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</div><h2>HL Paper 2</h2><div class="question">
<p>Boxes of mixed fruit are on sale at a local supermarket.</p>
<p>Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.</p>
<p>Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.</p>
<p>Box C contains 5 bananas and 4 kiwifruit and costs $3.00.</p>
<p>Find the cost of each type of fruit.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>let \(b\) be the cost of one banana, \(k\) the cost of one kiwifruit, and \(m\) the cost of one melon</p>
<p>attempt to set up three linear equations <strong><em>(M1)</em></strong></p>
<p>\(2b + 3k + 4m = 658\)</p>
<p>\(5b + 2k + 8m = 1232\)</p>
<p>\(5b + 4k = 300\) <strong><em>(A1)</em></strong></p>
<p>attempt to solve three simultaneous equations <strong><em>(M1)</em></strong></p>
<p>\(b = 36,{\text{ }}k = 30,{\text{ }}m = 124\)</p>
<p>banana costs ($)0.36, kiwifruit costs ($)0.30, melon costs ($)1.24 <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Given that \({\log _{10}}\left( {\frac{1}{{2\sqrt 2 }}\left( {p + 2q} \right)} \right) = \frac{1}{2}\left( {{{\log }_{10}}p + {{\log }_{10}}q} \right),{\text{ }}p > 0,{\text{ }}q > 0\), find \(p\) in terms of \(q\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>\({\log _{10}}\frac{1}{{2\sqrt 2 }}(p + 2q) = \frac{1}{2}({\log _{10}}p + {\log _{10}}q)\)</p>
<p>\({\log _{10}}\frac{1}{{2\sqrt 2 }}(p + 2q) = \frac{1}{2}{\log _{10}}pq\) <strong><em>(M1)</em></strong></p>
<p>\({\log _{10}}\frac{1}{{2\sqrt 2 }}(p + 2q) = {\log _{10}}{(pq)^{\frac{1}{2}}}\) <strong><em>(M1)</em></strong></p>
<p>\(\frac{1}{{2\sqrt 2 }}(p + 2q) = {(pq)^{\frac{1}{2}}}\) <strong><em>(A1)</em></strong></p>
<p>\({(p + 2q)^2} = 8pq\)</p>
<p>\({p^2} + 4pq + 4{q^2} = 8pq\)</p>
<p>\({p^2} - 4pq + 4{q^2} = 0\)</p>
<p>\({(p - 2q)^2} = 0\) <strong><em>M1</em></strong></p>
<p>hence \(p = 2q\) <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the method of mathematical induction to prove that \({5^{2n}} - 24n - 1\) is divisible by 576 for \(n \in {\mathbb{Z}^ + }\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(n):f(n) = {5^{2n}} - 24n - 1\) is divisible by 576 for \(n \in {\mathbb{Z}^ + }\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for \(n = 1,{\text{ }}f(1) = {5^2} - 24 - 1 = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Zero is divisible by 576, (as every non-zero number divides zero), and so P(1) is true. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for P(1) = 0 shown and zero is divisible by 576 not specified.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Ignore P(2) = 576 if P(1) = 0 is shown and zero is divisible by 576 is specified.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Assume \({\text{P}}(k)\) is true for some \(k{\text{ }}( \Rightarrow f(k) = N \times 576)\). </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Do not award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for statements such as “let </span><em style="font-family: 'times new roman', times; font-size: medium;">n</em><span style="font-family: 'times new roman', times; font-size: medium;"> = </span><em style="font-family: 'times new roman', times; font-size: medium;">k</em><span style="font-family: 'times new roman', times; font-size: medium;">”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">consider \({\text{P}}(k + 1):f(k + 1) = {5^{2(k + 1)}} - 24(k + 1) - 1\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 25 \times {5^{2k}} - 24k - 25\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 25 \times (24k + 1 + N \times 576) - 24k - 25\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 576k + 25 \times 576N\) which is a multiple of 576 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 25 \times {5^{2k}} - 600k - 25 + 600k - 24k\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 25({5^{2k}} - 24k - 1) + 576k\) (or equivalent) which is a multiple of 576 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P(1) is true and \({\text{P}}(k)\) true \( \Rightarrow {\text{P}}(k + 1)\) true, so \({\text{P}}(n)\) is true for all \(n \in {\mathbb{Z}^ + }\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> only if at least four prior marks have been awarded.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This proof by mathematical induction challenged most candidates. While most candidates were able to show that P(1) = 0, a significant number did not state that zero is divisible by 576. A few candidates started their proof by looking at P(2). It was pleasing to see that the inductive step was reasonably well done by most candidates. However many candidates committed simple algebraic errors. The most common error was to state that \({5^{2(k + 1)}} = 5{(5)^{2k}}\). The concluding statement often omitted the required implication statement and also often omitted that P(1) was found to be true.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\left| {{{\text{e}}^{{\text{i}}\theta }}} \right| = 1\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the geometric series \(1 + \frac{1}{3}{{\text{e}}^{{\text{i}}\theta }} + \frac{1}{9}{{\text{e}}^{2{\text{i}}\theta }} + \ldots {\text{ .}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the common ratio, <em>z</em>, of the series, and show that \(\left| z \right| = \frac{1}{3}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for the sum to infinity of this series.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, show that \(\sin \theta + \frac{1}{3}\sin 2\theta + \frac{1}{9}\sin 3\theta + \ldots = \frac{{9\sin \theta }}{{10 - 6\cos \theta }}\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 36.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{{\text{e}}^{{\text{i}}\theta }}} \right|{\text{ }}\left( { = \left| {\cos \theta + {\text{i}}\sin \theta } \right|} \right) = \sqrt {{{\cos }^2}\theta + {{\sin }^2}\theta } = 1\) <strong><em>M1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 36.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = \frac{1}{3}{{\text{e}}^{{\text{i}}\theta }}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| z \right| = \left| {\frac{1}{3}{{\text{e}}^{{\text{i}}\theta }}} \right| = \frac{1}{3}\) <strong><em>A1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_\infty } = \frac{a}{{1 - r}} = \frac{1}{{1 - \frac{1}{3}{{\text{e}}^{{\text{i}}\theta }}}}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_\infty } = \frac{1}{{1 - \frac{1}{3}\cos \theta - \frac{1}{3}{\text{i}}\sin \theta }}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{1 - \frac{1}{3}\cos \theta + \frac{1}{3}{\text{i}}\sin \theta }}{{\left( {1 - \frac{1}{3}\cos \theta - \frac{1}{3}{\text{i}}\sin \theta } \right)\left( {1 - \frac{1}{3}\cos \theta + \frac{1}{3}{\text{i}}\sin \theta } \right)}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{1 - \frac{1}{3}\cos \theta + \frac{1}{3}{\text{i}}\sin \theta }}{{{{\left( {1 - \frac{1}{3}\cos \theta } \right)}^2} + \frac{1}{9}{{\sin }^2}\theta }}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{1 - \frac{1}{3}\cos \theta + \frac{1}{3}{\text{i}}\sin \theta }}{{1 - \frac{2}{3}\cos \theta + \frac{1}{9}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_\infty } = \frac{1}{{1 - \frac{1}{3}{{\text{e}}^{{\text{i}}\theta }}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{1 - \frac{1}{3}{{\text{e}}^{ - {\text{i}}\theta }}}}{{\left( {1 - \frac{1}{3}{{\text{e}}^{{\text{i}}\theta }}} \right)\left( {1 - \frac{1}{3}{{\text{e}}^{ - {\text{i}}\theta }}} \right)}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{1 - \frac{1}{3}{{\text{e}}^{ - {\text{i}}\theta }}}}{{1 - \frac{1}{3}({{\text{e}}^{{\text{i}}\theta }} + {{\text{e}}^{ - {\text{i}}\theta }}) + \frac{1}{9}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{1 - \frac{1}{3}{{\text{e}}^{ - {\text{i}}\theta }}}}{{\frac{{10}}{9} - \frac{2}{3}\cos \theta }}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{1 - \frac{1}{3}(\cos \theta - {\text{i}}\sin \theta )}}{{\frac{{10}}{9} - \frac{2}{3}\cos \theta }}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">taking imaginary parts on both sides</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{3}\sin \theta + \frac{1}{9}\sin 2\theta + \ldots = \frac{{\frac{1}{3}\sin \theta }}{{\frac{{10}}{9} - \frac{2}{3}\cos \theta }}\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin \theta }}{{\frac{{10}}{9} - \frac{2}{3}\cos \theta }}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \sin \theta + \frac{1}{3}\sin 2\theta + \ldots = \frac{{9\sin \theta }}{{10 - 6\cos \theta }}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (b) and (c) were answered fairly well by quite a few candidates. In (a) many candidates failed to write the formula for the modulus of a complex number. (c) proved inaccessible for a large number of candidates. The algebraic manipulation required and the recognition of the imaginary and real parts in order to arrive at the necessary relationship were challenging for many candidates.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (b) and (c) were answered fairly well by quite a few candidates. In (a) many candidates failed to write the formula for the modulus of a complex number. (c) proved inaccessible for a large number of candidates. The algebraic manipulation required and the recognition of the imaginary and real parts in order to arrive at the necessary relationship were challenging for many candidates.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (b) and (c) were answered fairly well by quite a few candidates. In (a) many candidates failed to write the formula for the modulus of a complex number. (c) proved inaccessible for a large number of candidates. The algebraic manipulation required and the recognition of the imaginary and real parts in order to arrive at the necessary relationship were challenging for many candidates.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (b) and (c) were answered fairly well by quite a few candidates. In (a) many candidates failed to write the formula for the modulus of a complex number. (c) proved inaccessible for a large number of candidates. The algebraic manipulation required and the recognition of the imaginary and real parts in order to arrive at the necessary relationship were challenging for many candidates.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the system of equations</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[0.1x - 1.7y + 0.9z = - 4.4\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[ - 2.4x + 0.3y + 3.2z = 1.2\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[2.5x + 0.6y - 3.7z = 0.8.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Express the system of equations in matrix form.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 35.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the solution to the system of equations.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to express the system in matrix form <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> {0.1}&{ - 1.7}&{0.9} \\ <br> { - 2.4}&{0.3}&{3.2} \\ <br> {2.5}&{0.6}&{ - 3.7} <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 4.4} \\ <br> {1.2} \\ <br> {0.8} <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A1</em></strong> for a correct augmented matrix.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">either direct GDC use, attempting elimination or using an inverse matrix. <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 2.4} \\ <br> {1.6} \\ <br> { - 1.6} <br>\end{array}} \right)\) (correct to 2sf) or \(\left( {\begin{array}{*{20}{c}}<br> { - 2.40} \\ <br> {1.61} \\ <br> { - 1.57} <br>\end{array}} \right)\) (correct to 3sf) or \(\left( {\begin{array}{*{20}{c}}<br> { - \frac{{932}}{{389}}} \\ <br> {\frac{{628}}{{389}}} \\ <br> { - \frac{{612}}{{389}}} <br>\end{array}} \right)\) (exact) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was generally well done. In part (a), some candidates expressed the system of equations in the form \(XA = B\). In part (b), the overwhelming majority of candidates who used a direct GDC approach obtained the correct solution. Candidates who attempted matrix methods such as row reduction without a GDC were generally unsuccessful.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was generally well done. In part (a), some candidates expressed the system of equations in the form \(XA = B\). In part (b), the overwhelming majority of candidates who used a direct GDC approach obtained the correct solution. Candidates who attempted matrix methods such as row reduction without a GDC were generally unsuccessful.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Solve the equation \({z^3} = - 2 + 2{\text{i}}\), giving your answers in modulus-argument form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence</strong> show that one of the solutions is 1 + i when written in Cartesian form.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({z^3} = 2\sqrt 2 {{\text{e}}^{\frac{{3\pi {\text{i}}}}{4}}}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_1} = \sqrt 2 {{\text{e}}^{\frac{{\pi {\text{i}}}}{4}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">adding or subtracting \(\frac{{2\pi {\text{i}}}}{3}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_2} = \sqrt 2 {{\text{e}}^{\frac{{\pi {\text{i}}}}{4} + \frac{{2\pi {\text{i}}}}{3}}} = \sqrt 2 {{\text{e}}^{\frac{{11\pi {\text{i}}}}{{12}}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_3} = \sqrt 2 {{\text{e}}^{\frac{{\pi {\text{i}}}}{4} - \frac{{2\pi {\text{i}}}}{3}}} = \sqrt 2 {{\text{e}}^{ - \frac{{5\pi {\text{i}}}}{{12}}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> Accept equivalent solutions <em>e.g.</em> \({z_3} = \sqrt 2 {{\text{e}}^{\frac{{19\pi {\text{i}}}}{{12}}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Award marks as appropriate for solving \({(a + b{\text{i}})^3} = - 2 + 2{\text{i}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Accept answers in degrees.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\sqrt 2 {{\text{e}}^{\frac{{\pi {\text{i}}}}{4}}}{\text{ }}\left( { = \sqrt 2 \left( {\frac{1}{{\sqrt 2 }} + \frac{{\text{i}}}{{\sqrt 2 }}} \right)} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 1 + i <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept geometrical reasoning.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many students incorrectly found the argument of \({z^3}\) to be \(\arctan \left( {\frac{2}{{ - 2}}} \right) = - \frac{\pi }{4}\). Of those students correctly finding one solution, many were unable to use symmetry around the origin, to find the other two. In part (b) many students found the cube of 1 + <em>i</em> which could not be awarded marks as it was not “hence”.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Six people are to sit at a circular table. Two of the people are not to sit immediately beside each other. Find the number of ways that the six people can be seated.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">with no restrictions six people can be seated in \(5! = 120\) ways <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">we now count the number of ways in which the two restricted people will be </span><span style="font-family: times new roman,times; font-size: medium;">sitting next to each other</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">call the two restricted people \({p_1}\) and \({p_2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">they sit next to each other in two ways <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the remaining people can then be seated in \(4!\) ways <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the six may be seated \({p_1}\) and \({p_2}\) next to each other) in \(2 \times 4! = 48\) ways <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\therefore \) with </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({p_1}\)</span> and </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">( \({p_2}\)</span> not next to each other the number of ways \( = 120 - 48 = 72\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></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;">person </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({p_1}\)</span> seated at table in \(1\) way <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({p_2}\)</span></span> then sits in any of \(3\) seats (not next to </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({p_1}\)</span></span> ) <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the remaining \(4\) people can then be seated in \(4!\) ways <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\therefore \)</span> number ways with </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({p_1}\)</span></span> not next to \({p_2} = 3 \times 4! = 72\) ways <em><strong>A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> If candidate starts with \(6!\) instead of \(5!\), potentially leading to an answer of \(432\), do not penalise.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks</span>]</strong></em></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;">Very few candidates provided evidence of a clear strategy for solving such a question. The problem which was set in a circular scenario was no more difficult than an analogous linear one.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A metal rod 1 metre long is cut into 10 pieces, the lengths of which form a geometric sequence. The length of the longest piece is 8 times the length of the shortest piece. Find, to the nearest millimetre, the length of the shortest piece.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;">the pieces have lengths \(a,{\text{ }}ar,{\text{ ..., }}a{r^9}\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;">\(8a = a{r^9}{\text{ }}({\text{or }}8 = {r^9})\) <em><strong>A1</strong></em></span></p>
<p><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;">\(r = \sqrt[9]{8} = 1.259922...\) <em><strong>A1</strong></em></span></p>
<p><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;">\(a\frac{{{r^{10}} - 1}}{{r - 1}} = 1\,\,\,\,\,\left( {{\text{or }}a\frac{{{r^{10}} - 1}}{{r - 1}} = 1000} \right)\) <strong> <em>M1</em></strong></span></p>
<p><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;">\(a = \frac{{r - 1}}{{{r^{10}} - 1}} = 0.0286...\,\,\,\,\,\left( {{\text{or }}a = \frac{{r - 1}}{{{r^{10}} - 1}} = 28.6...} \right)\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;"><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;">a = 29 mm (accept 0.029 m or any correct answer regardless the units) <strong><em>A1</em></strong></span></span></p>
