJccrLIAEJbEtAx3Rb/qY+PwEd1PmZ7iVGxWkHNZ2u+syqL7vqKV7cUZyYuKQdFNsiSKAbAqVjyo0ji5nbld9N9e66IDqou67+qwuvOL0a3XY/xB2NEOXER4yyLYGLGoI0Y0d1R0PGdwnUR0DHtL46MUfzEtBBnZfnHmJTnDZSyxkvyknO56E7Gke0fG+kaGZTArsloGO626rfXcF1UHdX5TcVWHF6E77lfhx3NE9kOuWOlmNH7QZcrj6MWQJzEygdU7vy56ZrfDUSGDqort1bYy3VkSfFaR318Px40NIhLbvqyWZm08cddSJTJZVnNiQwkcD79+8Py0XxM4XpRHju3jSB0kF1RYqmq3LRzCtOF8V7OnKEZ9xRTlZe2ZZfZaxoOXZUdzR0fJdAmwQQprw43xWmbdahub6NQOmgcn1jAqBmy21Me/u14nTFGo0YzWQm3tlWBk7UOKN8VoyWdPwsgbYJ6Ji2XX/mfj4CDFn77rvvDtdArnU+bGI+tj3EpDhdsBbjhCJCM3Y020gW4ckr3fURowrSBSvFqCWwAQHOe4Qp40z5rGO6QSWYZHUEhgJVB7W6KtosQ4rTmdFz4Smd0aE7WorROKSK0ZkrwegkUBkBHkeKOCUoTCurHLOzKYFyYqAO6qZVUVXiX/3444+fEVS3hD2PFYEdM+lLQcq2MEV48uKk44UgzbZbmPtbCUigfgK0A4jSCNM3b97cvXv3rv6Mm0MJrEhAB3VF2I0k9dWnT58+Z0IO71MDwvTx8XFXg5kjRktBWnKLGI0zGkFa7uNnCUigfwKlY6ow7b++LeH1BEoHlWsmuoJrqWGfBJ679bnDpyHlACFwcJx6hB77cqfDvhw8vYtTyosgRbxT7lNrjsKMLjvF6D5PJkstgRAYtqcK05DxXQKnCeignmazt29+ngLH7Ys4ZTuu6Kk7FwQYLwQtDXFvIWIUdzSvsoywofzpquezQQISkAAEyht9hanHhATGEcDcITCLP8tN6aCOY9fbXs/ilIKdEqKnCs2BVIrZU/u1sD1uKHdunBT5P3mHDYJ0OHY03/suAQlIQMfUY0ACtxEoBSrG0MPDg+ug3oa0yV+/EKfXlABXANHWYijd0QjSshwRo3GJdUdLOn6WgASGBHRMh0T8XwLTCZQCVQd1Or8efnGzOMVJbCUgRiNII0bLIQmlO8rJgTid6ia3wsJ8SkAC8xHQMZ2PpTFJAAKlQNVB3d8xcZU4ReARWnBMOagRohk3mrynquOKIrIpj+5oyPguAQmMJaBjOpaU+0lgPIFSoMZB5UlSLWiP8aV0z2MErhKnHCQcHLccIBGJOA64k3kdy+SUbcQbQUo+ib90R5NvxCgH/lzpTsmj+0pAAn0Q0DHtox4tRb0Ecp1m7CnXdiZL+SSpeutrrpxdLU6//fbbq/LAwcWkoyzJRCQIRBxLDsJr4i2d0QjSMnPEjRiNS3qLqC7j9bMEJLBvAjqm+65/S78OAa7fCNJyFr8O6jrst0rlrDhFSCIcy8CTThCW14hIfsfBVTqZxM3/iEpe7HPuroh9ecUdZf/EwTv5RXxGkHJQ644eEPlHAhKYiQA9NAjTtD8uFzUTWKORwAkCOqgnwHS6+XkRfsqHOLy/vx9VVNYeQ/iNDaeE6bHfEy/xJ5RiFFHKi21l4DcRpOm6L7/3swQkIIE5CND2cJMdYcrjSBGnBglIYHkCpZbgmq+DujzzLVI46ZxS6cccTNY1zXOip2T4mGN66veIZFwJRCYHIi5Fxqjym9Id5W6KvNpVf4qm2yUggbkI0A6la5F2CGF6TS/SXPkxHgnsjcDQQc06qOgAQz8ETopTiojgowEuAw7B0LUsvz/2GYE59TdDAUw+OPjikPJ5mLdjabtNAhKQwBwEaMO4aebmmaAwnYOqcUhgOgF0QMag0pPKDaMO6nSONf/irDg9lfGpdyhpzE/Fd2o74piDMC/F6ClSbpeABJYkoGO6JF3jlsB0Ajqo05m19IufXZPZuJdjfzvVNSVehCjOBHdHOQjHpud+EpCABOYiQPuVrnzi1DGdi6zxSOA2AnFQ0QtxUMshgLfF7q+3JHCVOOVAWMPFXCONLeGbtgQkUDcBLnSMaaP3h/aIm2XHmNZdZ+ZuXwQwr+jS5/xEoGY91H1R6K+0V4nTqRiumazEgaY4nUra/SUggbkIlI4pbZGO6VxkjUcC8xLQQZ2XZw2xrSJOubOZGhjXOnVs69Q03F8CEpDAMQI6pseouE0C9RLQQa23bq7J2SriFJE5VaDadXZNdfobCUjgVgLHHNOp7detefD3EpDAdAI6qNOZ1fqLF+KU8RoJNNC85gp0iXHgjA0sJeXA5rG03E8CEpiDgI7pHBSNQwLbEdBB3Y79nCk/i1OEaZ54QgI00iy4P5dAZdxpZt6fKgDjunBMeWcCAgOb50r/VJpul4AEJAAB2prMys8YUx1Tjw0JtEdAB7W9Ohvm+KunWW6fzy2STyNNYDbcXCFCOE4taXAR4IDiM/nhIsHFIgfZNZOq5sqv8UhAAn0ToC2izeGdNujSjXTfNCydBPogUJpcDC/kvHYuSxt1+9Xnp1BjVnFteRoLApWDyac/1FhL5kkC7RNQmLZfh5ZAAqcIlGaXWuIUpfq2VytOQVUeVBkWMGXcan24zZEEJFATAYQpw4cYxkQbw02wzkpNNWReJHA7gdJBtTf2dp5rxFC1OAUABxXdbbl4PD4+Hi4ia8AxDQlIoF8CQ8cUYerNb7/1bcn2TaA0uzjPOd8zbHHfZOos/fOEqDqzd3e4WKRLH4F6f39/EKy15td8SUAC9ROIY8o7jik3vQrT+uvNHErgWgLMa2HMKYI0TiqawlAngX/6f0+hzqx9ydUvf/nLu1//+td3f/nLXw4OKu/87ySpL4z8JAEJjCMQx/Svf/3r4UL1xz/+8e6bb74Z92P3koAEmiWAbiDQBnD+8/rd735394tf/KLZMvWa8eq79UvwcTu423FGbUnGzxKQwBgCZRviGNMxxNxHAv0RGHbx46hqdtVVz004p0GGg4rDwQUGgYo1z+QFD6oQ8l0CEjhFgHaD8es6pqcIuV0C+yCgg7voTSAAACAASURBVFp/PTflnAYny0sx9pSLDQ4qT5/ycaeh47sEJDAkoGM6JOL/EpCADmq9x0BTzmkwMj4ExzQOKmNQEakuARNCvktAAiGgYxoSvktAAiUBHFS0AxrCMaglme0/N+mcBhsOKmsU0r1PwEF98+ZNvvZdAhLYOYHhUnSuY7rzA8LiS+AIAR3UI1A23tSkcxpmOKjMtGP8KXc9XIgYl6qDGkK+S2C/BHBMf//73x/aB9wRZ+Xv91iw5BI4R0AH9Rydbb6rfp3TS1i46OCGsIYZgQkPPPbUIAEJ7JcATgjj0rlxZcKk65ju91iw5BIYQ4B5K66DOobUOvs07ZyWiFhAOw4q40d++uknF9UuAflZAjshQA8KjinDfuhFwTG1N2UnlW8xJXADAR3UG+DN/NOmx5weY4Fz+uHDh8NXuRM6tp/bJCCB/giUY8cQpHm6XH8ltUQSkMBSBMp2BOPLdVCXIn063m6c0xSRdVBxTBhvxksHNWR8l0DfBIaOqcK07/q2dBJYioAO6lJkx8fbnXOaojPu9P3794d/y2fq5nvfJSCBfgiUToeOaT/1akkksCUB25Xt6HfnnAZlxphl/TLGo2LP+wzdEPJdAn0Q0DHtox4thQRqI1A6qJnTwgpB6ojla6pb5zToGH+Ki0pXP+KUrj5m+BskIIH2CZTOhud3+/VpCSRQI4GynbFnZp0a6tY5DT4OJMTon//858Nsfh3UkPFdAm0TGDqmzMpnnWODBCQggTkJDB1UemQZLqiDOifll3E1v87py+Ic/y+z9hGp3AFl/cPje7tVAhKonQA9IpzH6RFhHVPWMzVIQAISWIJAqSOYbK2OWILylzi7d05T1NJB/eGHHw5PlGJmv138IeS7BNogwA3mv//7vx9W4uC81jFto97MpQRaJ6CDul4Ndj/mdIiSrkDWQqV7P0+O0XEZUvJ/CdRJAMeU85fgGNM668hcSaB3AuiIh4eH5wd9uGzd/DW+i279EhsXNBbUJSBQI1TLffwsAQnURwDHNI8mxjHNowbry6k5koAEeiaAjnj37t2h55Uu/gjVnsu8dtl2J04BzIGVMWrcAX399dd3vBskIIE6CbBmcS4A5flbZ27NlQQk0DsBxqBm9R/HoM5f27sUp2CMg8qYUyZV4KAqUOc/wIxRArcSQJjGMfWBGrfS9PcSkMBcBHRQ5yL5Op7djTkdIuCOB2HKO0KVOyEOOIMEJLA9gaEw5fw0SEACEqiJgGNQ56+N3TqnQZmxa0yK0kENFd8lsD2BoTDNWPHtc2YOJCABCXwhoIP6hcVcn3bvnAZk6aCyDYeGLkSDBCSwPoFSmGZ9wfVzYYoSkIAExhPQQR3P6tKeu3dOAygOKu8EuvqZHWyQgATWJVAKU24QmRVrkIAEJFA7AR3U+WpI53TAkq59ZgVncpQO6gCQ/0pgQQJMfEKcEnRMFwRt1BKQwGIEhg4qQ5JifC2WaGcR65wOKjSTonIg4aCy8LdBAhJYlgCiNMJUx3RZ1sYuAQksR2DooKIjDNMIKE6P8EKgsg4qF8hMkspF88jubpKABG4kgGOa5aLK9QNvjNafS0ACEtiEQNmOZR1U9IRhHAHF6QlOCFSs+EyKKrsbT/zEzRKQwBUEdEyvgOZPJCCB6gngoOZJdunqrz7TlWTwn/7fU6gkL9Vl4xe/+MXzmqd/+ctfDuNQ2fbNN99Ul1czJIEWCXDT9x//8R+HrOM0/PGPf7zjHDNIQAIS6IHAr3/968Ma6n/+858Pj0xHS6AhMMAMpwkoTk+zOXzDhZLxp3/729/u/vrXvx4E6g8//HD3u9/97sIv/VoCEjhHAMc0wpQeiv/8z/9UmJ4D5ncSkECTBNAQaAnc02iJf/u3f2uyLGtlWnE6gjQHFfY8gbsexo8gVNmmyzMCoLtIoCDAuKvf//73d//1X/912KpjWsDxowQk0CUB3NJf/vKXdzqo46pXcTqO00GEIkZ/+umng0BFnHIHlDGpI6NxNwnsmgDClK78//7v/z5wePPmzR0ji7zJ2/VhYeElsAsCcVBjcKEj6IW1/Xtd/YrT10zObolbij3PgaWDehaXX0rgmUBWvogwZXF9hekzHj9IQAI7IJDxpnFQEarZtoPijy6i4nQ0qi87ciCVDioHl3c/X/j4SQJDAkPHFGGKa2qQgAQksDcCOqiXa1xxepnR0T1wUBk/whhU3FPvfo5icqMEntcKLh1ThakHhgQksGcCcUt1UI8fBYrT41xGbeXuh+UgIlB5Zwyq40dG4XOnHRDQMd1BJVtECUjgKgI6qKexffX5KZz+2m/GEPjTn/50x+PJuBDjqLLo7q9+9asxP3UfCXRLIGNMOT8IduV3W9UWTAISuIEAbeTDw8MhBjXE30HqnN5wQOWnWWQ3Dio2PWNQXWQ3hHzfGwEd073VuOWVgASuJYCGoMfVWfxfCOqcfmFx8ydm8OOgssQUzunHjx8PC/jfHLERSKAhAjqmDVWWWZWABKohoIP6pSp+9uWjn24lUNrxCFRset4NEtgLgTimduXvpcYtpwQkMBcB5qwwLJBe15hdtKl7DDqnC9Q61nyEKQ4qBxvC1SCBngnomPZcu5ZNAhJYi4AO6t2dzukCRxsz8DIpKg4qd0EGCfRKQGHaa81aLglIYG0COqh3dzqnCx51OKiMQeUdmx7B6uNOFwRu1JsQyA1YjnPWMHUd002qwkQlIIGOCOzZQVWcLnwgDwUqk6Ts4l8YutGvRgBhyg0YPQPegK2G3YQkIIGdEPjw4cPd27dvn5eqREPsYSUgxekKB/jQWWK9x2+//XaFlE1CAssRGB7X9gwsx9qYJSCB/RLYo4PqOqcrHO/c5TAOlYs5jzplHVQefco2gwRaJBDHlLV9dUxbrEHzLAEJtEKAdVDLx6WjI1hLveenUeqcrnh0OmlkRdgmtRgBHdPF0BqxBCQggZME9uSg6pyePAzm/4K7HMabxkFlnJ4O6vycjXE5Ajqmy7E1ZglIQALnCAwdVHqumGTdo4OqOD13JCzw3VCg0sX