<p><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;"><span style="font: normal normal normal 11px/normal Helvetica; font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done by most candidates. Some candidates recurred to a diagram to comprehend the nature of the problem but a few thought it was an arithmetic sequence.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates missed earning the final A1 mark because they did not read the question instructions fully and missed the accuracy instruction to give the answer correct to the nearest mm.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find, in its simplest form, the argument of \({\left( {\sin \theta + {\text{i}}(1 - \cos \theta )} \right)^2}\) where \(\theta \) is an acute angle.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\left( {\sin \theta + {\text{i}}(1 - \cos \theta )} \right)^2} = {\sin ^2}\theta - {(1 - \cos \theta )^2} + {\text{i}}2\sin \theta (1 - \cos \theta )\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\alpha \) be the required argument.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan \alpha = \frac{{2\sin \theta (1 - \cos \theta )}}{{{{\sin }^2}\theta - {{(1 - \cos \theta )}^2}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0px 0px 0px 30px; font: 23px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{2\sin \theta (1 - \cos \theta )}}{{(1 - {{\cos }^2}\theta ) - (1 - 2\cos \theta + {{\cos }^2}\theta )}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0px 0px 0px 30px; font: 23px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{2\sin \theta (1 - \cos \theta )}}{{2\cos \theta (1 - \cos \theta )}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0px 0px 0px 30px; font: 23px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \tan \theta \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\alpha = \theta \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few candidates scored more than the first two marks in this question. Some candidates had difficulty manipulating trigonometric identities. Most candidates did not get as far as defining the argument of the complex expression.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The complex numbers \({z_1}\) and \({z_2}\) have arguments between 0 and \(\pi \) radians. Given that \({z_1}{z_2} = - \sqrt 3 + {\text{i}}\) and \(\frac{{{z_1}}}{{{z_2}}} = 2{\text{i}}\), find the modulus and argument of \({z_1}\) and of \({z_2}\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{arg(}}{z_1}{z_2}) = \frac{{5\pi }}{6}\,\,\,\,\,(150^\circ )\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arg \left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \frac{\pi }{2}\,\,\,\,\,(90^\circ )\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \arg ({z_1}) + \arg ({z_2}) = \frac{{5\pi }}{6};{\text{ }}\arg ({z_1}) - \arg ({z_2}) = \frac{\pi }{2}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneously</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arg ({z_1}) = \frac{{2\pi }}{3}{\text{ }}(120^\circ ){\text{ and }}\arg ({z_2}) = \frac{\pi }{6}{\text{ }}(30^\circ )\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept decimal approximations of the radian measures.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{z_1}{z_2}} \right| = 2 \Rightarrow \left| {{z_1}} \right|\left| {{z_2}} \right| = 2;{\text{ }}\left| {\frac{{{z_1}}}{{{z_2}}}} \right| = 2 \Rightarrow \frac{{\left| {{z_1}} \right|}}{{\left| {{z_2}} \right|}} = 2\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneously</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{z_1}} \right| = 2;{\text{ }}\left| {{z_2}} \right| = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_1} = 2i{z_2}\,\,\,\,\,2iz_2^2 = - \sqrt 3 + i\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z_2^2 = \frac{{ - \sqrt 3 + i}}{{2i}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_2} = \sqrt {\frac{{ - \sqrt 3 + i}}{{2i}}} = \frac{{\sqrt 3 }}{2} + \frac{1}{2}i{\text{ or }}{e^{\frac{\pi }{6}i}}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(allow \(0.866 + 0.5i\) or \({{\text{e}}^{0.524{\text{i}}}}\))</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_1} = - 1 + \sqrt 3 i{\text{ or }}2{{\text{e}}^{\frac{{2\pi }}{3}i}} - \) (allow −1 + 1.73i or \(2{{\text{e}}^{2.09{\text{i}}}}\)) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_1}\,\,\,\,\,\)modulus = 2, argument \( = \frac{{2\pi }}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_2}\,\,\,\,\,\)modulus = 1, argument \( = \frac{\pi }{6}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept degrees and decimal approximations to radian measure.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates generally found this question challenging. Many candidates had difficulty finding the arguments of \({z_1}{z_2}\) and \({z_1}/{z_2}\). Among candidates who attempted to solve for \({z_1}\) and \({z_2}\) in Cartesian form, many had difficulty with the algebraic manipulation involved.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Ava and Barry play a game with a bag containing one green marble and two red marbles. Each player in turn randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ava wins on her first turn.</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">Find the probability that Barry wins on his first turn.</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 probability that Ava wins in one of her first three turns.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ava eventually wins.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P(Ava wins on her first turn)}} = \frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P(Barry wins on his first turn)}} = {\left( {\frac{2}{3}} \right)^2}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = \frac{4}{9}\;\;\;( = 0.444)\) <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">\(P\)(Ava wins in one of her first three turns)</p>
<p class="p2">\( = \frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>M1A1A1</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>for adding probabilities, award <strong><em>A1 </em></strong>for a correct second term and award <strong><em>A1 </em></strong>for a correct third term.</p>
<p class="p2">Accept a correctly labelled tree diagram, awarding marks as above.</p>
<p class="p3"> </p>
<p class="p2">\( = \frac{{103}}{{243}}\;\;\;( = 0.424)\) <span class="Apple-converted-space"> </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>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P(Ava eventually wins)}} = \frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \ldots \) <strong><em>(A1)</em></strong></p>
<p>using \({S_\infty } = \frac{a}{{1 - r}}\) with \(a = \frac{1}{3}\) and \(r = \frac{2}{9}\) <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for using \({S_\infty } = \frac{a}{{1 - r}}\) and award <strong><em>(A1) </em></strong>for \(a = \frac{1}{3}\) and \(r = \frac{2}{9}\).</p>
<p> </p>
<p>\( = \frac{3}{7}\;\;\;( = 0.429)\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong><em>Total [11 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) were straightforward and were well done.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) were straightforward and were well done.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (c) and (d) were also reasonably well done.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (c) and (d) were also reasonably well done. A pleasingly large number of candidates recognized that an infinite geometric series was required in part (d).</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(z = r(\cos \alpha + {\text{i}}\sin \alpha )\), where \(\alpha \) is measured in degrees, be the solution of \({z^5} - 1 = 0\) which has the smallest positive argument.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Use the binomial theorem to expand \({(\cos \theta + {\text{i}}\sin \theta )^5}\).</p>
<p>(ii) Hence use De Moivre’s theorem to prove</p>
<p>\[\sin 5\theta = 5{\cos ^4}\theta \sin \theta - 10{\cos ^2}\theta {\sin ^3}\theta + {\sin ^5}\theta .\]</p>
<p>(iii) State a similar expression for \(\cos 5\theta \) in terms of \(\cos \theta \) and \(\sin \theta \).</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>Find the value of \(r\) and the value of \(\alpha \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using (a) (ii) and your answer from (b) show that \(16{\sin ^4}\alpha - 20{\sin ^2}\alpha + 5 = 0\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Hence express \(\sin 72^\circ \) </span>in the form \(\frac{{\sqrt {a + b\sqrt c } }}{d}\) where \(a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(i) \({(\cos \theta + {\text{i}}\sin \theta )^5}\)</p>
<p>\( = {\cos ^5}\theta + 5{\text{i}}{\cos ^4}\theta \sin \theta + 10{{\text{i}}^2}{\cos ^3}\theta {\sin ^2}\theta + \)</p>
<p>\(10{{\text{i}}^3}{\cos ^2}\theta {\sin ^3}\theta + 5{{\text{i}}^4}\cos \theta {\sin ^4}\theta + {{\text{i}}^5}{\sin ^5}\theta \) <strong><em>A1A1</em></strong></p>
<p>\(( = {\cos ^5}\theta + 5{\text{i}}{\cos ^4}\theta \sin \theta - 10{\cos ^3}\theta {\sin ^2}\theta - \)</p>
<p>\(10{\text{i}}{\cos ^2}\theta {\sin ^3}\theta + 5\cos \theta {\sin ^4}\theta + {\text{i}}{\sin ^5}\theta )\)</p>
<p> </p>
<p><strong>Note:</strong> Award first <strong><em>A1</em></strong> for correct binomial coefficients.</p>
<p> </p>
<p>(ii) \({({\text{cis}}\theta )^5} = {\text{cis}}5\theta = \cos 5\theta + {\text{i}}\sin 5\theta \) <strong><em>M1</em></strong></p>
<p>\( = {\cos ^5}\theta + 5{\text{i}}{\cos ^4}\theta \sin \theta - 10{\cos ^3}\theta {\sin ^2}\theta - 10{\text{i}}{\cos ^2}\theta {\sin ^3}\theta + \)</p>
<p>\(5\cos \theta {\sin ^4}\theta + {\text{i}}{\sin ^5}\theta \) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Previous line may be seen in (i)</p>
<p> </p>
<p>equating imaginary terms <strong><em>M1</em></strong></p>
<p>\(\sin 5\theta = 5{\cos ^4}\theta \sin \theta - 10{\cos ^2}\theta {\sin ^3}\theta + {\sin ^5}\theta \) <strong><em>AG</em></strong></p>
<p>(iii) equating real terms</p>
<p>\(\cos 5\theta = {\cos ^5}\theta - 10{\cos ^3}\theta {\sin ^2}\theta + 5\cos \theta {\sin ^4}\theta \) <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({(r{\text{cis}}\alpha )^5} = 1 \Rightarrow {r^5}{\text{cis}}5\alpha = 1{\text{cis}}0\) <strong><em>M1</em></strong></p>
<p>\({r^5} = 1 \Rightarrow r = 1\) <strong><em>A1</em></strong></p>
<p>\(5\alpha = 0 \pm 360k,{\text{ }}k \in \mathbb{Z} \Rightarrow a = 72k\) <strong><em>(M1)</em></strong></p>
<p>\(\alpha = 72^\circ \) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1A0</em></strong> if final answer is given in radians.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of \(\sin (5 \times 72) = 0\) <strong>OR</strong> the imaginary part of \(1\) is \(0\) <strong><em>(M1)</em></strong></p>
<p>\(0 = 5{\cos ^4}\alpha \sin \alpha - 10{\cos ^2}\alpha {\sin ^3}\alpha + {\sin ^5}\alpha \) <strong><em>A1</em></strong></p>
<p>\(\sin \alpha \ne 0 \Rightarrow 0 = 5{(1 - {\sin ^2}\alpha )^2} - 10(1 - {\sin ^2}\alpha ){\sin ^2}\alpha + {\sin ^4}\alpha \) <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1</em></strong> for replacing \({\cos ^2}\alpha \).</p>
<p> </p>
<p>\(0 = 5(1 - 2{\sin ^2}\alpha + {\sin ^4}\alpha ) - 10{\sin ^2}\alpha + 10{\sin ^4}\alpha + {\sin ^4}\alpha \) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1</em></strong> for any correct simplification.</p>
<p> </p>
<p>so \(16{\sin ^4}\alpha - 20{\sin ^2}\alpha + 5 = 0\) <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\sin ^2}\alpha = \frac{{20 \pm \sqrt {400 - 320} }}{{32}}\) <strong><em>M1A1</em></strong></p>
<p>\(\sin \alpha = \pm \sqrt {\frac{{20 \pm \sqrt {80} }}{{32}}} \)</p>
<p>\(\sin \alpha = \frac{{ \pm \sqrt {10 \pm 2\sqrt 5 } }}{4}\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1</em></strong> regardless of signs. Accept equivalent forms with integral denominator, simplification may be seen later.</p>
<p> </p>
<p>as \(72 > 60\), \(\sin 72 > \frac{{\sqrt 3 }}{2} = 0.866 \ldots \) we have to take both positive signs (or equivalent argument) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow verification of correct signs with calculator if clearly stated</p>
<p> </p>
<p>\(\sin 72 = \frac{{\sqrt {10 + 2\sqrt 5 } }}{4}\) <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<p><strong><em>Total [19 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (i) many candidates tried to multiply it out the binomials rather than using the binomial theorem. In parts (ii) and (iii) many candidates showed poor understanding of complex numbers and made no attempt to equate real and imaginary parts. In a some cases the correct answer to part (iii) was seen although it was unclear how it was obtained.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was poorly done. Very few candidates made a good attempt to apply De Moivre’s theorem and most of them could not even equate the moduli to obtain \(r\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was poorly done. From the few candidates that attempted it, many candidates started by writing down what they were trying to prove and made no progress.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Very few made a serious attempt to answer this question. Also very few realised that they could use the answers given in part (c) to attempt this part.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the first three terms in the expansion of \({(1 - 2x)^5}{(1 + x)^7}\) in ascending powers of <em>x</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">constant term: \(\left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 0 <br>\end{array}} \right){( - 2x)^0}\left( {\begin{array}{*{20}{c}}<br> 7 \\ <br> 0 <br>\end{array}} \right){x^0} = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">term in <em>x</em>: \(\left( {\begin{array}{*{20}{c}}<br> 7 \\ <br> 1 <br>\end{array}} \right)x + \left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 1 <br>\end{array}} \right)( - 2x) = - 3x\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">term in \({x^2}\) : \(\left( {\begin{array}{*{20}{c}}<br> 7 \\ <br> 2 <br>\end{array}} \right){x^2} + \left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 2 <br>\end{array}} \right){( - 2x)^2} + \left( {\begin{array}{*{20}{c}}<br> 7 \\ <br> 1 <br>\end{array}} \right)x\left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 1 <br>\end{array}} \right)( - 2x) = - 9{x^2}\) <strong><em>M1A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(1 - 2x)^5}{(1 + x)^7} = \left( {1 + 5( - 2x) + \frac{{5 \times 4{{( - 2x)}^2}}}{{2!}} + ...} \right)\left( {1 + 7x + \frac{{7 \times 6}}{2}{x^2} + ...} \right)\) <strong><em>M1M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (1 - 10x + 40{x^2} + ...)(1 + 7x + 21{x^2} + …)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 + 7x + 21{x^2} - 10x - 70{x^2} + 40{x^2} + …\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 - 3x - 9{x^2} + …\) <strong><em>A1A1A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Although the majority of the candidates understood the question and attempted it, excessive time was spent on actually expanding the expression without consideration of the binomial theorem. A fair amount of students confused “ascending order”, giving the last three instead of the first three terms.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The seventh, third and first terms of an arithmetic sequence form the first three terms of a geometric sequence.</p>
<p class="p1">The arithmetic sequence has first term <em>\(a\) </em>and non-zero common difference <em>\(d\)</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(d = \frac{a}{2}\).</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">The seventh term of the arithmetic sequence is \(3\). The sum of the first \(n\) terms in the arithmetic sequence exceeds the sum of the first <em>\(n\) </em>terms in the geometric sequence by at least \(200\).</p>
<p class="p1">Find the least value of <em>\(n\) </em>for which this occurs.</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>using \(r = \frac{{{u_2}}}{{{u_1}}} = \frac{{{u_3}}}{{{u_2}}}\) to form \(\frac{{a + 2d}}{{a + 6d}} = \frac{a}{{a + 2d}}\) <strong><em>(M1)</em></strong></p>
<p>\(a(a + 6d) = {(a + 2d)^2}\) <strong><em>A1</em></strong></p>
<p>\(2d(2d - a) = 0\;\;\;\)(or equivalent) <strong><em>A1</em></strong></p>
<p>since \(d \ne 0 \Rightarrow d = \frac{a}{2}\) <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 class="p1">substituting \(d = \frac{a}{2}\) into \(a + 6d = 3\) and solving for \(a\) and \(d\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(a = \frac{3}{4}\) and \(d = \frac{3}{8}\) <span class="Apple-converted-space"> </span><span class="s1"><em>(</em><strong><em>A1)</em></strong></span></p>
<p class="p1">\(r = \frac{1}{2}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(\frac{n}{2}\left( {2 \times \frac{3}{4} + (n - 1)\frac{3}{8}} \right) - \frac{{3\left( {1 - {{\left( {\frac{1}{2}} \right)}^n}} \right)}}{{1 - \frac{1}{2}}} \ge 200\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">attempting to solve for \(n\) <span class="Apple-converted-space"> </span><span class="s1">(<strong><em>M1)</em></strong></span></p>
<p class="p1">\(n \ge 31.68 \ldots \)</p>
<p class="p1">so the least value of \(n\) is 32 <strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<p class="p1"><strong><em>Total [9 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was reasonably well done. A number of candidates used \(r = \frac{{{u_1}}}{{{u_2}}} = \frac{{{u_2}}}{{{u_3}}}\) rather than \(r = \frac{{{u_2}}}{{{u_1}}} = \frac{{{u_3}}}{{{u_2}}}\). This invariably led to candidates obtaining \(r = 2\) in part (b).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), most candidates were able to correctly find the first term and the common difference for the arithmetic sequence. However a number of candidates either obtained \(r = 2\) via means described in part (a) or confused the two sequences and used \({u_1} = \frac{3}{4}\) for the geometric sequence.</p>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The arithmetic sequence \(\{ {u_n}:n \in {\mathbb{Z}^ + }\} \) has first term \({u_1} = 1.6\) and common difference <em>d</em> = 1.5. The geometric sequence \(\{ {v_n}:n \in {\mathbb{Z}^ + }\} \) has first term \({v_1} = 3\) and common ratio <em>r</em> = 1.2.</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \({u_n} - {v_n}\) in terms of <em>n</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the set of values of <em>n</em> for which \({u_n} > {v_n}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the greatest value of \({u_n} - {v_n}\). Give your answer correct to four significant figures.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({u_n} - {v_n} = 1.6 + (n - 1) \times 1.5 - 3 \times {1.2^{n - 1}}{\text{ }}( = 1.5n + 0.1 - 3 \times {1.2^{n - 1}})\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve \({u_n} > {v_n}\) numerically or graphically. <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 2.621 \ldots ,9.695 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">So \(3 \leqslant n \leqslant 9\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The greatest value of \({u_n} - {v_n}\) is 1.642. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Do not accept 1.64.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (a), most candidates were able to express \({u_n}\) and \({v_n}\) correctly and hence obtain a correct expression for \({u_n} - {v_n}\). Some candidates made careless algebraic errors when unnecessarily simplifying \({u_n}\) while other candidates incorrectly stated \({v_n}\) as \(3{(1.2)^n}\).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In parts (b) and (c), most candidates treated <em>n</em> as a continuous variable rather than as a discrete variable. Candidates should be aware that a GDC’s table feature can be extremely useful when attempting such question types.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In parts (b) and (c), most candidates treated <em>n</em> as a continuous variable rather than as a discrete variable. Candidates should be aware that a GDC’s table feature can be extremely useful when attempting such question types. In part (c), a number of candidates attempted to find the maximum value of <em>n </em>rather than attempting to find the maximum value of \({u_n} - {v_n}\).</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Each time a ball bounces, it reaches 95 % of the height reached on the previous bounce.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Initially, it is dropped from a height of 4 metres.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">What height does the ball reach after its fourth bounce?</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">How many times does the ball bounce before it no longer reaches a height of 1 metre?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">What is the total distance travelled by the ball?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">height \( = 4 \times {0.95^4}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">= 3.26 (metres) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(4 \times {0.95^n} < 1\) (<strong><em>M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({0.95^n} < 0.25\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow n > \frac{{\ln 0.25}}{{\ln 0.95}}\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow n > 27.0\)</span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not penalize improper use of inequalities.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow n = 28\) <strong> <em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>If candidates have used <em>n</em> – 1 rather than <em>n</em> throughout penalise in part (a) and treat as follow through in parts (b) and (c).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">recognition of geometric series with sum to infinity, first term of \(4 \times 0.95\) and common ratio 0.95 <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">recognition of the need to double this series and to add 4 <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">total distance travelled is \(2\left( {\frac{{4 \times 0.95}}{{1 - 0.95}}} \right) + 4 = 156{\text{ (metres)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>If candidates have used <em>n</em> – 1 rather than <span style="font: 12.0px Helvetica;"><em>n </em></span>throughout penalise in part (a) and treat as follow through in parts (b) and (c).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">recognition of a geometric series with sum to infinity, first term of 4 and common ratio 0.95 <strong>M1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">recognition of the need to double this series and to subtract 4 <strong>M1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">total distance travelled is \({\text{2}}\left( {\frac{4}{{1 - 0.95}}} \right) - 4 = 156{\text{ (metres)}}\) <strong>A1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of candidates were able to start this question and gain some marks, but only better candidates gained full marks. In part (a) the common error was to assume the wrong number of bounces and in part (b) many candidates lost marks due to rounding the inequality in the wrong direction. Part (c) was found difficult with only a limited number recognising the need for the sum to infinity of a geometric sequence and many of those did not recognise how to link the sum to infinity to the total distance travelled.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of candidates were able to start this question and gain some marks, but only better candidates gained full marks. In part (a) the common error was to assume the wrong number of bounces and in part (b) many candidates lost marks due to rounding the inequality in the wrong direction. Part (c) was found difficult with only a limited number recognising the need for the sum to infinity of a geometric sequence and many of those did not recognise how to link the sum to infinity to the total distance travelled.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of candidates were able to start this question and gain some marks, but only better candidates gained full marks. In part (a) the common error was to assume the wrong number of bounces and in part (b) many candidates lost marks due to rounding the inequality in the wrong direction. Part (c) was found difficult with only a limited number recognising the need for the sum to infinity of a geometric sequence and many of those did not recognise how to link the sum to infinity to the total distance travelled.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The three planes having Cartesian equations \(2x + 3y - z = 11,{\text{ }}x + 2y + z = 3\) and \(5x - y - z = 10\) meet at a point \(P\). Find the coordinates of \(P\)<span class="s1">.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>using technology and/or by elimination (<em>eg </em>ref on \(GDC\)) <strong><em>(M1)</em></strong></p>
<p>\(x = 1.89{\text{ }}\left( { = \frac{{17}}{9}} \right),{\text{ }}y = 1.67{\text{ }}\left( { = \frac{5}{3}} \right),{\text{ }}z = - 2.22{\text{ }}\left( { = \frac{{ - 20}}{9}} \right)\) <strong><em>A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1A1A0 </em></strong>for a set of correct answers not given exactly or to three significant figures.</p>
<p> </p>
<p><strong><em>[4 marks]<br></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Show that the complex number i is a root of the equation\[{x^4} - 5{x^3} + 7{x^2} - 5x + 6 = 0{\text{ }}.\](b) Find the other roots of this equation.</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;">(a) \({{\text{i}}^4} - 5{{\text{i}}^3} + 7{{\text{i}}^2} - 5{\text{i}} + 6 = 1 + 5{\text{i}} - 7 - 5{\text{i}} + 6\) <em><strong>M1A1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 0\) <em><strong>AG N0</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) </span><span style="font-family: times new roman,times; font-size: medium;">\({\text{i}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> root \( \Rightarrow \) \( - {\text{i}}\) is second root <em><strong>(M1)A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">moreover, \({x^4} - 5{x^3} + 7{x^2} - 5x + 6 = \left( {x - {\text{i}}} \right)\left( {x + {\text{i}}} \right)q(x)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">where \(q(x) = {x^2} - 5x + 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding roots of \(q(x)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the other two roots are \(2\) and \(3\) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Final <em><strong>A1A1</strong></em> is independent of previous work.</span></p>
<p><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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">A surprising number of candidates solved the question by dividing the expression by \(1 - i\) rather than substituting \(l\) into the expression. Many students were not aware that complex roots occur in conjugate pairs, and many did not appreciate the difference between a factor and a root. Generally the question was well done.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">From a group of five males and six females, four people are chosen.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine how many possible groups can be chosen.</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">Determine how many groups can be formed consisting of two males and two females.</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">Determine how many groups can be formed consisting of at least one female.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(\left( \begin{array}{l}<br>11\\<br>4<br>\end{array} \right)\) \( = \frac{{11 \times 10 \times 9 \times 8}}{{4 \times 3 \times 2 \times 1}} = 330\) <span class="p1"><strong><em>(M1)A1</em></strong></span></p>
<p><span class="p1"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"> </p>
<p class="p1"> </p>
<p>\(\left( \begin{array}{l}<br>5\\<br>2<br>\end{array} \right) \times \left( \begin{array}{l}<br>6\\<br>2<br>\end{array} \right) = \frac{{5 \times 4}}{{2 \times 1}} \times \frac{{6 \times 5}}{{2 \times 1}}\)<span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1">\( = 150\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p2">number of ways all men \( = \) \(\left( \begin{array}{l}<br>5\\<br>4<br>\end{array} \right) = 5\)</p>
<p class="p2">\(330 - 5 = 325\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p3"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Allow FT from answer obtained in part (a).</p>
<p class="p1"><em><strong>[2 marks]</strong></em></p>
<p class="p3"> </p>
<p class="p1"><strong>METHOD 2</strong></p>
<p>\(\left( \begin{array}{l}<br>6\\<br>1<br>\end{array} \right)\left( \begin{array}{l}<br>5\\<br>3<br>\end{array} \right) + \left( \begin{array}{l}<br>6\\<br>2<br>\end{array} \right)\left( \begin{array}{l}<br>5\\<br>2<br>\end{array} \right) + \left( \begin{array}{l}<br>6\\<br>3<br>\end{array} \right)\left( \begin{array}{l}<br>5\\<br>1<br>\end{array} \right) + \left( \begin{array}{l}<br>6\\<br>4<br>\end{array} \right)\) <em><strong>M1</strong></em></p>
<p class="p1">\( = 325\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following system of equations</p>
<p>\[2x + y + 6z = 0\]</p>
<p>\[4x + 3y + 14z = 4\]</p>
<p>\[2x - 2y + (\alpha - 2)z = \beta - 12.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find conditions on \(\alpha \) and \(\beta \) <span class="s1">for which</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>the system has no solutions;</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>the system has only one solution;</p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>the system has an infinite number of solutions.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">In the case where the number of solutions is infinite, find the general solution of the system of equations in Cartesian form.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(2x + y + 6z = 0\)</p>
<p>\(4x + 3y + 14z = 4\)</p>
<p>\(2x - 2y + (\alpha - 2)z = \beta - 12\)</p>
<p>attempt at row reduction <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\;\;\;{R_2} - 2{R_1}{\text{ and }}{R_3} - {R_1}\)</p>
<p>\(2x + y + 6z = 0\)</p>
<p>\(y + 2z = 4\)</p>
<p>\( - 3y + (\alpha - 8)z = \beta - 12\) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;{R_3} + 3{R_2}\)</p>
<p>\(2x + y + 6z = 0\)</p>
<p>\(y + 2z = 4\) <strong><em>A1</em></strong></p>
<p>\((\alpha - 2)z = \beta \)</p>
<p>(i) no solutions if \(\alpha = 2,{\text{ }}\beta \ne 0\) <strong><em>A1</em></strong></p>
<p>(ii) one solution if \(\alpha \ne 2\) <strong><em>A1</em></strong></p>
<p>(iii) infinite solutions if \(\alpha = 2,{\text{ }}\beta = 0\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept alternative methods e.g. determinant of a matrix</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1A1A0</em></strong> if all three consistent with their reduced form, <strong><em>A1A0A0</em></strong> if two or one answer consistent with their reduced form.</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>\(y + 2z = 4 \Rightarrow y = 4 - 2z\)</p>
<p>\(2x = - y - 6z = 2z - 4 - 6z = - 4z - 4 \Rightarrow x = - 2z - 2\) <strong><em>A1</em></strong></p>
<p>therefore Cartesian equation is \(\frac{{x + 2}}{{ - 2}} = \frac{{y - 4}}{{ - 2}} = \frac{z}{1}\) or equivalent <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p><strong><em>Total [9 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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;">One root of the equation \({x^2} + ax + b = 0\) is \(2 + 3{\text{i}}\) where \(a,{\text{ }}b \in \mathbb{R}\). Find the value of \(a\) and the value of \(b\).</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">substituting</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 5 + 12{\text{i}} + a(2 + 3{\text{i}}) + b = 0\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">equating real or imaginary parts <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(12 + 3a = 0 \Rightarrow a = - 4\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 5 + 2a + b = 0 \Rightarrow b = 13\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">other root is \(2 - 3{\text{i}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">considering either the sum or product of roots or multiplying factors <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(4 = - a\) (sum of roots) so \(a = - 4\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(13 = b\) (product of roots) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 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" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the set of values of <em>x</em> for which the series \(\sum\limits_{n = 1}^\infty {{{\left( {\frac{{2x}}{{x + 1}}} \right)}^n}} \) has a finite sum.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the sum in terms of <em>x</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for the series to have a finite sum, \(\left| {\frac{{2x}}{{x + 1}}} \right| < 1\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(sketch from gdc or algebraic method) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_\infty }\) exists when \( - \frac{1}{3} < x < 1\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for bounds and </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for strict inequalities.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_\infty } = \frac{{\frac{{2x}}{{x + 1}}}}{{1 - \frac{{2x}}{{x + 1}}}} = \frac{{2x}}{{1 - x}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A large number of candidates omitted the absolute value sign in the inequality in (a), or the use of the correct double inequality. Among candidates who had the correct statement, those who used their GDC were the most successful. The algebraic solution of the inequality was difficult for some candidates. In (b), quite a number of candidates found the sum of the first <em>n</em> terms of the geometric series, rather than the infinite sum of the series.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A large number of candidates omitted the absolute value sign in the inequality in (a), or the use of the correct double inequality. Among candidates who had the correct statement, those who used their GDC were the most successful. The algebraic solution of the inequality was difficult for some candidates. In (b), quite a number of candidates found the sum of the first <em>n</em> terms of the geometric series, rather than the infinite sum of the series.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The system of equations</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[2x - y + 3z = 2\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[3x + y + 2z = - 2\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[ - x + 2y + az = b\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is known to have more than one solution. Find the value of <em>a</em> and the value of <em>b</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">using row reduction (or attempting to eliminate a variable) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{ccc|c}}<br> 2&{ - 1}&3&2 \\ <br> 3&1&2&{ - 2} \\ <br> { - 1}&2&a&b <br>\end{array}} \right)\begin{array}{*{20}{l}}<br> {} \\ <br> { \to 2R2 - 3R1} \\ <br> { \to 2R3 + R1} <br>\end{array}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{ccc|c}}<br> 2&{ - 1}&3&2 \\ <br> 0&5&{ - 5}&{ - 10} \\ <br> 0&3&{2a + 3}&{2b + 2} <br>\end{array}} \right)\begin{array}{*{20}{l}}<br> {} \\ <br> { \to R2/5} \\ <br> {} <br>\end{array}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> For an algebraic solution award <strong><em>A1</em></strong> for <strong>two</strong> correct equations in two variables.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="vertical-align: bottom; font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{ccc|c}}<br> 2&{ - 1}&3&2 \\ <br> 0&1&{ - 1}&{ - 2} \\ <br> 0&3&{2a + 3}&{2b + 2} <br>\end{array}} \right)\begin{array}{*{20}{l}}<br> {} \\ <br> {} \\ <br> { \to R3 - 3R2} <br>\end{array}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{ccc|c}}<br> 2&{ - 1}&3&2 \\ <br> 0&1&{ - 1}&{ - 2} \\ <br> 0&0&{2a + 6}&{2b + 8} <br>\end{array}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept alternative correct row reductions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">recognition of the need for 4 zeroes <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so for multiple solutions <em>a</em> = – 3 and <em>b</em> = – 4 <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\begin{array}{*{20}{c}}<br> 2&{ - 1}&3 \\ <br> 3&1&2 \\ <br> { - 1}&2&a <br>\end{array}} \right| = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 2(a - 4) + (3a + 2) + 3(6 + 1) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 5a + 15 = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow a = - 3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\begin{array}{*{20}{c}}<br> 2&{ - 1}&2 \\ <br> 3&1&{ - 2} \\ <br> { - 1}&2&b <br>\end{array}} \right| = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 2(b + 4) + (3b - 2) + 2(6 + 1) = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 5b + 20 = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow b = - 4\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates attempted an algebraic approach that used excessive time but still allowed few to arrive at a solution. Of those that recognised the question should be done by matrices, some were unaware that for more than one solution a complete line of zeros is necessary.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider \(w = \frac{z}{{{z^2} + 1}}{\text{ where }}z = x + {\text{i}}y{\text{ , }}y \ne 0{\text{ and }}{z^2} + 1 \ne 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\operatorname{Im} w = 0\), show that \(\left| z \right| = 1\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Substituting \(z = x + {\text{i}}y\) to obtain \(w = \frac{{x + y{\text{i}}}}{{{{(x + y{\text{i}})}^2} + 1}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(w = \frac{{x + y{\text{i}}}}{{{x^2} - {y^2} + 1 + 2xy{\text{i}}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use of \(({x^2} - {y^2} + 1 + 2xy{\text{i)}}\) to make the denominator real. <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{ = }}\frac{{(x + y{\text{i)}}({x^2} - {y^2} + 1 - 2xy{\text{i)}}}}{{{{({x^2} - {y^2} + 1)}^2} + 4{x^2}{y^2}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\operatorname{Im} w = \frac{{y({x^2} - {y^2} + 1) - 2{x^2}y}}{{{{({x^2} - {y^2} + 1)}^2} + 4{x^2}{y^2}}}\) <strong>(A1)</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{y(1 - {x^2} - {y^2})}}{{{{({x^2} - {y^2} + 1)}^2} + 4{x^2}{y^2}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\operatorname{Im} w = 0 \Rightarrow 1 - {x^2} - {y^2} = 0\) <em>i.e.</em> \(\left| z \right| = 1{\text{ as }}y \ne 0\) <strong><em>R1AG</em></strong> <strong><em>N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(w({z^2} + 1) = z\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(w({x^2} - {y^2} + 1 + 2{\text{i}}xy) = x + yi\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Equating real and imaginary parts</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(w({x^2} - {y^2} + 1) = x{\text{ and }}2wx = 1,{\text{ }}y \ne 0\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Substituting \(w = \frac{1}{{2x}}\) to give \(\frac{x}{2} - \frac{{{y^2}}}{{2x}} + \frac{1}{{2x}} = x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - \frac{1}{{2x}}({y^2} - 1) = \frac{x}{2}\) or equivalent <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({x^2} + {y^2} = 1\), <em>i.e.</em> \(\left| z \right| = 1{\text{ as }}y \ne 0\) <strong><em>R1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a difficult question that troubled most candidates. Most candidates were able to substitute <em>z</em> = <em>x</em> + <em>y</em>i into <em>w</em> but were then unable to make any further meaningful progress. Common errors included not expanding \({(x + {\text{i}}y)^2}\) correctly or not using a correct complex conjugate to make the denominator real. A small number of candidates produced correct solutions by using \(w = \frac{1}{{z + {z^{ - 1}}}}\).</span></p>