/008/OUlqAdZGOS+BOKZ25c/L1dgkIAEJjCXAcEDWT8dF/eGHHw6rAX3zzTfdTZJSnI49ImbcbyhQudgjUDnoerwDmhGdUW1EQMd0I/AmKwEJSGBAYA8OqmNOB5W+9r8sw8NSEQRm8DPj2SCBmgjEMc06ps7Kr6l2zIsEJLBXAj2PQVWcbnxUM0nq/fv3hxdZyQLme1jHbGP0qyVPHfO6JXA8bHFMKExvqTV/KwEJSGBZAuU6qPS+Pj4+rnqt4BoxJeQ6lvdTv1WcniKz8vbSQS0fXbZyNkxuAQJMfOPFScyd7jWBmxbWx10zKEzXpG1aEpCABK4jsKWDmgcEkIcxJgzjZXkxOZzXqSU1FafXHQuz/2rooCpQZ0dcRYQPDw/PApWT8twwDo4JTnjujNce8jF8spld+VUcPmZCAhKQwFECWzuopI/JloChMnxcOyYN1zSuLwTcU/bj+vbKSf1sqIrAkzv2+amSPj/V2+enu4rPP/74Y1X5MzO3EXgSeYe6Tf2Oie3p5P38dJKP2XWWfT59+vT56c72kE/enx6XN0u8RiIBCUhAAssRoK3m2pLry/fff79cYoOYSSvXDdI/l3Z5HWRfrnHD8LOnLwwVEeAuIt233GVwJzJ1TEdFxTErMxCg62OtwDGHu8sxR9cLz3Ee3v2ulRfTkYAEJCCB8QTKHlfa8nS5j4/htj1fuZ8nosMpReskMO9mqHMUp6FT0TsVh0ClojOWZFhxFWXXrCxMAHG6hkBMV34pTE+NB1q4yEYvAQlIQAJXECiHgKEfMLjGjAW9IqmbfkI+MUASyGsZFKcljYo+c1fBOD8EKqIBNyvjNCrKpllZicDS4vSYY6owXalyTUYCEpDAjATioBJlrQIVbVM6rUMDTnE64wExd1Q5wCJQ7eKfm3Dd8XG3Ozxhl8hxOXwkXfkK0yVIG6cEJCCBdQi04qCGRilU2fbzfOF7nQQQqFRanFPecVQVD3XW15y5wilHoJZdH1Pj5/eIT14RusTHUAFeduVPJer+EpCABNoggEDlGsCYThxUPjOPIEKQ/7ku8E5gO9eHfL9kKbn25JpEOq/mVgxnSPl/nQSYhfd0wBxm4fH+tNBunRk1V2cJlLMUn07Gs/syg5H9rw3Mun+6iXmevfl0/r/4zHc5pp4apM/sb5CABCQggb4IlNedJ8PrMJOelYGOXR+4Ll1z3WF2fhnfudn6rEJU7svnYdA5XfLWYMa4cVC5o8lMarr4nw6g13cbM6ZpVMsS4G6Vu8djd6msGccrKzdMzUnZVX/qt6RNIH3upp8aiFO7ul0CEpCABBolMHRQuT7ELR0WKT1tXB8yMXu4z5j/iWfohuKUsh0XN64p+6BlhkFxOiRS8f+IB0QEwpQDB6HK/8MDoOIimLWCAHX49ddfF1vm+UijwxIiOfkvxYo4PSaQL/3O7yUgAQlIoA0CTLKmnR87ex9zhHBMOI4pMelcCuW8muG+TogaEqn8/whUXFRECAfAcAmGyotg9v5BgJuKp670u6fuj1evNCTXwOJ4iCs65veI2DREY/Z3HwlIQAISaI/AVBOC68KUa0lJ5Gno4avrGmZauYQULur9/f3R64/OaUmzkc8I02MOKnchhrYInOpKv7U7ZSoFGgmDBCQgAQn0S+AaIwuBeo17ik7hVQb+R6dgiGCscd1B/MbNxZRJUJyGRGPviBruTOjap4JTudyVGPogQB2fGhd0qoTsP7Y7v4wjjcXUO+syDj9LQAISkECdBLg2XGNCcG3gt3NeG2KwRb9AjBUFuOZlmKLitM7jaFSuOFi4o8kdCO9s00Edha/6na4Rp9cWisbHrv1r6fk7CUhAAn0S4NqwRECr4JRGMOcapDhdgvYGcQ7vQBCo2OTXzvLeoAgmeYIAJy+vqeHa36RRmJqe+0tAAhKQQP0EEIJTxeY115OxJIbXnLLXT+d0LMWK9+PgwUFlhjZjSrDH2VaO36g4+2ZtJIEMTMdRPRWod77P3eip/YbbaSQYx2yQgAQkIIH+CCBKy270sSXkmsJrjVCm42z9NYivkEYc1Iw5RajyMvRBgIaFm44xgWEd5Uk+5jfeyIyh5D4SkIAE2iTANeGaIX/RFEuUemiilE6q4nQJ4hvGSXd+DibEzFhBs2GWTXoEAeqRLg9uQi4FnNMpYpN9z7mxl9LzewlIQAISqJ8A4nTMNSQlQSyWgjHbT71PGTLA9aw00MhXKZ4Vp6coN7qduyO6+CNQqXys/CkHTaNFbyLbZT3wufz/WAFyAjNZaYojiuC8tBwVxwr7TRGyx/LoNglIQAISqJ9A9AHvlwKidMrcleG1jGvXcBtpsg3HFF2SoWrJVymcHXN6qYYa/T7CBMeNcahUfrY1WqTms82JWK4zx/9MYCtPyLKQnMTsw4v6m3IHSzxxREmTxoDGIoE0OR7KO9V857sEJCABCfRJ4JhgHJYUc2uKXiBODJTyGoMxxnWmvL6V17SkyXWN9IbXt68+P4Xs5Ht/BBCnsc4RIriqY+6a+iOxXYkQh7w4Mcc0DMdySrc7dXdtSLrkI4KYdXLLhuPauP2dBCQgAQnUT4DrAI/MRkSiBzAwSuOC6wwicex1gfi4nhBKYXqJBPGTDumdGlKmOL1EsYPvS4E69Y6og+JbhIIADQiPi+Od2fk6pwUcP0pAAhLomECpBXhs9lgRugUSx5xuQX3lNLk7wqInYL1HnKycDZOrgEDuWMnKcKZkBdkzCxKQgAQksAABhoflQStogpqFKcVXnC5wENQYJQdjuoUz1nGKDV9jmczTdQTSjYI4TXf/dTH5KwlIQAISaIEArinXfNr/TJiuOd+K05prZ+a8cUDSlcuYU4QJs+UUqDNDbiA6xvpwDFD33KgYJCABCUigXwJc75lvQGjBNSWfilMo7ChkUhRF1kHdUcUXRaU7J+5pGqziaz9KQAISkEAnBOgdwzUl0O63Ms9AcdrJATilGBycpYPKGFQdtCkE2983DZRd++3XpSWQgAQkcIoA40xp5+kty9C+U/vWtF1xWlNtrJiXUqDSvctyEArUFStg46TStc9dtfW