<p> </p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The equations of three planes, are given by\[ax + 2y + z = 3\]\[ - x + \left( {a + 1} \right)y + 3z = 1\]\[ - 2x + y + \left( {a + 2} \right)z = k\]</span><span style="font-family: times new roman,times; font-size: medium;">where \(a \in \mathbb{R}\) .</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;">Given that \(a = 0\) , show that the three planes intersect at a point.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of a such that the three planes do not meet at a point.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given a such that the three planes do not meet at a point, find the value of \(k\) such that the planes meet in one line and find an equation of this line in the form \[\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {{x_0}} \\ <br> {{y_0}} \\ <br> {{z_0}} <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> l \\ <br> m \\ <br> n <br>\end{array}} \right).\]</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 0&2&1 \\ <br> { - 1}&1&3 \\ <br> { - 2}&1&2 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 1 \\ <br> k <br>\end{array}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left| {\begin{array}{*{20}{c}}<br> 0&2&1 \\ <br> { - 1}&1&3 \\ <br> { - 2}&1&2 <br>\end{array}} \right| = 0 - 2\left( { - 2 + 6} \right) + \left( { - 1 + 2} \right) = - 7\) <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">since determinant \( \ne 0 \Rightarrow \) unique solution to the system <em><strong>R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">planes intersect in a point <em><strong>AG</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> For any method, including row reduction, leading to the explicit solution \(\left( {\frac{{6 - 5k}}{7},\frac{{10 + k}}{7},\frac{{1 - 2k}}{7}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;">, award <em><strong>M1</strong></em> for an attempt at a correct method, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for two correct coordinates and <em><strong>A1</strong></em> for a third correct coordinate.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 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;">\(\left| {\begin{array}{*{20}{c}}<br> a&2&1 \\ <br> { - 1}&{a + 1}&3 \\ <br> { - 2}&1&{a + 2} <br>\end{array}} \right| = a\left( {\left( {a + 1} \right)\left( {a + 2} \right) - 3} \right) - 2\left( { - 1\left( {a + 2} \right) + 6} \right) + \left( { - 1 + 2\left( {a + 1} \right)} \right)\) <em><strong>M1(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">planes not meeting in a point \( \Rightarrow \) no unique solution <em>i.e.</em> determinant \(= 0\) <em><strong>(M1</strong></em></span><em><strong>)</strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a\left( {{a^2} + 3a - 1} \right) + \left( {2a - 8} \right) + \left( {2a + 1} \right) = 0\)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({a^3} + 3{a^2} + 3a - 7 = 0\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(</span><span style="font-family: times new roman,times; font-size: medium;">a =1\) <em><strong>A1</strong></em></span></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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&1&3 \\ <br> 0&4&4&4 \\ <br> { - 2}&1&3&k <br>\end{array}} \right)\begin{array}{*{20}{c}}<br> {{r_1} + {r_2}} \\ <br> {} \\ <br> {} <br>\end{array}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">M1</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&1&3 \\ <br> 0&4&4&4 \\ <br> 0&5&5&{6 + k} <br>\end{array}} \right)\begin{array}{*{20}{c}}<br> {2{r_1} + {r_3}} \\ <br> {} \\ <br> {} <br>\end{array}\) </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;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&1&3 \\ <br> 0&4&4&4 \\ <br> 0&0&0&{4 + 4k} <br>\end{array}} \right)\begin{array}{*{20}{c}}<br> {4{r_3} + 5{r_2}} \\ <br> {} \\ <br> {} <br>\end{array}\) </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;">for an infinite number of solutions to exist, \(4 + 4k = 0 \Rightarrow k = - 1\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x + 2y + z = 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(y + z = 1\)</span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 1 \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 1} \\ <br> 1 <br>\end{array}} \right)\) <em><strong>A</strong></em></span><em><strong><span style="font-family: times new roman,times; font-size: medium;">1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept methods involving elimination.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept any equivalent form <em>e.g. </em></span><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 0 \\ <br> 1 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> { - 1} <br>\end{array}} \right)\) or </span><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 2 \\ <br> { - 1} <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 1} \\ <br> 1 <br>\end{array}} \right)\).<br></span><span style="font-family: times new roman,times; font-size: medium;">Award <em><strong>A0</strong></em> if </span><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right)= \) </span><span style="font-family: times new roman,times; font-size: medium;">or \(r =\) is absent.</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">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">It was disappointing to see that a significant number of candidates did not appear to be well prepared for this question and made no progress at all. There were a number of schools where no candidate made any appreciable progress with the question. This was of concern since this is quite a standard problem in Mathematics HL exams. Parts (a) and (b) were intended to be answered by the use of determinants, but many candidates were not aware of this technique and used elimination. Whilst a valid method, elimination led to a long and cumbersome solution when a much more straightforward solution was available using determinants.</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;">It was disappointing to see that a significant number of candidates did not appear to be well prepared for this question and made no progress at all. There were a number of schools where no candidate made any appreciable progress with the question. This was of concern since this is quite a standard problem in Mathematics HL exams. Parts (a) and (b) were intended to be answered by the use of determinants, but many candidates were not aware of this technique and used elimination. Whilst a valid method, elimination led to a long and cumbersome solution when a much more straightforward solution was available using determinants.<br></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">It was disappointing to see that a significant number of candidates did not appear to be well prepared for this question and made no progress at all. There were a number of schools where no candidate made any appreciable progress with the question. This was of concern since this is quite a standard problem in Mathematics HL exams. Parts (a) and (b) were intended to be answered by the use of determinants, but many candidates were not aware of this technique and used elimination. Whilst a valid method, elimination led to a long and cumbersome solution when a much more straightforward solution was available using determinants. Part (c) was also a standard question but more challenging. Very few candidates made progress on (c).</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Express the sum of the first <em>n</em> positive odd integers using sigma notation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that the sum stated above is \({n^2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Deduce the value of the difference between the sum of the first 47 positive odd integers and the sum of the first 14 positive odd integers.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A number of distinct points are marked on the circumference of a circle, forming a polygon. Diagonals are drawn by joining all pairs of non-adjacent points.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show on a diagram all diagonals if there are 5 points.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that the number of diagonals is \(\frac{{n(n - 3)}}{2}\) if there are n points, where \(n > 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Given that there are more than one million diagonals, determine the least number of points for which this is possible.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable \(X \sim B(n,{\text{ }}p)\) has mean 4 and variance 3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine <em>n</em> and <em>p</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the probability that in a single experiment the outcome is 1 or 3.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\sum\limits_{k = 1}^n {(2k - 1)} \) (or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for \(\sum\limits_{n = 1}^n {(2n - 1)} \) or equivalent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) </span><strong style="font-family: 'times new roman', times; font-size: medium;">EITHER</strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2 \times \frac{{n(n + 1)}}{2} - n\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{n}{2}\left( {2 + (n - 1)2} \right){\text{ (using }}{S_n} = \frac{n}{2}\left( {2{u_1} + (n - 1)d} \right))\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{n}{2}(1 + 2n - 1){\text{ (using }}{S_n} = \frac{n}{2}({u_1} + {u_n}))\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {n^2}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({47^2} - {14^2} = 2013\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a pentagon and five diagonals <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">five diagonals (circle optional) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Each point joins to <em>n</em> – 3 other points. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a correct argument for \({n(n - 3)}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a correct argument for \(\frac{{n(n - 3)}}{2}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) attempting to solve \(\frac{1}{2}n(n - 3) > 1\,000\,000\) for <em>n</em>. <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n > 1415.7\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 1416\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) <em>np</em> = 4 and <em>npq</em> = 3 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve for <em>n</em> and <em>p</em> <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 16\) and \(p = \frac{1}{4}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(X \sim B(16,0.25)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(X = 1) = 0.0534538...( = \left( {\begin{array}{*{20}{c}}<br> {16} \\ <br> 1 <br>\end{array}} \right)(0.25){(0.75)^{15}})\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(X = 3) = 0.207876...( = \left( {\begin{array}{*{20}{c}}<br> {16} \\ <br> 3 <br>\end{array}} \right){(0.25)^3}{(0.75)^{13}})\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 1) + {\text{P}}(X = 3)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.261 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[8 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (a) (i), a large number of candidates were unable to correctly use sigma notation to express the sum of the first <em>n </em>positive odd integers. Common errors included summing \(2n - 1\) from 1 to <em>n </em>and specifying sums with incorrect limits. Parts (a) (ii) and (iii) were generally well done.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (b) (i) and (iii) were generally well done. In part (b) (iii), many candidates unnecessarily simplified their quadratic when direct GDC use could have been employed. A few candidates gave \(n > 1416\) as their final answer. While some candidates displayed sound reasoning in part (b) (ii), many candidates unfortunately adopted a ‘proof by example’ approach.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (c) was generally well done. In part (c) (ii), some candidates multiplied the two probabilities rather than adding the two probabilities.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(z = \frac{{2 - {\text{i}}}}{{1 + {\text{i}}}} - \frac{{6 + 8{\text{i}}}}{{u + {\text{i}}}}\), find the values of <em>u</em>, <em>u</em> \( \in \mathbb{R}\), such that \(\operatorname{Re} z = \operatorname{Im} z\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{2 - {\text{i}}}}{{1 + {\text{i}}}} = \frac{{1 - 3{\text{i}}}}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{6 + 8{\text{i}}}}{{u + {\text{i}}}} \times \frac{{u - {\text{i}}}}{{u - {\text{i}}}} = \frac{{6u + 8 + (8u - 6){\text{i}}}}{{{u^2} + 1}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \frac{{2 - {\text{i}}}}{{1 + {\text{i}}}} - \frac{{6 + 8u}}{{u + {\text{i}}}} = \frac{1}{2} - \frac{{6u + 8}}{{{u^2} + 1}} - \left( {\frac{3}{2} + \frac{{8u - 6}}{{{u^2} + 1}}} \right){\text{i}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\operatorname{Im} z = \operatorname{Re} z\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \frac{1}{2} - \frac{{6u + 8}}{{{u^2} + 1}} = - \frac{3}{2} - \frac{{8u - 6}}{{{u^2} + 1}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(sketch from gdc, or algebraic method) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>u</em> = −3; <em>u</em> = 2 <strong><em>A1A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{2 - {\text{i}}}}{{1 + {\text{i}}}} - \frac{{6 + 8{\text{i}}}}{{u + {\text{i}}}} = \frac{{(2 - {\text{i}})(u + {\text{i}}) - (1 + {\text{i}})(6 + 8{\text{i}})}}{{(u - 1) + {\text{i}}(u + 1)}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{(2 - {\text{i}})(u + {\text{i}}) - (1 + {\text{i}})(6 + 8{\text{i}})}}{{(u - 1) + {\text{i}}(u + 1)}} \cdot \frac{{(u - 1) - {\text{i}}(u + 1)}}{{(u - 1) - {\text{i}}(u + 1)}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{u^2} - 12u - 15 + {\text{i}}( - 3{u^2} - 16u + 9)}}{{2({u^2} + 1)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\operatorname{Re} z = \operatorname{Im} z \Rightarrow {u^2} - 12u - 15 = - 3{u^2} - 16u + 9\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>u</em> = −3; <em>u</em> = 2 <strong><em>A1A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates failed to access their GDC early enough to avoid huge algebraic manipulations, often carried out with many errors. Some candidates failed to separate and equate the real and imaginary parts of the expression obtained.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The fourth term in an arithmetic sequence is 34 and the tenth term is 76.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the first term and the common difference.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The sum of the first <em>n</em> terms exceeds 5000. Find the least possible value of <em>n</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(34 = a + 3d\) and \(76 = a + 9d\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(d = 7\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 13\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(76 = 34 + 6d\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(d = 7\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(34 = a + 21\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 13\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\frac{n}{2}\left( {26 + 7(n - 1)} \right) > 5000\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(n > 36.463 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A1A1 </em></strong>for using either an equation, a graphical approach or a numerical approach.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 37\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Both parts were very well done. In part (a), a few candidates made a careless algebraic error when attempting to find the value of <em>a</em> or <em>d</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b), a few candidates attempted to find the value of <em>n</em> for which \({u_n} > 5000\). Some candidates used the incorrect formula \({S_n} = \frac{n}{2}\left[ {{u_1} + (n - 1)d} \right]\). A number of candidates unnecessarily attempted to simplify \({S_n}\). Most successful candidates in part (b) adopted a graphical approach and communicated their solution effectively. A few candidates did not state their value of <em>n</em> as an integer.</span></p>
</div>
<br><hr><br><div class="specification">
<p>It is known that the number of fish in a given lake will decrease by 7% each year unless some new fish are added. At the end of each year, 250 new fish are added to the lake.</p>
<p>At the start of 2018, there are 2500 fish in the lake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there will be approximately 2645 fish in the lake at the start of 2020.</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>Find the approximate number of fish in the lake at the start of 2042.</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><strong>EITHER</strong></p>
<p>2019: 2500 × 0.93 + 250 = 2575    <em><strong>(M1)A1</strong></em></p>
<p>2020: 2575 × 0.93 + 250   <em><strong> M1</strong></em></p>
<p><strong>OR</strong></p>
<p>2020: 2500 × 0.93<sup>2</sup> + 250(0.93 + 1)   <em><strong>M1M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for starting with 2500, <em><strong>M1</strong></em> for multiplying by 0.93 and adding 250 twice. <em><strong>A1</strong></em>Â for correct expression. Can be shown in recursive form.</p>
<p><strong>THEN</strong></p>
<p>(= 2644.75)Â = 2645Â Â Â Â <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2020: 2500 × 0.93<sup>2</sup> + 250(0.93 + 1)<br>2042: 2500 × 0.93<sup>24</sup> + 250(0.93<sup>23</sup> + 0.93<sup>22</sup> + … + 1)  <em> <strong>(M1)(A1)</strong></em></p>
<p>\( = 2500 \times {0.93^{24}} + 250\frac{{\left( {{{0.93}^{24}} - 1} \right)}}{{\left( {0.93 - 1} \right)}}\)Â Â Â Â <em><strong>(M1)(A1)</strong></em></p>
<p>=3384Â Â Â <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> If recursive formula used, award <em><strong>M1</strong> </em>for <em>u<sub>n</sub></em> = 0.93 <em>u<sub>n</sub></em><sub>−1</sub> <strong>and</strong> <em>u</em><sub>0</sub> or <em>u</em><sub>1</sub> seen (can be awarded if seen in part (a)). Then award <em><strong>M1A1</strong> </em>for attempt to find <em>u</em><sub>24</sub> or <em>u</em><sub>25</sub> respectively (different term if other than 2500 used) (<em><strong>M1A0</strong></em> if incorrect term is being found) and <em><strong>A2</strong> </em>for correct answer.</p>
<p><strong>Note:</strong> Accept all answers that round to 3380.</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="specification">
<p class="p1">It has been suggested that in rowing competitions the time, \(T\) <span class="s1">seconds taken to complete a 2000 m </span>race can be modelled by an equation of the form \(T = a{N^b}\), where \(N\) is the number of rowers in the boat and \(a\) and \(b\) <span class="s1">are constants for rowers of a similar standard.</span></p>
<p class="p1"><span class="s1">To test this model the times for the finalists in all the 2000 m </span>men’s races at a recent Olympic games were recorded and the mean calculated.</p>
<p class="p1">The results are shown in the following table for \(N = 1\) and \(N = 2\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-26_om_06.21.32.png" alt="M16/5/MATHL/HP2/ENG/TZ1/07"></p>
</div>
<div class="specification">
<p class="p1">It is now given that the mean time in the final for boats with <span class="s1">8 </span>rowers was <span class="s1">342.08 </span>seconds.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use these results to find estimates for the value of \(a\) and the value of \(b\). Give your answers to five significant figures.</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">Use this model to estimate the mean time for the finalists in an Olympic race for boats with <span class="s1">8 </span>rowers. Give your answer correct to two decimal places.</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">Calculate the error in your estimate as a percentage of the actual value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Comment on the likely validity of the model as \(N\) <span class="s1">increases beyond 8</span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(a = 420.65\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p2">\(390.94 = a \times {2^b}\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2">\({2^b} = \frac{{390.94}}{{420.65}} = 0.929 \ldots \) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">\(b = Â - 0.10567\)Â <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(N = 8\,\,\,T = 337.67\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p3"><span class="s1"><strong>Note: <span class="Apple-converted-space">Â Â </span></strong></span>Accept 5sf answers between 337.44 and 337.67.</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"><span class="s1">\(N = 8\) </span>Percentage error 1.29% <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong>Note: <span class="Apple-converted-space">Â Â </span></strong></span>Accept negative values of the above.</p>
<p class="p4"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">likely not to be a good fit for larger values of \(N\) <span class="Apple-converted-space">Â Â </span><strong><em>R1</em></strong></p>
<p class="p2">likely to be quite a good fit for values close to 8 <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>R1</em></strong></span></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) to (c) were generally well done, although far too much inaccuracy with basic calculations.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) to (c) were generally well done, although far too much inaccuracy with basic calculations.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) to (c) were generally well done, although far too much inaccuracy with basic calculations.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) to (c) were generally well done, although far too much inaccuracy with basic calculations. Part (d) caused more difficulties as candidates frequently had insufficient analysis to gain the two marks.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The three planes</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(2x - 2y - z = 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(4x + 5y - 2z = - 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(3x + 4y - 3z = - 7\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">intersect at the point with coordinates (<em>a</em>, <em>b</em>, <em>c</em>).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of each of <em>a</em>, <em>b</em> and <em>c</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The equations of three planes are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(2x - 4y - 3z = 4\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( - x + 3y + 5z = - 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(3x - 5y - z = 6\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find a vector equation of the line of intersection of these three planes.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) use GDC or manual method to find <em>a</em>, <em>b</em> and <em>c</em> <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">obtain \(a = 2,{\text{ }}b = - 1,{\text{ }}c = 3\) (in any identifiable form) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">use GDC or manual method to solve second set of equations <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">obtain \(x = \frac{{4 - 11t}}{2};{\text{ }}y = \frac{{ - 7t}}{2};{\text{ }}z = t\) (or equivalent) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br> { - 5.5} \\ <br> { - 3.5} \\ <br> 1 <br>\end{array}} \right)\) (accept equivalent vector forms) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Final <strong><em>A1</em></strong> requires <em>r</em> = or equivalent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally well done.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Moderate success here. Some forgot that an equation must have an = sign.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(z = \cos \theta + {\text{i}}\sin \theta \) show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\operatorname{Im} \left( {{z^n} + \frac{1}{{{z^n}}}} \right) = 0,{\text{ }}n \in {\mathbb{Z}^ + }\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\operatorname{Re} \left( {\frac{{z - 1}}{{z + 1}}} \right) = 0,{\text{ }}z \ne - 1\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) using de Moivre’s theorem</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z^n} + \frac{1}{{{z^n}}} = \cos n\theta + {\text{i}}\sin n\theta + \cos n\theta - {\text{i}}\sin n\theta {\text{ }}( = 2\cos n\theta )\), imaginary part of which is 0 <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(\operatorname{Im} \left( {{z^n} + \frac{1}{{{z^n}}}} \right) = 0\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\frac{{z - 1}}{{z + 1}} = \frac{{\cos \theta + {\text{i}}\sin \theta - 1}}{{\cos \theta + {\text{i}}\sin \theta + 1}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{(\cos \theta - 1 + {\text{i}}\sin \theta )(\cos \theta + 1 - {\text{i}}\sin \theta )}}{{(\cos \theta + 1 + {\text{i}}\sin \theta )(\cos \theta + 1 - {\text{i}}\sin \theta )}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1</em></strong> for an attempt to multiply numerator and denominator by the complex conjugate of their denominator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \operatorname{Re} \left( {\frac{{z - 1}}{{z + 1}}} \right) = \frac{{(\cos \theta - 1)(\cos \theta + 1) + {{\sin }^2}\theta }}{{{\text{real denominator}}}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1</em></strong> for multiplying out the numerator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{{\cos }^2}\theta + {{\sin }^2}\theta - 1}}{{{\text{real denominator}}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong> </span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part(a) - The majority either obtained full marks or no marks here.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part(b) - This question was algebraically complex and caused some candidates to waste their efforts for little credit.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express the binomial coefficient \(\left( \begin{gathered}<br> 3n + 1 \hfill \\<br> 3n - 2 \hfill \\ <br>\end{gathered} \right)\) as a polynomial in \(n\).</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>Hence find the least value of \(n\) for which \(\left( \begin{gathered}<br> 3n + 1 \hfill \\<br> 3n - 2 \hfill \\ <br>\end{gathered} \right) > {10^6}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(\left( \begin{gathered}<br> 3n + 1 \hfill \\<br> 3n - 2 \hfill \\ <br>\end{gathered} \right) = \frac{{\left( {3n + 1} \right){\text{!}}}}{{\left( {3n - 2} \right){\text{!}}3{\text{!}}}}\)Â Â Â <em><strong>(M1)</strong></em></p>
<p>\( = \frac{{\left( {3n + 1} \right)3n\left( {3n - 1} \right)}}{{3{\text{!}}}}\)Â Â Â <strong>A1</strong></p>
<p>\( = \frac{9}{2}{n^3} - \frac{1}{2}n\) or equivalent    <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve \( = \frac{9}{2}{n^3} - \frac{1}{2}n > {10^6}\)   <em><strong>(M1)</strong></em></p>
<p>\(n > 60.57 \ldots \)Â Â Â <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Allow equality.</p>
<p>\( \Rightarrow n = 61\)Â Â Â <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Three Mathematics books, five English books, four Science books and a dictionary are to be placed on a student’s shelf so that the books of each subject remain together.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) In how many different ways can the books be arranged?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) In how many of these will the dictionary be next to the Mathematics books?</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) There are 3! ways of arranging the Mathematics books, 5! ways of arranging the English books and 4! ways of arranging the Science books. <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Then we have 4 types of books which can be arranged in 4! ways. <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(3! \times 5! \times 4 \times 4! = 414\,720\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) There are 3! ways of arranging the subject books, and for each of these there are 2 ways of putting the dictionary next to the Mathematics books. <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(3! \times 5! \times 4! \times 3! \times 2 = 207\,360\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many students added instead of multiplying. There were, however, quite a few good answers to this question.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="text-align: left;">Twelve students are to take an exam in advanced combinatorics.<br>The exam room is set out in three rows of four desks, with the invigilator at the front of the room, as shown in the following diagram.</p>
<p style="text-align: center;">INVIGILATOR</p>
<p style="text-align: left;">\[\begin{array}{*{20}{l}} {{\text{Desk 1}}}&{{\text{Desk 2}}}&{{\text{Desk 3}}}&{{\text{Desk 4}}} \\ {{\text{Desk 5}}}&{{\text{Desk 6}}}&{{\text{Desk 7}}}&{{\text{Desk 8}}} \\ {{\text{Desk 9}}}&{{\text{Desk 10}}}&{{\text{Desk 11}}}&{{\text{Desk 12}}} \end{array}\]</p>
</div>
<div class="specification">
<p>Two of the students, Helen and Nicky, are suspected of cheating in a previous exam.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of ways the twelve students may be arranged in the exam hall.</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>Find the number of ways the students may be arranged if Helen and Nicky must sit so that one is directly behind the other (with no desk in between). For example Desk 5 and Desk 9.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of ways the students may be arranged if Helen and Nicky must not sit next to each other in the same row.</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>\(12!{\text{ }}( = 479001600)\) <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>\(8 \times 2 = 16\) ways of sitting Helen and Nicky, 10! ways of sitting everyone else <strong><em>(A1)</em></strong></p>
<p>\(16 \times 10!\)</p>
<p>\( = 58060800\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\(8 \times 1 \times 10!{\text{ }}( = 29030400)\) ways if Helen sits in the front or back row</p>
<p>\(4 \times 2 \times 10!{\text{ }}( = 29030400)\) ways if Helen sits in the middle row <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for one correct value.</p>
<p> </p>
<p>\(2 \times 29030400\)</p>
<p>\( = 58060800\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>\(9 \times 2 \times 0!{\text{ }}( = 65318400)\) ways if Helen and Nicky sit next to each other <strong><em>(A1)</em></strong></p>
<p>attempt to subtract from total number of ways <strong><em>(M1)</em></strong></p>
<p>\(12! - 9 \times 2 \times 10!\)</p>
<p>\( = 413683200\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\(6 \times 10 \times 10!{\text{ }}( = 217728000)\) ways if Helen sits in column 1 or 4 <strong><em>(A1)</em></strong></p>
<p>\(6 \times 9 \times 10!{\text{ }}( = 195955200)\) ways if Helen sits in column 2 or 3 <strong><em>(A1)</em></strong></p>
<p>\(217728000 + 195955200\)</p>
<p>\( = 413683200\) <strong><em>A1</em></strong></p>
<p><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;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Three boys and three girls are to sit on a bench for a photograph.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of ways this can be done if the three girls must sit together.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of ways this can be done if the three girls must all sit apart.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the three girls can sit together in 3! = 6 ways <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">this leaves 4 ‘objects’ to arrange so the number of ways this can be done is 4! <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so the number of arrangements is \(6 \times 4! = 144\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Finding more than one position that the girls can sit <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Counting exactly four positions <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">number of ways \( = 4 \times 3! \times 3! = 144\) <strong><em>M1A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some good solutions to part (a) and certainly fewer completely correct answers to part (b). Many candidates were able to access at least partial credit, if they were showing their reasoning.</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;">Some good solutions to part (a) and certainly fewer completely correct answers to part (b). Many candidates were able to access at least partial credit, if they were showing their reasoning.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Prove, by mathematical induction, that \({7^{8n + 3}} + 2,{\text{ }}n \in \mathbb{N}\), is divisible by 5.</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">if \(n = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({7^3} + 2 = 345\) which is divisible by 5, hence true for \(n = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A0 </em></strong>for using \(n = 1\) but do not penalize further in question.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">assume true for \(n = k\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Only award the <strong><em>M1 </em></strong>if truth is assumed.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">so \({7^{8k + 3}} + 2 = 5p,{\text{ }}p \in \bullet \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">if \(n = k + 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({7^{8(k + 1) + 3}} + 2\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {7^8}{7^{8k + 3}} + 2\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {7^8}(5p - 2) + 2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {7^8}.5p - {2.7^8} + 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {7^8}.5p - 11\,529\,600\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 5({7^8}p - 2\,305\,920)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">hence if true for \(n = k\), then also true for \(n = k + 1\). Since true for \(n = 0\), then true for all \(n \in \bullet \) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Only award the <strong><em>R1 </em></strong>if the first two <strong><em>M1</em></strong>s have been awarded.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the arithmetic sequence 8, 26, 44, \( \ldots \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find an expression for the \({n^{{\text{th}}}}\) term.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Write down the sum of the first <em>n</em> terms using sigma notation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Calculate the sum of the first 15 terms.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(18n - 10\,\,\,\,\,\)(or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\sum\limits_1^n {(18r - 10)\,\,\,\,\,} \)(or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) by use of GDC or algebraic summation or sum of an AP <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sum\limits_1^{15} {(18r - 10) = 2010} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">An easy starter question, but few candidates seem to be familiar with the conventions of sigma notation.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the sum of all three-digit natural numbers that are not exactly divisible by 3.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((100 + 101 + 102 + \ldots + 999) - (102 + 105 + \ldots + 999)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{900}}{2}(100 + 999) - \frac{{300}}{2}(102 + 999)\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 329\,400\) <strong><em>A1 N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> A variety of other acceptable methods may be seen including for example \(\frac{{300}}{2}(201 + 1995)\) or \(\frac{{600}}{2}(100 + 998)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">There were many good solutions seen by a variety of different methods.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">On the day of her birth, 1st January 1998, Mary’s grandparents invested \(\$ x\) in a savings account. They continued to deposit \(\$ x\) on the first day of each month thereafter.</p>