+cWWYvAQkIIEFCJSuKcP60mO2QFKzR6k4nR1pOxEiULLWJQKlfGJDO6Uwp9cQ4C6ahorGi7tqgwQkIAEJ9EWANc5p4xnKxVjTloLitKXaWiCvCJR08eOgIlAVKwuArixKxCk3JwS79iurHLMjAQlI4EYCtOuZU4Awpc1vKShOW6qthfKKQMVB5T0CNQf1QkkabQUE0rWPa069GyQgAQlIoA8CcU1p5zPHoKWSKU5bqq0F8xoHlXe6ARiDqkBdEHgFUVPXWYhZt7yCCjELEpCABGYgwLU7cwl4AE9rrikIFKczHAi9RIFQoYufOy0EKl38CtReavd4OdK1bz0f5+NWCUhAAi0RoBcM15TQ2iSokrPitKTh54OTxnITuGoEHNQ88kw8/RFIdw8NWu60+yulJZKABCSwDwJcr2nPMZtaeBLUqVpRnJ4is+PtQwcVgZpFfHeMpcuiU9cZymHXfpdVbKEkIIGdEMBgyLWaSVAxmVosvuK0xVpbIc+lQCU5Dvgc9CskbxIrEciSUiSnc7oSdJORgAQksACBXKMRpS27pqBRnC5wgPQSJcKFMagc5IxBZRxLDv5eymg57p5ncjLu1Fn7HhESkIAE2iNAdz5tONdtJkG1HhSnrdfgwvnPgZ6xiQjUDLZeOGmjX4kAd9k45QTd05Wgm4wEJCCBmQhgHsU4YpJrJrrOFP0m0ShON8HeVqIIVCZJ5QkTnASMQ+WEMLRPgPpNY+bkt/br0xJIQAL7IsA1mV6vmEk9lF5x2kMtrlCGHPQZx4KI0UFdAfxKSUSc0sDZtb8SdJORgAQkcCMBeruyFCDX5/SC3Rjt5j9XnG5eBW1lAAc141kQqDqobdXfqdymax9hatf+KUpul4AEJFAXAUwi2m3a8PRu1pXD63KjOL2O265/xQlQOqgK1PYPhywpRUlcUqr9+rQEEpBA/wRoq9Net/okqFO1pDg9RcbtZwngoPKiu58uBZ4mZXfwWWTVf5muferT8cTVV5cZlIAEdkyA622G1tF2p/3uBYnitJea3KAcuKfpRuDuTQd1g0qYMUkaN242EKZ27c8I1qgkIAEJzEwAE4F2OvNBZo5+8+gUp5tXQdsZQJzGQUWg4qAqbNqs07JrPwPs2yyJuZaABCTQLwFc0ywdxTWY8aa9BcVpbzW6QXmOOah28W9QETMkmfVsudGwa38GoEYhAQlIYGYCCFPaZwyFzP+YOYnNo1Ocbl4FfWSAu7fHx8dDFwPO6f39vQ5qg1Vr136DlWaWJSCB3RCgVyvrUfc2CaqsRMVpScPPNxFA2KSLH+eUMag6qDchXf3HjF+ii4i78swCXT0TJigBCUhAAkcJpDuf6y2vXoPitNea3ahcdAt//PjxhYOqyNmoMq5IFnGaBs+u/SsA+hMJSEACCxFAmGZOR8+uKfgUpwsdRHuO9piDmhNqz1xaKTv1h0ilzhx32kqtmU8JSKBnAvRCpjufcaY9ToIq609xWtLw82wEcFA/ffp0OIE4qRiDqoM6G95FI6LRyyPw0hgumqCRS0ACEpDAWQK9PgnqVKEVp6fIuP1mAggcxqDyjgPnMlM3I10tgrJrf7VETUgCEpCABF4RoBcry/vhmsY8eLVjRxsUpx1VZo1FwYVjFn8m2eCg6sbVWFMv85QlpXC9HZLxko3/SUACEliLAMZOngTFdbTXpaOGPBWnQyL+PzuBOKgRqJxouQucPTEjnIUAdZb6cjjGLEiNRAISkMBkAlwraYN7fRLUKSCK01Nk3D4rAYQODirdxdwJssxUlsSYNSEjm4WAs/ZnwWgkEpCABK4mULqmOKYZbnV1hA39UHHaUGW1nlUED2NQI1ARpzqo9dYqNxQEu/XrrSNzJgEJ9EuAayQClWvnXrrzU5uK05DwfRUCdBezDmoEKpOkMp5mlQyYyGgC1BH1RePoTcRobO4oAQlI4GYCdOVnfgZPYKQt3lNQnO6ptispaxzUTLrh7tAu/koqp8gG9RT3VHFagPGjBCQggYUJxDWlDd6bawpaxenCB5jRHycQBzUnHe6pDupxVltuTf0wa5+XQQISkIAEliWQSVCkkkeCL5tifbErTuurk13liEewRQBxp6hArav6uYng5ZJSddWLuZGABPokQFubnkSGVqX3qs/Sni6V4vQ0G79ZgUC6+EuByjhUxjkatieAMKWBJNi1v319mAMJSKBvAowzZRIqbS/mzV6D4nSvNV9ZuTkJGfRNQAThoCpQ66ik3LnTYFonddSJuZCABPojgGuaSVAYNml7+yvp5RJ99fkpXN7NPSSwDoGya58JU3sdb7MO7XGp0GB+/fXXB2GatWrH/dK9JCABCUhgLAF6DTFncE2///77sT/rcj+d0y6rtd1C4Z6mK4OTlMX6EUeG7QjQUOYO3q797erBlCUggX4J0LbmaXy5BvZb2sslU5xeZuQeKxOIQGU8Kicsd5MK1JUrYZBclv2i8bRrfwDHfyUgAQncQIA2NUPZaGvT3t4QZfM/VZw2X4V9FgCBmi59xjrqoG5bzzin3CzQiPrEqG3rwtQlIIG+CDDONAZM5l70VcLppVGcTmfmL1YiUI45xbHTQV0J/JFkMmsfcZqupyO7uUkCEpCABCYQ4GY/k6AQphlCNSGKLndVnHZZrf0UqhSonMQIVJ279eu3fFqUXfvr8zdFCUigTwJxTRGlWVKxz5JOK5XidBov996AQAQq7p0CdYMK+EeSrHeKSKUO0gW1XW5MWQISkEDbBLjRL11TrnGGvxNQnHokNEEAgfrx48eDOEIYMQZVB3XdquPOPo2nXfvrsjc1CUigLwIMkSqfBMU1zvCFgOL0Cws/VU4AccQ6mzqo21WUT4vajr0pS0AC/RDAMeUmn94oJ0G9rlfF6WsmbqmYAAIVBxWBioOaRYsrznJXWcvdPex1rruqWgsjAQmsRKB0TWlTc9O/UvJNJKM4baKazGRJIAKV93Txuzh8SWi5z9wUwN1Z+8sxNmYJSKBvAlnTlPbUBfeP17Xi9DgXt1ZOoBSoCCXGoCpQl680uqByl69zujxvU5CABPoiQFd+rlXMzqdNNbwmoDh9zcQtjRDgrpMufsSSAnW9Sos4pYHFuTZIQAISkMA4AnFNaUddOuo0M8XpaTZ+0wABBCpPkkpXMw5qZkA2kP0mswhruBOctd9kFZppCUhgAwLc0KfHie58XdPTlaA4Pc3GbxohEAeVgeU4qNyZKlCXq7yyaz/dU8ulZswSkIAE2idALxPXJgKOKTf5htMEFKen2fhNQwTioGY2OeJUgbpcBaZrnwbXrv3lOBuzBCTQB4E8CYprld35l+tUcXqZkXs0QgBHjy7+oYOKm2qYl0C69hGmdu3Py9bYJCCBvgjQlR+zhDVNdU0v16/i9DIj92iIQClQyTbdKOlKaagY1WeVu/80sBlDVX2mzaAEJCCBDQhEmNJm6pqOqwDF6ThO7tUQAQQqs/jz1A26U5gopYM6byWma59xp7Kdl62xSUACfRDg+kMbyXUp16Q+SrZsKRSny/I19g0JMBsyjQENBA6qImq+CkGc0uDCVPd0Pq7GJAEJ9EGAtjGuKe1l5kT0UbplS6E4XZavsW9MQIG6XAWUXfvO2l+OszFLQAJtEkCYMi6fm3ifBDWtDhWn03i5d4MEhgL14eFBB3WmeowTwKQoXemZoBqNBCTQPAF6k3LTzjhTbuYN4wkoTsezcs+GCSBQmclPQEghUF0C6fYKLbv2nbV/O09jkIAE+iDAMDKuMUyCyvCyPkq2TikUp+twNpUKCHD3ikCliwUh5SSp2ysFljS+jju9naUxSEACfRDAMc3NOsKUdtIwjYDidBov926cAAI1Y39oPO7v753Mc0Od0uhm1j487dq/AaY/lYAEmifgJKh5qlBxOg9HY2mIQOmgMi4IB9Uu/usrMF37sJTj9Rz9pQQk0D4BVoahLeTGPUZI+6VavwSK0/WZm2IFBBCorIVKA0JDwhhU3g3TCdCtn8H+6cqaHou/kIAEJNA2AW7Os3QU1xjaRsN1BBSn13HzVx0QwPFDoCKsEKY4qArU6yo2XfuZnXpdLP5KAhKQQLsEEKZ063NNcRLUbfWoOL2Nn79unACiKpOk4qDq/k2v1CwphXOgwJ/Oz19IQAJtE+C6kZtzJ0HdXpeK09sZGkPjBEoHFXFFF78CdVql4hRk1r7sprFzbwlIoH0CeQIh15PcrLdfqu1KoDjdjr0pV0SABuXx8fHQHUO3DF38iqzxFeSs/fGs3FMCEuiLQCZBUSomQdEeGm4joDi9jZ+/7ogA7h9jUHEAdVCnVywCn0C3vktKTefnLyQggfYIOAlqmTpTnC7D1VgbJYAwjUBFYNHFz12x4TIB2CHw4abrfJmXe0hAAu0TYBIUAtVJUPPWpeJ0Xp7G1gGBOKg4geniV6Bertiyaz8TAy7/yj0kIAEJtEmAXqJcG5gExbXDMA8Bxek8HI2lMwKlQKVoDHZPI9RZUWctTiYC4CTwMkhAAhLokQDGBdcFAr1GrGtqmI+A4nQ+lsbUGQGcQJaZQnDFQc0Cy50VdbbiIOp5IUzt2p8NqxFJQAKVEaB3iDaO64RPgpq/chSn8zM1xo4IILQQqJnsw51y7pY7KuZsRYEXLgLB9U5nw2pEEpBARQQwK2JUYF7k+lBRFpvPiuK0+Sq0AEsT4M6YSVLptqFRUqCepp6ufVwFGnGDBCQggZ4IZBIU1wafBLVMzSpOl+FqrJ0RSBe/AvVyxeKcwouufd3Ty7zcQwISaIeAk6DWqSvF6TqcTaUTAowtKgUqi/XrDr6s3LJr31n7L9n4nwQk0DYBes1o8+nKz7Wg7RLV1VSCowAAGBNJREFUmXvFaZ31Yq4qJTB0UJnBr0B9XVl27b9m4hYJSKBtApkERSnozud6YFiGgOJ0Ga7G2jkBJkllhiYNlgL1ZYXjKtBw4zDYtf+Sjf9JQALtEWCYkpOg1qs3xel6rE2pMwLcOStQj1cqwpSxp4hTl5Q6zsitEpBAOwToJeNGm7bNSVDL15vidHnGptAxgaFA5XGnLj5/d2jAs7yKs/Y7PgEsmgR2QIA2PQ9hoc3Pcnk7KPpmRVScbobehHshEIHKHTVCTIH695pN1z5ug4K9l6PdckhgfwQyCYrJnk6CWqf+FafrcDaVzgkgUBmHikBFjClQ//5IPxpzgrP2Oz8BLJ4EOiVQToJiGBdtvGF5AorT5Rmbwk4IMENdgfqysstZ+y+/8T8JSEAC9RNgEhRj52nLMlSp/ly3n0PFaft1aAkqIjAUqPf397ueEJTGnG59Z+1XdKCaFQlI4CIBhGkmQemaXsQ16w6K01lxGpkE7g532DiodGkjylhmaq/CLAvyO2vfM0MCEmiJAG12JkFhOmSIUktlaDmvitOWa8+8V0uAxuzjx4/PApUxqHtcUonxWXFP91j+ag9QMyYBCZwlgDDFXGBmPnMKDOsSUJyuy9vUdkSARm3ooO5RoEWc4kQ4a39HJ4BFlUCjBErXlNn5uqbrV6TidH3mprgjAgizoYO6t5nriHQad7v2d3TgW1QJNEqAdoqlowi03y4dtU1FKk634W6qOyKAOEOg8k7DxxjUPQnUsmt/T+Xe0SFuUSXQDYEsHUW7ZXf+dtWqON2OvSnviMAxgZrB9nvAwBhcAt36du3vocYtowTaI1C6pjimGZLUXknaz7HitP06tASNEKBru3RQ6Trai0Cl7LwQpnscd9vIIWo2JbBrAlnTVNd0+8NAcbp9HZiDHRGIQOWOPF38NIi9B8odF2Kvy2r1XseWTwItE+CmOWaBa5puX5OK0+3rwBzsjEAEarq6cVD3IFAZ2kDgIoAwN0hAAhKohUBcU26i0zbXkrc95kNxusdat8ybE6DbiGWm0gjuQaDS6FNuuvZ1Tzc/BM2ABCTwDwKZBMW/TIKinTJsS0Bxui1/U98xgWMCNUuY9IgFxzjuqbP2e6xhyySB9ghws5yeK8yCDD9qryR95Vhx2ld9WprGCCBQmSSVtfRoJFlqqtdu7zjFdu03dqCaXQl0SoA2l54cbp5dOqqeSlac1lMX5mTHBOjiT8PIoPxeHdR07SO+7drf8QFv0SVQAQFc0/TiYBCkZ6eCrO0+C4rT3R8CAqiFADNES4Hao4OKU4xARZy6pFQtR575kMD+CNAGYQLwjihN27s/EnWWWHFaZ72Yq50S6F2gIk7jTti1v9OD3GJLoAICtD9xTRWmFVTIIAuK0wEQ/5XA1gQQqFlnj8bz4eGhq6cqpWufbn261QwSkIAE1iQQ15Q06c7PWPg182Ba5wkoTs/z8VsJbEKAO3kEKoE7/HQ/bZKZmRPFOWXyAcGu/ZnhGp0EJHCRAOP6c2Osa3oR1yY7KE43wW6iErhMgDt6JkrRFd6bg5rlWtKtdpmGe0hAAhK4nQA9Nlk6CgMgN8q3x2wMcxJQnM5J07gkMDMBBGrpoPbSxZ9uNNwLZ+3PfNAYnQQkcJIAwjSToNIOndzZLzYjoDjdDL0JS2AcAQQqa6Fyh4+Qu7+/b17QURa697lI2LU/7jhwLwlI4DYC9NSkt4bufF3T23gu+WvF6ZJ0jVsCMxHgDj9d/LiNLDOVMVMzJbFqNFlSikQRp4hUgwQkIIGlCNDGMNaUwLAiXdOlSM8Tr+J0Ho7GIoHFCdCgDh3Ull3HzNpXnC5+6JiABHZPAGFKW8ONcYZK7R5KxQAUpxVXjlmTwJAAgm7ooLYqUCNOKaPjToc17f8SkMBcBHBNMwmKYVJZa3mu+I1nfgKK0/mZGqMEFiWAqPv06dOhgaVrn0lSLQtUYKW7bVFwRi4BCeySQJbiY4ypS0e1cQgoTtuoJ3MpgRcEaGRxUHnHFWAMaosCNeO+ENktj6F9UTn+IwEJVEOAdrGcBEW3vqF+AorT+uvIHErgKAG6ph4fH184qK05kFmQH2Haorg+WjFulIAEqiEQ15Qb4dwMV5M5M3KSgOL0JBq/kED9BOKgZlkmGuK4BPXn/u4wOSHjvxx32kKNmUcJtEOAm/W0K3Tn65q2U3eK03bqypxK4CiBOKi4AnTxMwa1JQc1bgaimvwbJCABCdxKgN4YJ0HdSnG73ytOt2NvyhKYjQCOAGNQmSxFwEFtRaAirsk/wjQux2xgjEgCEtglAYQpApXeJWboG9oioDhtq77MrQROEkDgsQ5qHFQmSSFSaw9cPNK134qgrp2p+ZPAnglwk5u2hO78tC97ZtJa2RWnrdWY+ZXAGQJxUNNVjnuQrq0zP9v8q+SXi4pd+5tXhxmQQLMEaD/S5umaNluNd4rTduvOnEvgKIE4qFnPD/e0dgc1C/JzYXHW/tFqdaMEJDCCAGPXeeVGfcRP3KVCAorTCivFLElgDgKI04y1wkmoWaByIaHrzXGnc9S8cUhgnwRK15Qb3ozB3yeNtkutOG27/sy9BE4SiHMQBxWBykx+GvDaAnlN176z9murHfMjgTYI0MYxCYr25N27d21k2lweJaA4PYrFjRLohwDiNAIV4cdEqRpDZu1zceFlkIAEJDCWwHASFONNDe0SUJy2W3fmXAKjCMRFYKkpAgK1RgcVcZoLCnk0SEACEhhLgGFL9ArRlZ/hTGN/6371EVCc1lcn5kgCixCgwU5XVxzU2rr407XvpKhFDgEjlUCXBGgv0mbQznFDbmibgOK07foz9xKYRIDu/dJBvb+/r6oLPRMY6NZ3Qf5JVevOEtglAdqKTPbk5jY3uLuE0VGhFacdVaZFkcAYAjgLCFTcBQRgTV38WZAfRzdOyJgyuY8EJLBPAvQC0Y7RnmVs/T5J9FVqxWlf9WlpJDCKAAKVp0lFoNbioJKfuKeI09qGHYyC604SkMAqBErXFGHqk6BWwb5KIorTVTCbiATqI4AIHDqoNXSlk6+IZsVpfceNOZJALQR8ElQtNTF/PhSn8zM1Rgk0Q4DxWTiodKeni3/r7vQsKWXXfjOHkRmVwOoE6M7Pqh5M9OSG1tAPAcVpP3VpSSRwFYE4qAhUuslYB3VLB7Xs2s/F56qC+SMJSKBbArim3MByg52hQN0WdocFU5zusNItsgSGBGjc46AiUJkktaWDmhm35IWXQQISkEAIIEzLSVC6piHTz7vitJ+6tCQSuIkA3emMQS0d1K0EKnlIPrbKw00w/bEEJLAIAW5WP3z4cIibm1gnQS2CefNIFaebV4EZkEA9BOKg0uDHQd2iax1hmq46xWk9x4c5kcDWBFjTlLaJNsqlo7aujeXSV5wux9aYJdAkgTiovDOmizGoWwhU0ifQfees/SYPJTMtgVkJ0BakLWI5PG5iDX0SUJz2Wa+WSgI3EUAYPj4+HtzLCNR0pd0U8YQf45wylgyXRPd0Ajh3lUCHBGiH8iQo2ifEqaFfAorTfuvWkkngJgIIQ8agxkHlwpB1BW+KeOSPcUXinipOR0JzNwl0SgDHlHaAdomlowx9E1Cc9l2/lk4CNxFAIOKgMvEgzsWaAjXuCBclu/Zvqkp/LIFmCaTtoQAuHdVsNU7KuOJ0Ei53lsD+CMRBzfJOazqoOKekz8Vpy7VX91frllgC9RDImqa0BU6CqqdelsyJ4nRJusYtgU4IcFFgHdRSoGb815JFzKx9xKld+0uSNm4J1EmA8z7j3enOdxJUnfU0d64Up3MTNT4JdEyAMagRqLgZzORfOpTjTu3aX5q28UugLgJxTZkgmbanrhyamyUIKE6XoGqcEuiUQBzUdK3haCBQlxSNmbVPtz4z9w0SkMA+CNC+pMeENof2x7APAorTfdSzpZTArAToXisF6pJd/Din6crLGoezFsbIJCCB6ghwI5rufBzTPJSjuoyaoUUIKE4XwWqkEuifwFCgPjw8LOagpjsvLkr/dC2hBPZNAGFKbwk3pi4dtb9jQXG6vzq3xBKYjUApUHE1l+rij2uCm+Ks/dmqz4gkUCWB0jX1SVBVVtHimVKcLo7YBCTQNwEEKhOlGA+GQMVBnXtsKO4J3fvO2u/7WLJ0EoAAw4Q41znnM3xIMvsioDjdV31bWgksQgB3A5GKQKXrfW4HlXjjnhL/khOwFgFkpBKQwCgCeRIUOytMRyHrcifFaZfVaqEksD6BoUC9v7+ftQs+s/YVp+vXrSlKYA0C3HTGNWWcecaar5G2adRFQHFaV32YGwk0TaAUqIwNxUGda4woXXxZSsaJUU0fJmZeAkcJMAmKIUGc506