<p class="p1">The account paid a fixed rate of <span class="s1">0.4% </span>interest per month. The interest was calculated on the last day of each month and added to the account.</p>
<p class="p1">Let \(\$ {A_n}\) be the amount in Mary’s account on the last day of the \(n{\text{th}}\) month, immediately after the interest had been added.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \({A_1}\)Â <span class="s1">and show that \({A_2} = {1.004^2}x + 1.004x\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space">Â Â </span>Write down a similar expression for \({A_3}\) and \({A_4}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space">  </span>Hence show that the amount in Mary’s account the day before she turned <span class="s1">10 </span>years old is given by \(251({1.004^{120}} - 1)x\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Write down an expression for \({A_n}\)Â </span>in terms of \(x\) on the day before Mary turned <span class="s2">18 </span>years old showing clearly the value of \(n\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Mary’s grandparents wished for the amount in her account to be at least \(\$ 20\,000\) </span>the day before she was <span class="s2">18</span>. Determine the minimum value of the monthly deposit \(\$ x\) required to achieve this. Give your answer correct to the nearest dollar.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">As soon as Mary was </span><span class="s2">18 </span>she decided to invest \(\$ 15\,000\) of this money in an account of the same type earning <span class="s2">0.4% </span>interest per month. She withdraws \(\$ 1000\) every year on her birthday to buy herself a present. Determine how long it will take until there is no money in the account.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({A_1} = 1.004x\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({A_2} = 1.004(1.004x + x)\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = {1.004^2}x + 1.004x\) Â Â </span><strong><em>AG</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Accept an argument in words for example, first deposit has been in for two months and second deposit has been in for one month.</p>
<p class="p2">Â </p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) Â Â \({A_3} = 1.004({1.004^2}x + 1.004x + x) = {1.004^3}x + {1.004^2}x + 1.004x\)Â <span class="Apple-converted-space">Â Â </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({A_4} = {1.004^4}x + {1.004^3}x + {1.004^2}x + 1.004x\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space">  \({A_{120}} = ({1.004^{120}} + {1.004^{119}} +  \ldots  + 1.004)x\)</span> <span class="Apple-converted-space">  </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{{{1.004}^{120}} - 1}}{{1.004 - 1}} \times 1.004x\) Â Â </span><strong><em>M1A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = 251({1.004^{120}} - 1)x\) Â Â </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[6 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">\({A_{216}} = 251({1.004^{216}} - 1)x{\text{ }}\left( { = x\sum\limits_{t = 1}^{216} {{{1.004}^t}} } \right)\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(251({1.004^{216}} - 1)x = 20\,000 \Rightarrow x = 58.22 \ldots \) Â Â </span>(<strong><em>A1)(M1)(A1)</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for \(251({1.004^{216}} - 1)x > 20\,000\), <strong><em>(M1) </em></strong>for attempting to solve and <strong><em>(A1) </em></strong>for \(x > 58.22 \ldots \).</p>
<p class="p2">Â </p>
<p class="p1"><span class="Apple-converted-space">\(x = 59\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Accept \(x = 58\). Accept \(x \geqslant 59\).</p>
<p class="p2">Â </p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(r = {1.004^{12}}{\text{ }}( = 1.049 \ldots )\) Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(15\,000{r^n} - 1000\frac{{{r^n} - 1}}{{r - 1}} = 0 \Rightarrow n = 27.8 \ldots \) Â Â </span>(<strong><em>A1)(M1)(A1)</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for the equation (with their value of \(r\)), <strong><em>(M1) </em></strong>for attempting to solve for \(n\) and <strong><em>(A1) </em></strong>for \(n = 27.8 \ldots \)</p>
<p class="p1">Â </p>
<p class="p1"><span class="Apple-converted-space">\(n = 28\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Accept \(n = 27\).</p>
<p class="p1">Â </p>
<p class="p1"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The complex numbers</span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"> </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">\(u\) and \(v\)</span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"> are represented by point A and point B respectively on an Argand diagram.</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Point A is rotated through \(\frac{\pi }{2}\) in the anticlockwise direction about the origin O to become point \({\text{A}}'\). Point B is rotated through \(\frac{\pi }{2}\) in the clockwise direction about O to become point \({\text{B}}'\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider \(z = r(\cos \theta + {\text{i}}\sin \theta ),{\text{ }}z \in \mathbb{C}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use mathematical induction to prove that \({z^n} = {r^n}(\cos n\theta + {\text{i}}\sin n\theta ),{\text{ }}n \in {\mathbb{Z}^ + }\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given \(u = 1 + \sqrt 3 {\text{i}}\) and \(v = 1 - {\text{i}}\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) express \(u\) and \(v\) in modulus-argument form;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) hence find \({u^3}{v^4}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plot point A and point B on the Argand diagram.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the area of triangle O\({\text{A}}'\)\({\text{B}}'\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">Given that \(u\) and \(v\) are roots of the equation \({z^4} + b{z^3} + c{z^2} + dz + e = 0\), where \(b,{\text{ }}c,{\text{ }}d,{\text{ }}e \in \mathbb{R}\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">find the values of \(b,{\text{ }}c,{\text{ }}d\) and \(e\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">let \({\text{P}}(n)\) be the proposition \({z^n} = {r^n}(\cos n\theta + {\rm{i}}\sin n\theta ),n \in {¢^ + }\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">let \(n = 1 \Rightarrow \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{LHS}} = r(\cos \theta + {\text{i}}\sin \theta )\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{RHS}} = r(\cos \theta + {\text{i}}\sin \theta ),{\text{ }}\therefore {\text{P}}(1)\) is true <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">assume true for \(n = k \Rightarrow {r^k}{(\cos \theta + {\text{i}}\sin \theta )^k} = {r^k}\left( {\cos (k\theta ) + {\text{i}}\sin (k\theta )} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:<em> </em></strong>Only award the <strong><em>M1 </em></strong>if truth is assumed.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">now show \(n = k\) true implies \(n = k + 1\) also true</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({r^{k + 1}}{(\cos \theta + {\text{i}}\sin \theta )^{k + 1}} = {r^{k + 1}}{(\cos \theta + {\text{i}}\sin \theta )^k}(\cos \theta + {\text{i}}\sin \theta )\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {r^{k + 1}}\left( {\cos (k\theta ) + {\text{i}}\sin (k\theta )} \right)(\cos \theta + {\text{i}}\sin \theta )\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {r^{k + 1}}\left( {\cos (k\theta )\cos \theta - \sin (k\theta )\sin \theta + {\text{i}}\left( {\sin (k\theta )\cos \theta + \cos (k\theta )\sin \theta } \right)} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {r^{k + 1}}\left( {\cos (k\theta + \theta ) + {\text{i}}\sin (k\theta + \theta )} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {r^{k + 1}}\left( {\cos (k + 1)\theta + {\text{i}}\sin (k + 1)\theta } \right) \Rightarrow n = k + 1\) is true <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(k)\) true implies \({\text{P}}(k + 1)\) true and \({\text{P}}(1)\) is true, therefore by mathematical induction statement is true for \(n \geqslant 1\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Only award the final <strong><em>R1 </em></strong>if the first 4 marks have been awarded.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(u = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = \sqrt 2 {\text{cis}}\left( { - \frac{\pi }{4}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> Accept 3 sf answers only. Accept equivalent forms.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> Accept \(2{e^{\frac{\pi }{3}i}}\) and \(\sqrt 2 {e^{ - \frac{\pi }{4}i}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({u^3} = {2^3}{\text{cis}}(\pi ) = - 8\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({v^4} = 4{\text{cis}}( - \pi ) = - 4\) <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;">\({u^3}{v^4} = 32\) <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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> Award <strong><em>(M1) </em></strong>for an attempt to find \({u^3}\) and \({v^4}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> Accept equivalent forms.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-15_om_17.58.50.png" alt> <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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong>A1 </strong>if A or \({\text{1 + }}\sqrt 3 i\) and B or \(1 - i\) are in their correct quadrants, are aligned vertically and it is clear that \(\left| u \right| > \left| v \right|\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Area \( = \frac{1}{2} \times 2 \times \sqrt 2 \times \sin \left( {\frac{{5\pi }}{{12}}} \right)\) <strong><em>M1A1</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;">\( = 1.37{\text{ }}\left( { = \frac{{\sqrt 2 }}{4}\left( {\sqrt 6 + \sqrt 2 } \right)} \right)\) <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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> Award <strong><em>M1A0A0 </em></strong>for using \(\frac{{7\pi }}{{12}}\)<span style="font: 21.5px 'Times New Roman';">.</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\((z - 1 + {\text{i}})(z - 1 - {\text{i}}) = {z^2} - 2z + 2\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1 </em></strong>for recognition that a complex conjugate is also a root.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {z - 1 - \sqrt 3 {\text{i}}} \right)\left( {z - 1 + \sqrt 3 {\text{i}}} \right) = {z^2} - 2z + 4\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {{z^2} - 2z + 2} \right)\left( {{z^2} - 2z + 4} \right) = {z^4} - 4{z^3} + 10{z^2} - 12z + 8\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1 </em></strong>for an attempt to expand two quadratics.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Phil takes out a bank loan of $150 000 to buy a house, at an annual interest rate of 3.5%. The interest is calculated at the end of each year and added to the amount outstanding.</p>
</div>
<div class="specification">
<p>To pay off the loan, Phil makes annual deposits of $<em>P </em>at the end of every year in a savings account, paying an annual interest rate of 2% . He makes his first deposit at the end of the first year after taking out the loan.</p>
</div>
<div class="specification">
<p>David visits a different bank and makes a single deposit of $<em>Q </em>, the annual interest rate being 2.8%.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the total value of Phil’s savings after 20 years is \(\frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}\).</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>Given that Phil’s aim is to own the house after 20 years, find the value for \(P\) to the nearest dollar.</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>David wishes to withdraw $5000 at the end of each year for a period of \(n\) years. Show that an expression for the minimum value of \(Q\) is</p>
<p>\(\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find the minimum value of \(Q\) that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(150000 \times {1.035^{20}}\) <strong><em>(M1)(A1)</em></strong></p>
<p>\( = \$ 298468\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only accept answers to the nearest dollar. Accept $298469.</p>
<p> </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>attempt to look for a pattern by considering 1 year, 2 years <em>etc </em><strong><em>(M1)</em></strong></p>
<p>recognising a geometric series with first term \(P\) and common ratio 1.02 <strong><em>(M1)</em></strong></p>
<p><strong>EITHER</strong></p>
<p>\(P + 1.02P + \ldots + {1.02^{19}}P{\text{ }}\left( { = P(1 + 1.02 + \ldots + {{1.02}^{19}})} \right)\) <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>explicitly identify \({u_1} = P,{\text{ }}r = 1.02\) and \(n = 20\) (may be seen as \({S_{20}}\)). <strong><em>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p>\({s_{20}} = \frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}\) <strong><em>AG</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>\(24.297 \ldots P = 298468\) <strong><em>(M1)(A1)</em></strong></p>
<p>\(P = 12284\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept answers which round to 12284.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>\(Q({1.028^n}) = 5000(1 + 1.028 + {1.028^2} + {1.028^3} + \ldots + {1.028^{n - 1}})\) <strong><em>M1A1</em></strong></p>
<p>\(Q = \frac{{5000\left( {1 + 1.028 + {{1.028}^2} + {{1.028}^3} + ... + {{1.028}^{n - 1}}} \right)}}{{{{1.028}^n}}}\) <strong><em>A1</em></strong></p>
<p>\( = \frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>the initial value of the first withdrawal is \(\frac{{5000}}{{1.028}}\) <strong><em>A1</em></strong></p>
<p>the initial value of the second withdrawal is \(\frac{{5000}}{{{{1.028}^2}}}\) <strong><em>R1</em></strong></p>
<p>the investment required for these two withdrawals is \(\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}}\) <strong><em>R1</em></strong></p>
<p>\(Q = \frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[3 Marks]</em></strong></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>sum to infinity is \(\frac{{\frac{{5000}}{{1.028}}}}{{1 - \frac{1}{{1.028}}}}\) <strong><em>(M1)(A1)</em></strong></p>
<p>\( = 178571.428 \ldots \)</p>
<p>so minimum amount is $178572 <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept answers which round to $178571 or $178572.</p>
<p> </p>
<p><strong><em>[3 Marks]</em></strong></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</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;">A system of equations is given below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(x + 2y - z = 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(2x + y + z = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \( - x + 4y + az = 4\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the value of <em>a </em>so that the system does not have a unique solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that the system has a solution for any value of <em>a</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\left[ \begin{array}{l}x + 2y - z = 2\\2x + y + z = 1\\ - x + 4y + az = 4\end{array} \right.\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </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;">\( \to \left[ \begin{array}{l}x + 2y - z = 2\\ - 3y + 3z = - 3\\6y + (a - 1)z = 6\end{array} \right.\) <strong><em>M1A1</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;">\( \to \left[ \begin{array}{l}x + 2y - z = 2\\ - 3y + 3z = - 3\\(a + 5)z = 0\end{array} \right.\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(or equivalent)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">if not a unique solution then \(a = - 5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> The first <strong><em>M1 </em></strong>is for attempting to eliminate a variable, the first <strong><em>A1 </em></strong>for obtaining two expression in just two variables (plus <em>a</em>), and the second <strong><em>A1 </em></strong>for obtaining an expression in just <em>a </em>and one other variable</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[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) if \(a = - 5\) there are an infinite number of solutions as last equation always true <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and if \(a \ne - 5\) there is a unique solution <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence always a solution <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>[2 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;"><strong><em>Total [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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\omega = \cos \theta + {\text{i}}\sin \theta \) . Find, in terms of \(\theta \) , the modulus and argument of \({(1 - {\omega ^2})^ * }\) .<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"> </p>
<p> </p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px;"> </span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(1 - {\omega ^2})^ * } = {(1 - {\text{cis}}\,2\theta )^ * } = {\left( {(1 - \cos 2\theta ) - {\text{i}}\sin 2\theta } \right)^ * }\) <em style="font-style: italic;"><strong style="font-weight: bold;">M1A1</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (1 - \cos 2\theta ) + {\text{i}}\sin 2\theta \) <em style="font-style: italic;"><strong style="font-weight: bold;">A1</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{{(1 - {\omega ^2})}^ * }} \right| = \sqrt {{{(1 - \cos 2\theta )}^2} + {{\sin }^2}2\theta } \left( { = \sqrt {{{(2{{\sin }^2}\theta )}^2} + {{(2\sin \theta \cos \theta )}^2}} } \right)\) <strong style="font-weight: bold;"> </strong><em style="font-style: italic;"><strong style="font-weight: bold;">M1</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left| {2\sin \theta } \right|\) <strong style="font-weight: bold;"> </strong><em style="font-style: italic;"><strong style="font-weight: bold;">A1</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arg \left( {{{(1 - {\omega ^2})}^ * }} \right) = \alpha \Rightarrow \tan \alpha = \cot (\theta )\) <strong><em>M1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\alpha = \frac{\pi }{2} - \theta \) <strong><em>A1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore:</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">modulus is \(2\left| {\sin \theta } \right|\) and argument is \(\frac{\pi }{2} - \theta {\text{ or }}\frac{\pi }{2} - \theta \pm \pi \)</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 11px;"> </span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept modulus is </span><span style="font-size: medium;"><span style="font-family: 'times new roman', times;"><span style="font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; font-family: 'times new roman', times;">2sin</span></span><em><span style="font-family: 'times new roman', times;"><span style="font-variant: normal; font-weight: normal; line-height: normal; font-family: Helvetica;"><span style="font-family: 'times new roman', times;">θ</span></span></span></em></span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font: 12.5px Helvetica;"> </span>and argument is</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{\pi }{2} - \theta \)</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(1 - {\varpi ^2})^ * } = {(1 - {\text{cis}}\,2\theta )^ * } = {\left( {(1 - \cos 2\theta ) - {\text{i}}\sin 2\theta } \right)^ * }\) <em style="font-style: italic;"><strong style="font-weight: bold;">M1A1</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (1 - \cos 2\theta ) + {\text{i}}\sin 2\theta \) <em style="font-style: italic;"><strong style="font-weight: bold;">A1</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (1 - 1 + 2{\sin ^2}\theta ) + 2{\text{i}}\sin \theta \cos \theta \) <em style="font-style: italic;"><strong style="font-weight: bold;">M1</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(1 - {\varpi ^2})^ * } = {\left( {1 - {{(\cos \theta + {\text{i}}\sin \theta )}^2}} \right)^ * }\) <strong><em>M1A1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {(1 - {\cos ^2}\theta + {\sin ^2}\theta - 2{\text{i}}\sin \theta \cos \theta )^ * }\) <strong> <em>A1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2{\sin ^2}\theta + 2{\text{i}}\sin \theta \cos \theta \) <strong> <em>M1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2\sin \theta (\sin \theta + {\text{i}}\cos \theta )\) <em style="font-style: italic;"><strong style="font-weight: bold;">(M1)</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2\sin \theta \left( {\cos \left( {\frac{\pi }{2} - \theta } \right) + {\text{i}}\sin \left( {\frac{\pi }{2} - \theta } \right)} \right)\) <strong><em>A1A1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2\sin \theta \,{\text{cis}}\left( {\frac{\pi }{2} - \theta } \right)\)</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>t</strong>herefore:</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">modulus is \(2\left| {\sin \theta } \right|\) and argument is \(\frac{\pi }{2} - \theta {\text{ or }}\frac{\pi }{2} - \theta \pm \pi \)</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Accept modulus is </span><span style="font-family: 'times new roman', times;"><span>2sin</span></span><em style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: medium;"><span style="font-family: 'times new roman', times;"><span style="font-variant: normal; font-weight: normal; line-height: normal; font-family: Helvetica;"><span style="font-family: 'times new roman', times;">θ</span></span></span></em><span style="font-family: 'times new roman', times; font-size: medium;"> and argument is \(\frac{\pi }{2} - \theta \) .</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"> </p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was the most challenging question in part A with just a few candidates scoring full marks. This question showed that many candidates have difficulties with algebraic manipulations, application of De Moivre’s theorem and use of trigonometric identities. Although some candidates managed to calculate the square of a complex number, many failed to write down its conjugate or made algebraic errors which lead to wrong results in many cases<span style="color: #3600fc;">. </span>Just a few candidates were able to calculate the modulus and the argument of the complex number.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Fifteen boys and ten girls sit in a single line.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In how many ways can they be seated in a single line so that the boys and girls are in two separate groups?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two boys and three girls are selected to go the theatre. In how many ways can this selection be made?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">number of arrangements of boys is \(15!\) and number of arrangements of girls is \(10!\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">total number of arrangements is \(15! \times 10! \times 2( = 9.49 \times {10^{18}})\) <strong><em>M1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>If 2 is omitted, award <strong><em>(A1)M1A0</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">number of ways of choosing two boys is \(\left( {\begin{array}{*{20}{c}}<br> {15} \\ <br> 2 <br>\end{array}} \right)\) and the number of ways of choosing three girls is </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> {10} \\ <br> 3 <br>\end{array}} \right)\)</span> <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">number of ways of choosing two boys and three girls is \(\left( {\begin{array}{*{20}{c}}<br> {15} \\ <br> 2 <br>\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}<br> {10} \\ <br> 3 <br>\end{array}} \right) = 12600\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A good number of correct answers were seen to this question, but a significant number of candidates forgot to multiply by 2 in part (a) and in part (b) the most common error was to add the combinations rather than multiply them.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A good number of correct answers were seen to this question, but a significant number of candidates forgot to multiply by 2 in part (a) and in part (b) the most common error was to add the combinations rather than multiply them.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z_1} = {(1 + {\text{i}}\sqrt 3 )^m}{\text{ and }}{z_2} = {(1 - {\text{i}})^n}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the modulus and argument of \({z_1}\) and \({z_2}\) in terms of <em>m</em> and <em>n</em>, respectively.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence</strong>, find the smallest positive integers <em>m</em> and <em>n</em> such that \({z_1} = {z_2}\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\left| {1 + {\text{i}}\sqrt 3 } \right| = 2{\text{ or }}\left| {1 - {\text{i}}} \right| = \sqrt 2 \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arg (1 + {\text{i}}\sqrt 3 ) = \frac{\pi }{3}{\text{ or }}\arg (1 - {\text{i}}) = - \frac{\pi }{4}\,\,\,\,\,\left( {{\text{accept }}\frac{{7\pi }}{4}} \right)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{z_1}} \right| = {2^m}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{z_2}} \right| = {\sqrt 2 ^n}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arg ({z_1}) = m\arctan \sqrt 3 = m\frac{\pi }{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arg ({z_2}) = n\arctan ( - 1) = n\frac{{ - \pi }}{4}\,\,\,\,\,\left( {{\text{accept }}n\frac{{7\pi }}{4}} \right)\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({2^m} = {\sqrt 2 ^n} \Rightarrow n = 2m\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(m\frac{\pi }{3} = n\frac{{ - \pi }}{4} + 2\pi k\) , where <em>k</em> is an integer <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow m\frac{\pi }{3} + n\frac{\pi }{4} = 2\pi k\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow m\frac{\pi }{3} + 2m\frac{\pi }{4} = 2\pi k\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{5}{6}m\pi = 2\pi k\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow m = \frac{{12}}{5}k\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The smallest value of <em>k</em> such that <em>m</em> is an integer is 5, hence</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>m</em> =12 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>n</em> = 24. <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [14 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) of this question was answered fairly well by candidates who attempted this question. The main error was the sign of the argument of \({z_2}\). Few candidates attempted part (b), and of those who did, most scored the first two marks for equating the modulii. Only a very small number equated the arguments correctly using \(2\pi k\). </span></p>
</div>
<br><hr><br><div class="question">
<p>Use mathematical induction to prove that \({\left( {1 - a} \right)^n} > 1 - na\) for \(\left\{ {n\,{\text{:}}\,n \in {\mathbb{Z}^ + },\,n \geqslant 2} \right\}\) where \(0 < a < 1\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>Let \({{\text{P}}_n}\) be the statement: \({\left( {1 - a} \right)^n} > 1 - na\) for some \({n \in {\mathbb{Z}^ + },\,n \geqslant 2}\) where \(0 < a < 1\) consider the case \(n = 2{\text{:}}\,\,\,{\left( {1 - a} \right)^2} = 1 - 2a + {a^2}\)   <em><strong>M1</strong></em></p>
<p>\( > 1 - 2a\) because \({a^2} < 0\). Therefore \({{\text{P}}_2}\) is true   <em><strong>R1</strong></em></p>
<p>assume \({{\text{P}}_n}\) is true for some \(n = k\)</p>
<p>\({\left( {1 - a} \right)^k} > 1 - ka\)Â Â Â <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Assumption of truth must be present. Following marks are not dependent on this <em><strong>M1</strong></em>.</p>
<p><strong>EITHER</strong></p>
<p>consider \({\left( {1 - a} \right)^{k + 1}} = \left( {1 - a} \right){\left( {1 - a} \right)^k}\)   <em><strong>M1</strong></em></p>
<p>\( > 1 - \left( {k + 1} \right)a + k{a^2}\)Â Â Â <em><strong>A1</strong></em></p>
<p>\( > 1 - \left( {k + 1} \right)a \Rightarrow {{\text{P}}_{k + 1}}\) is true (<strong>as</strong> \(k{a^2} > 0\))Â Â Â <em><strong>R1</strong></em></p>
<p><strong>OR</strong></p>
<p>multiply both sides by \(\left( {1 - a} \right)\) (which is positive)   <em><strong>M1</strong></em></p>
<p>\({\left( {1 - a} \right)^{k + 1}} > \left( {1 - ka} \right)\left( {1 - a} \right)\)</p>
<p>\({\left( {1 - a} \right)^{k + 1}} > 1 - \left( {k + 1} \right)a + k{a^2}\)Â Â Â <em><strong>A1</strong></em></p>
<p>\({\left( {1 - a} \right)^{k + 1}} > 1 - \left( {k + 1} \right)a \Rightarrow {{\text{P}}_{k + 1}}\)Â is true (<strong>as</strong>Â \(k{a^2} > 0\))Â Â Â <em><strong>R1</strong></em></p>
<p><strong>THEN</strong></p>
<p>\({{\text{P}}_2}\) is true \({{\text{P}}_k}\) is true \( \Rightarrow {{\text{P}}_{k + 1}}\) is true so \({{\text{P}}_n}\) true for all \(n > 2\) (or equivalent)   <em><strong>R1</strong></em></p>
<p><strong>Note</strong>: Only award the last <em><strong>R1</strong> </em>if at least four of the previous marks are gained including the <em><strong>A1</strong></em>.</p>
<p><em><strong>[7 marks]</strong></em></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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the constant term in the expansion of \({\left( {x - \frac{2}{x}} \right)^4}{\left( {{x^2} + \frac{2}{x}} \right)^3}\).</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;">\({\left( {x - \frac{2}{x}} \right)^4} = {x^4} - 8{x^2} + 24 - \frac{{32}}{{{x^2}}} + \frac{{16}}{{{x^4}}}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\left( {{x^2} + \frac{2}{x}} \right)^3} = {x^6} + 6{x^3} + 12 + \frac{8}{{{x^3}}}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p> <strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Accept unsimplified or uncalculated coefficients in the constant term</span></p>
<p> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = 24 \times 12\) <strong><em>(M1)(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = 288\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[7 marks]</span><br></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many correct answers were seen, although most candidates used rather inefficient methods (e.g. expanding the brackets in multiple steps). In a very few cases candidates used the binomial theorem to obtain the answer quickly.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the polynomial \(p(x) = {x^4} + a{x^3} + b{x^2} + cx + d\), where <em>a</em>, <em>b</em>, <em>c</em>, <em>d</em> \( \in \mathbb{R}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that 1 + i and 1 − 2i are zeros of \(p(x)\), find the values of <em>a</em>, <em>b</em>, <em>c</em> and <em>d</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">1 + i is a zero \( \Rightarrow \) 1 – i is a zero <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">1 – 2i is a zero \( \Rightarrow \) 1 + 2i is a zero <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {x - (1 - {\text{i}})} \right)\left( {x - (1 + {\text{i}})} \right) = ({x^2} - 2x + 2)\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {x - (1 - 2{\text{i}})} \right)\left( {x - (1 + 2{\text{i}})} \right) = ({x^2} - 2x + 5)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p(x) = ({x^2} - 2x + 2)({x^2} - 2x + 5)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {x^4} - 4{x^3} + 11{x^2} - 14x + 10\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = - 4,{\text{ }}b = 11,{\text{ }}c = - 14,{\text{ }}d = 10\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p(1 + {\text{i}}) = - 4 + ( - 2 + 2{\text{i}})a + (2{\text{i}})b + (1 + {\text{i}})c + d\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p(1 + {\text{i}}) = 0 \Rightarrow \left\{ {\begin{array}{*{20}{c}}<br> { - 4 - 2a + c + d = 0} \\ <br> {2a + 2b + c = 0} <br>\end{array}} \right.\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p(1 - 2{\text{i}}) = - 7 + 24{\text{i}} + ( - 11 + 2{\text{i}})a + ( - 3 - 4{\text{i}})b + (1 - 2{\text{i}})c + d\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p(1 - 2{\text{i}}) = 0 \Rightarrow \left\{ {\begin{array}{*{20}{c}}<br> { - 7 - 11a - 3b + c + d = 0} \\ <br> {24 + 2a - 4b - 2c = 0} <br>\end{array}} \right.\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> a \\ <br> b \\ <br> c \\ <br> d <br>\end{array}} \right) = {\left( {\begin{array}{*{20}{c}}<br> { - 2}&0&1&1 \\ <br> 2&2&1&0 \\ <br> { - 11}&{ - 3}&1&1 \\ <br> 2&{ - 4}&{ - 2}&0 <br>\end{array}} \right)^{ - 1}}\left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 0 \\ <br> 7 \\ <br> { - 24} <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 4} \\ <br> {11} \\ <br> { - 14} \\ <br> {10} <br>\end{array}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = - 4,{\text{ }}b = 11,{\text{ }}c = - 14,{\text{ }}d = 10\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates attempted this question, using different approaches. The most successful approach was the method of complex conjugates and the product of linear factors. Candidates who used this method were in general successful whereas candidates who attempted direct substitution and separation of real and imaginary parts to obtain four equations in four unknowns were less successful because either they left the work incomplete or made algebraic errors that led to incorrect answers.</span></p>
</div>
<br><hr><br><div class="specification">
<p>The 3rd term of an arithmetic sequence is 1407 and the 10th term is 1183.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the first term and the common difference of the sequence.</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>Calculate the number of positive terms in the sequence.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>u</em><sub>1</sub> + 2<em>d</em> = 1407,  <em>u</em><sub>1</sub> + 9<em>d</em> = 1183  <em><strong>(M1)(A1)</strong></em></p>
<p><em>u</em><sub>1</sub> = 1471, <em>d</em> = −32  <em><strong> A1A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>1471 + (<em>n</em> − 1)(−32) > 0   <em><strong>(M1)</strong></em></p>
<p>⇒ <em>n</em> < \(\frac{{1471}}{{32}} + 1\)</p>
<p><em>n</em> < 46.96…   <em><strong>(A1)</strong></em></p>
<p>so 46 positive terms   <em><strong> A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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 class="p1">The sum of the second and third terms of a geometric sequence is 96.</p>
<p class="p1">The sum to infinity of this sequence is 500<span class="s1">.</span></p>
<p class="p2">Find the possible values for the common ratio, \(r\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><span class="Apple-converted-space">\(ar + a{r^2} = 96\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><strong>Note: <span class="Apple-converted-space">Â Â </span></strong>Award <strong><em>A1 </em></strong>for any valid equation involving \(a\) and \(r\), <em>eg</em>, \(\frac{{a(1 - {r^3})}}{{1 - r}} - a = 96\).</p>
<p class="p3"><span class="Apple-converted-space">\(\frac{a}{{1 - r}} = 500\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p3"><strong>EITHER</strong></p>
<p class="p3">attempting to eliminate \(a\) to obtain \(500r(1 - {r^2}) = 96\) (or equivalent in unsimplified form) Â Â Â <span class="s2">(</span><strong><em>M1</em></strong><span class="s2">)</span></p>
<p class="p3"><strong>OR</strong></p>
<p class="p3">attempting to obtain \(a = \frac{{96}}{{r + {r^2}}}\) and \(a = 500(1 - r)\)Â <span class="Apple-converted-space">Â Â </span><span class="s2">(</span><strong><em>M1</em></strong><span class="s2">)</span></p>
<p class="p3"><strong>THEN</strong></p>
<p class="p3">attempting to solve for \(r\) <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p3"><span class="s3">\(r = 0.2{\text{ }}\left( { = \frac{1}{5}} \right)\)Â </span>or \(r = 0.885{\text{ }}\left( { = \frac{{\sqrt {97} Â - 1}}{{10}}} \right)\)Â <span class="Apple-converted-space">Â Â </span><strong><em>A1A1</em></strong></p>