COopoNxsVp7upagsqgXUIIFB5mhTjRBGmjEGdQ6CWXfsuKbVOXZqKBNYiQBuRpaNoQ7J83Frpm05dBBSnddWHuZFAFwTogs8kKZyQuRzUdPNxIZt70lUX4C2EBBolwJOgOKfpIUGcGvZNQHG67/q39BJYjAACdeig3todz4ULR8VZ+4tVmxFLYHUCtAvpDWESlK7p6lVQXYKK0+qqxAxJoB8CcVC52MRBvUWgll37t8TTD2FLIoG2CXCjiWtKoL1I70jbpTL3txJQnN5K0N9LQAJnCXDBeXx8PHTXIVAZg3qLsMQ9JRCHS0qdRe+XEqieAONMOZedBFV9Va2aQcXpqrhNTAL7JIBzShc/whJByRjUdONNJYLY5UJGPHNMtJqavvtLQALzEOBmNa4p40xz4zlP7MbSMgHFacu1Z94l0BCBUqCmiz+zc6cUg3hyEbvm91PScl8JSGA5AghTbjI5p11wfznOLcasOG2x1syzBBolEIGK+8lFiQW3rxGYGZeGc0o8BglIoC0CdOWn9wRhSm+IQQIhoDgNCd8lIIFVCAwFKl38UwVq2bV/y/jVVQpsIhKQwCsCcU05l3Oz+WonN+yWgOJ0t1VvwSWwHQFcEsagcmEiTHVQ+X3GrzrudLt6NGUJXEOAm9HcVOqaXkOw/98oTvuvY0sogSoJlAKVrnkcVETqmMBvI2y5yNm1P4aa+0hgewLDSVA5j7fPmTmoiYDitKbaMC8S2BmBCNQ8EYauvrEClYsav/dpUTs7aCxu0wRwTRGoToJquhoXz7zidHHEJiABCZwjgMDkUadTBSrd+lzgCJlYcS4dv5OABLYlwI1klo7ySVDb1kXtqStOa68h8yeBnRAYClS6+S9112ciRcav7QSVxZRAkwQiTLmxzM1okwUx04sTUJwujtgEJCCBsQQQqFnvkO6/SwI149XoJnRi1FjK7ieB9QlwPqeHI+f4+rkwxVYIKE5bqSnzKYGdEHj37t0dLwIXs3MClW79zNrXPd3JAWIxmyNAD0hcUxzT9Hg0VxAzvBoBxelqqE1IAhIYSwBnZYxAddb+WKLuJ4HtCCBM6d3gfNU13a4eWkpZcdpSbZlXCeyIwFCgPjw8HC5wQwTlrP1LY1SHv/V/CUhgWQIMtym78zOJcdlUjb11AorT1mvQ/EugYwIIVMah4rjQbX9MoNKtz/cIU7v2Oz4YLFqTBOKaOgmqyerbLNOK083Qm7AEJDCGAGPUIlBxYYYCtezaj0MzJl73kYAEliXA+ZhzkmE6nKsGCYwhoDgdQ8l9JCCBTQkwgeKcQM0EC8a18TJIQALbEignQTH0JitrbJsrU2+FgOK0lZoynxLYOYGhQL2/v39ePoouQ8ayIUzt2t/5gWLxqyBAdz49HZyXmdxYRcbMRBMEFKdNVJOZlIAEIIBA/fjx47MQpYufC2DZta849ViRwLYEuElkXVMC5yw3jwYJTCHw1eenMOUH7isBCUhgawLl2FOcGbr86UZErPI/E6kQrLz4n5dBAhKYnwDnXTmchnMN15Sxpnz+9OmTY03nx959jIrT7qvYAkqgTwJDgcqYtrg1ZYm5QOLeRLCW3/lZAhK4ngDnGy/OxWOBm0YfU3qMjNsuEVCcXiLk9xKQQLUESoF6KZN0LWZIwKV9/V4CEjhNALeUJ7dlJv6pPec450iLUE50TK8I7wnsV/6f7b63SUBx2ma9mWsJVEmA7rxcTMZkEFezfI35zXAfLpLHHNPhfvyPu4pA9SJ2jI7bJDCOwNu3b58fR3rpF+VExkv7lt/TjiB+GUOOMC3dWc7fTIKkR4Q2hHaA85v0DO0TUJy2X4eWQALVEOCCwguRWgpGLiZ5JbMRsXlP9zsXFz6PCVywmLWfOMb8hpnDXNAMEpDAdAKIRcZ2TznnpnbvI0oRwIjSY+0C5z35YL/kg/ep6Uwvvb9YjQATogwSkIAE5iTw/ffff366qDDZ8vB6EoSf2VaGH3/88fPj4+PnpzFpn5+E6/O+/I7tY8KTyHz+XdK69P7kuHwmbYMEJDCdAOfrpXNs+P2Tozk6ofKcJq1z5yptSpkf2hlDHwR+/nQQGSQgAQnMSgCXFMcj48T4zKsM7EM3HC+cTLrl0oWHM4PD+XThKX/y6nPZ1ffqyxMbyFMcmRO7uFkCEjhCAHfymnNu7G/K4QK0CbxoJ04F2hTcUvaZOqToVJxur4OA4rSOejAXEtg1AS4yjAVFlCJQuQhyoWJcGa9TIeL31PfHthM36Zy76B37ndskIIG7q8Qp59ylG0KGASEwCZzzl4RpWRfsm3aj3O7ndgkoTtutO3Muga4IIBZxQRhDyoWMCxpu6lMX/0khea3AvEbUdgXbwkhgRQKcp8OekzJ5zvUIU7ZPEabsT/z0siBQDX0QUJz2UY+WQgJdEOACxoUJUUrIxIdTM3BxWMZ2GQZQ0rhW2CYe3yWwRwI4nFNF4DlhCkPizA0j+57rLTnFnDYicZzax+3tEFCctlNX5lQCuyDARYYufdwUAjNyT4lT3JJyVYAxgIjr0ljWMfG4jwT2SmCqOD13vnGec44nMAb9kpjNvuU7N5uMUzf0QeBnfRTDUkhAAr0Q4CJTOic4oxGqwzKy37kL33B/LnpT9h/+3v8lsHcCiMdTN4vH2HCOntsft7N0PK8RpsfSdVvbBBSnbdefuZdAlwRKccqF65Q4pfC4JVwwLwUueoxp9eJ3iZTfS+A0AW4eOY/Kc/TU3pxrlx56Mbz59Pw8RXNf2xWn+6pvSyuBJggMx4OeE6fsywXw3CQKnBv2GSNimwBkJiWwIYHynDuVDXoomMx4SWwOz+3huX8qfrf3TcAxp33Xr6WTwC4IcEHDQc2SMpkkxYXx2jFsuwBnISVwJQHOraxFzPmWcw5HldclUZpkFach4XtJQHFa0vCzBCTQNAFEKi7puTFuTRfQzEugMgKIUF6ec5VVTOPZsVu/8Qo0+xLokYBuSo+1apkk8JrA0GEdnvuvf+GWPRBQnO6hli2jBBojkC5Csk0XoePQGqtAsyuBkQQQp+X5Xc7cHxmFu3VIQHHaYaVaJAm0TKAcv0Y5GDNaXrxaLpt5l4AEXhIYitPyxvTlnv63JwKK0z3VtmWVQAMEWJA7XXuIUmfYN1BpZlECVxJAnJbLUiFOc/5fGaU/64CA4rSDSrQIEuiFAE+eKZ/4xHI0itNeatdySOA4AVbZSECcTn0CVX5bPgY123xvk4DitM16M9cS6I4AY83Kx5bippQXre4KbIEkIIEDAc718slt79+/n+yeImhxXIcTrETcJgHFaZv1Zq4lUD2BKV1zdOXf398/r5WIW3rpyTLVAzCDEpDAaALlk95wT7/77rsXjzU9FxH704aUAvfc/n5XPwHFaf11ZA4l0BwBhGk565aLB68y8D0XlIeHhxcXIi4wPma0JOVnCfRPgPHl5WNRaRsQqMN2Y0iCrnyc1nNPiBv+xv/rJ/DV56dQfzbNoQQk0AKBdK1xsbh0UaE8XJDywi1FmNot10JNm0cJLEOAG9sIzvS+0O3PIv/lxCnaGtqYDP+hHTH0Q0Bx2k9dWhIJbE4AtyMXlDGZQYjmNWZ/95GABPZBAOGJAI0ILdsVbmQRpbzzMvRHQHHaX51aIglIQAISkIAEJNAsAcecNlt1ZlwCEpCABCQgAQn0R0Bx2l+dWiIJSEACEpCABCTQLAHFabNVZ8YlIAEJSEACEpBAfwQUp/3VqSWSgAQkIAEJSEACzRJQnDZbdWZcAhKQgAQkIAEJ9EdAcdpfnVoiCUhAAhKQgAQk0CwBxWmzVWfGJSABCUhAAhKQQH8EFKf91aklkoAEJCABCUhAAs0SUJw2W3VmXAISkIAEJCABCfRH4P8D6Vb8VrSxhvcAAAAASUVORK5CYII=" alt> <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">because it has vertices which are not of even degree. <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">for example \({\text{D}} \to {\text{C}} \to {\text{B}} \to {\text{E}} \to {\text{A}} \to {\text{F}} \to {\text{D}}\) <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">\(G\) </span>is planar. <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">either note from original graph in part (a) that there are no edges crossing or a redrawing with statement that there are no edges crossing. <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong>Note: </strong>Do not accept an argument based on \(e \leqslant 3v - 6\)</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">the graph is not bipartite. <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p1">the graph contains a triangle <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">to be bipartite according to line 1 of the table <span class="s1">A</span>, <span class="s1">C </span>and <span class="s1">D </span>need to be one set as <span class="s1">A </span>connects to <span class="s1">B</span>, <span class="s1">E </span>and <span class="s1">F</span>. Since <span class="s1">C </span>and <span class="s1">D </span>are connected, this is contradicted. <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><img src="data:image/png;base64,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" alt> <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A1 </em></strong>if extra line seen or missed out.</p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general. In (d), some candidates stated that G is planar because \(e \leqslant 3v - 6\). It is however important to realise that this condition is necessary for a graph to be planar but not sufficient. Some candidates stated that G is planar ‘because I have drawn it as a planar graph’ or even ‘see graph in (a)’. Candidates were expected to state that G is planar because it can be drawn with no edges crossing.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general.</p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was very well answered in general.</p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br>