<p class="p3"><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Reasonably well done. Quite a number of candidates included a solution outside \( - 1 < r < 1\).</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the set of values of <em>k</em> for which the following system of equations has</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">no solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>x</em> + 2<em>y</em> − 3<em>z</em> = <em>k</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">3<em>x</em> + <em>y</em> + 2<em>z</em> = 4</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">5<em>x</em> + 7<em>z</em> = 5</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Describe the geometrical relationship of the three planes represented by this system of equations.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&2&{ - 3}&k \\ <br> 3&1&2&4 \\ <br> 5&0&7&5 <br>\end{array}} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({R_1} - 2{R_2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> { - 5}&0&{ - 7}&{k - 8} \\ <br> 3&1&2&4 \\ <br> 5&0&7&5 <br>\end{array}} \right)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({R_1} + {R_3}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 0&0&0&{k - 3} \\ <br> 3&1&2&4 \\ <br> 5&0&7&5 <br>\end{array}} \right)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence no solutions if \(k \in \mathbb{R}\), \(k \ne 3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Two planes meet in a line and the third plane is parallel to that line.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><em><strong><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates realised that some form of row operations was appropriate here but arithmetic errors were fairly common. Many candidates whose arithmetic was correct gave their answer as <em>k</em> = 3 instead of \(k \ne 3\) . Very few candidates gave a correct answer to (b) with most failing to realise that stating that there was no common point was not enough to answer the question.</span></p>
</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;">A complex number <em>z</em> is given by \(z = \frac{{a + {\text{i}}}}{{a - {\text{i}}}},{\text{ }}a \in \mathbb{R}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine the set of values of <em>a </em>such that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) <em>z </em>is real;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) <em>z </em>is purely imaginary<em>.</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that \(\left| z \right|\) is constant for all values of <em>a</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\frac{{a + {\text{i}}}}{{a - {\text{i}}}} \times \frac{{a + {\text{i}}}}{{a + {\text{i}}}}\) <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;">\( = \frac{{{a^2} - 1 + 2a{\text{i}}}}{{{a^2} + 1}}{\text{ }}\left( { = \frac{{{a^2} - 1}}{{{a^2} + 1}} + \frac{{2a}}{{{a^2} + 1}}{\text{i}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) <em>z</em> is real when \(a = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) <em>z</em> is purely imaginary when \(a = \pm 1\) <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: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A0A1A0 </em></strong>for \(\frac{{{a^2} - 1 + 2a{\text{i}}}}{{{a^2} - 1}}{\text{ }}\left( { = 1 + \frac{{2a}}{{{a^2} - 1}}{\text{i}}} \right)\) leading to \(a = 0\) in (i).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to find either \(\left| z \right|\) or \({\left| z \right|^2}\) by expanding and simplifying</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\left| z \right|^2} = \frac{{{{\left( {{a^2} - 1} \right)}^2} + 4{a^2}}}{{{{\left( {{a^2} + 1} \right)}^2}}} = \frac{{{a^4} + 2{a^2} + 1}}{{{{\left( {{a^2} + 1} \right)}^2}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{{\left( {{a^2} + 1} \right)}^2}}}{{{{\left( {{a^2} + 1} \right)}^2}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\left| z \right|^2} = 1 \Rightarrow \left| z \right| = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| z \right| = \frac{{\left| {a + {\text{i}}} \right|}}{{\left| {a - {\text{i}}} \right|}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| z \right| = \frac{{\sqrt {{a^2} + 1} }}{{\sqrt {{a^2} + 1} }} \Rightarrow \left| z \right| = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was reasonably well done. When multiplying and dividing by the conjugate of \(a - {\text{i}}\), some candidates incorrectly determined their denominator as \({a^2} - 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b), a significant number of candidates were able to correctly expand and simplify \(\left| z \right|\) although many candidates appeared to not understand the definition of \(\left| z \right|\).</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the term in \({x^5}\) in the expansion of \((3x + A){(2x + B)^6}\).</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 class="p1"><span style="font-family: 'times new roman', times; font-size: medium;">Mina and Norbert each have a fair cubical die with faces labelled 1, 2, 3, 4, 5 and 6; they throw</span></p>
<p class="p1"><span style="font-family: 'times new roman', times; font-size: medium;">it to decide if they are going to eat a cookie.</span></p>
<p class="p2"><span style="font-family: 'times new roman', times; font-size: medium;">Mina throws her die just once and she eats a cookie if she throws a four, a five or a six.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: 20px; background-color: #f7f7f7;">Norbert throws his die six times and each time eats a cookie if he throws a five or a six.</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the probability that five cookies are eaten.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {A\left( \begin{array}{l}6\\5\end{array} \right){2^5}B + 3\left( \begin{array}{l}6\\4\end{array} \right){2^4}{B^2}} \right){x^5}\) <strong><em>M1A1A1</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;">\( = \left( {192AB + 720{B^2}} \right){x^5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \frac{1}{6},{\text{ }}A = \frac{3}{6}\left( { = \frac{1}{2}} \right),{\text{ }}B = \frac{4}{6}\left( { = \frac{2}{3}} \right)\) <strong><em>A1A1A1</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;">probability is \(\frac{4}{{81}}{\text{ }}( = 0.0494)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</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;">P (5 eaten) =P (M eats 1) P (N eats 4) + P (M eats 0) P (N eats 5) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{2}\left( \begin{array}{l}6\\4\end{array} \right){\left( {\frac{1}{3}} \right)^4}{\left( {\frac{2}{3}} \right)^2} + \frac{1}{2}\left( \begin{array}{l}6\\5\end{array} \right){\left( {\frac{1}{3}} \right)^5}\left( {\frac{2}{3}} \right)\) <strong><em>(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{4}{{81}}{\text{ }}( = 0.0494)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>In a trial examination session a candidate at a school has to take 18 examination papers including the physics paper, the chemistry paper and the biology paper. No two of these three papers may be taken consecutively. There is no restriction on the order in which the other examination papers may be taken.</p>
<p>Find the number of different orders in which these 18 examination papers may be taken.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>consideration of all papers</p>
<p>all papers may be sat in 18! ways <strong><em>A1</em></strong></p>
<p>number of ways of positioning “pairs” of science subjects \( = 3! \times 17!\) <strong><em>A1</em></strong></p>
<p>but this includes two copies of each “triple” <strong><em>(R1)</em></strong></p>
<p>number of ways of positioning “triplets” of science subjects \( = 3! \times 16!\) <strong><em>A1</em></strong></p>
<p>hence number of arrangements is \(18! - 3! \times 17! + 3! \times 16!\) <strong><em>M1A1</em></strong></p>
<p>\(( = 4.39 \times {10^{15}})\)</p>
<p><strong>METHOD 2</strong></p>
<p>consideration of all the non-science papers <strong><em>(M1)</em></strong></p>
<p>hence all non-science papers can be sat in 15! ways <strong><em>A1</em></strong></p>
<p>there are \(16 \times 15 \times 14{\text{ }}( = 3360)\) ways of positioning the three science papers <strong><em>(M1)A1</em></strong></p>
<p>hence the number of arrangements is \(16 \times 15 \times 14 \times 15!{\text{ (}} = 4.39 \times {10^{15}})\) <strong><em>(M1)A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>consideration of all papers</p>
<p>all papers may be sat in 18! ways <strong><em>A1</em></strong></p>
<p>number of ways of positioning exactly two science subjects \( = 3! \times 15! \times 16 \times 15\) <strong><em>M1A1</em></strong></p>
<p>number of ways of positioning “triplets” of science subjects \( = 3! \times 16!\) <strong><em>A1</em></strong></p>
<p>hence number of arrangements is \(18! - 3! \times 16! - 3! \times 15! \times 16 \times 15\) <strong><em>M1A1</em></strong></p>
<p>\(( = 4.39 \times {10^{15}})\)</p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The interior of a circle of radius 2 cm is divided into an infinite number of sectors. The areas of these sectors form a geometric sequence with common ratio <em>k</em>. The angle of the first sector is \(\theta \) radians.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(\theta = 2\pi (1 - k)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The perimeter of the third sector is half the perimeter of the first sector.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>k</em> and of \(\theta \).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) the area of the first sector is \(\frac{1}{2}{2^2}\theta \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the sequence of areas is \(2\theta ,{\text{ }}2k\theta ,{\text{ }}2{k^2}\theta \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the sum of these areas is \(2\theta (1 + k + {k^2} + \ldots )\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{2\theta }}{{1 - k}} = 4\pi \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \(\theta = 2\pi (1 - k)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept solutions where candidates deal with angles instead of area.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the perimeter of the first sector is \(4 + 2\theta \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the perimeter of the third sector is \(4 + 2{k^2}\theta \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the given condition is \(4 + 2{k^2}\theta = 2 + \theta \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">which simplifies to \(2 = \theta (1 - 2{k^2})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">eliminating \(\theta \), obtain cubic in <em>k</em>: \(\pi (1 - k)(1 - 2{k^2}) - 1 = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">or equivalent</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solve for <em>k</em> = 0.456 and then \(\theta = 3.42\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [12 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a disappointingly answered question.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part(a) - Many candidates correctly assumed that the areas of the sectors were proportional to their angles, but did not actually state that fact.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part(b) - Few candidates seem to know what the term ‘perimeter’ means.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the following system of equations.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\[{\log _{x + 1}}y = 2\]\[{\log _{y + 1}}x = \frac{1}{4}\]<br></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;">\({\log _{x + 1}}y = 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _{y + 1}}x = \frac{1}{4}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so \({\left( {x + 1} \right)^2} = y\)</span><span style="font-family: times new roman,times; font-size: medium;"> </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;">\({\left( {y + 1} \right)^{\frac{1}{4}}} = x\) <em><strong>A1</strong></em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({x^4} - 1 = {\left( {x + 1} \right)^2}\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = - 1\)</span><span style="font-family: times new roman,times; font-size: medium;">, not possible <em><strong>R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 1.70\), \(y = 7.27\) <em><strong>A1A1</strong></em></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;">1</span><br><span style="font-family: times new roman,times; font-size: medium;">\({\left( {{x^2} + 2x + 2} \right)^{\frac{1}{4}}} - x = 0\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve or graph of LHS <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x =1.70\), \(y = 7.27\) <em><strong>A1A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></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;">This question was well answered by a significant number of candidates. There was evidence of good understanding of logarithms. The algebra required to solve the problem did not intimidate candidates and the vast majority noticed the necessity of technology to solve the final equation. Not all candidates recognized the extraneous solution and there were situations where a rounded value of \(x\) was used to calculate the value of \(y\) leading to an incorrect solution.</span></p>
</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 number of ways in which seven different toys can be given to three children, if the youngest is to receive three toys and the others receive two toys each.</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">the number of ways of allocating presents to the first child is \(\left( \begin{array}{c}7\\3\end{array} \right)\left( {{\rm{or }}\left( \begin{array}{l}7\\2\end{array} \right)} \right)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">multiplying by \(\left( \begin{array}{c}4\\2\end{array} \right)\left( {{\rm{or}}\left( \begin{array}{l}5\\3\end{array} \right){\rm{or}}\left( \begin{array}{l}5\\2\end{array} \right)} \right)\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1 </em></strong>for multiplication of combinations.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( \begin{array}{c}7\\3\end{array} \right)\left( \begin{array}{c}4\\2\end{array} \right) = 210\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p class="p1">Two distinct roots for the equation \({z^4} - 10{z^3} + a{z^2} + bz + 50 = 0\) are \(c + {\text{i}}\) <span class="s1">and \(2 + {\text{i}}d\) </span>where \(a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{R},{\text{ }}d > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the other two roots in terms of \(c\) and \(d\).</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">Find the value of \(c\) and the value of \(d\).</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 class="p1">other two roots are \(c - {\text{i}}\) and \(2 - {\text{i}}d\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">use of sum of roots <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p2">\(2c + 4 = 10\)</p>
<p class="p2">\(c = 3\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">use of product of roots <span class="Apple-converted-space">Â Â </span><strong><em>M1</em></strong></p>
<p class="p1">product is \((c + {\text{i}})(c - {\text{i}})(2 + {\text{i}}d)(2 - {\text{i}}d)\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1">\(({c^2} + 1)(4 + {d^2})\left[ { = 10(4 + {d^2})} \right] = 50\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space">Â Â </span></strong>The line above can be awarded if they have used their value of \(c\).</p>
<p class="p1">\(d = 1\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">\({z^4} - 10{z^3} + a{z^2} + bz + 50 = ({z^2} - 2cz + {c^2} + 1)({z^2} - 4z + 4 + {d^2})\) <span class="Apple-converted-space">Â Â </span><strong><em>M1A1</em></strong></p>
<p class="p1">compare constant terms or coefficients of \({z^3}\) <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p4">\(4 + 2c = 10\)</p>
<p class="p4">\(({c^2} + 1)(4 + {d^2}) = 50\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">\(c = 3,{\text{ }}d = 1\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1A1</em></strong></span></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most students using the sum and product of roots were able to work this problem through. There were many candidates who were attempting to multiply out, with varying degrees of success.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most students using the sum and product of roots were able to work this problem through. There were many candidates who were attempting to multiply out, with varying degrees of success.</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;">When \({\left( {1 + \frac{x}{2}} \right)^2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(n \in \mathbb{N}\) , is expanded in ascending powers of \(x\) , the coefficient of \({x^3}\) is \(70\).</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Find the value of \(n\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Hence, find the coefficient of \({x^2}\) .</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;">(a) coefficient of </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({x^3}\)</span> is \(\left( {\begin{array}{*{20}{c}}<br> n \\ <br> 3 <br>\end{array}} \right){\left( {\frac{1}{2}} \right)^3} = 70\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1(A1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{n!}}{{3!\left( {n - 3} \right)!}} \times \frac{1}{8} = 70\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow \frac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{{48}} = 70\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 16\)</span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b)</span> <span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> {16} \\ <br> 2 <br>\end{array}} \right){\left( {\frac{1}{2}} \right)^2} = 30\) </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;"> </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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to answer this question well.</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">Find the constant term in the expansion of \({\left( {4{x^2} - \frac{3}{{2x}}} \right)^{12}}\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">attempting a valid method to obtain the required term in the expansion <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Valid methods include an attempt to expand, noting the behaviour of the powers of \(x\), use of the general binomial expansion term, use of a ratio etc.</p>
<p class="p2">Â </p>
<p class="p1">identifying the correct term <span class="Apple-converted-space">Â Â </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\left( {\begin{array}{*{20}{c}} {12} \\ 8 \end{array}} \right) \times {4^4} \times {\left( { - \frac{3}{2}} \right)^8}{\text{ }}\left( { = 495 \times {4^4} \times {{\left( { - \frac{3}{2}} \right)}^8}} \right)\) Â Â </span><strong><em>M1A1</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong><span class="s1">Accept \(\left( {\begin{array}{*{20}{c}} {12} \\ 4 \end{array}} \right)\)</span><span class="s1">.</span></p>
<p class="p3">Â </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>M1 </em></strong>for the product of a binomial coefficient, a power of 4 and either a power of \( - \frac{3}{2}\) or \(\frac{3}{2}\).</p>
<p class="p2">Â </p>
<p class="p1"><span class="Apple-converted-space">\( = 3\,247\,695\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the values of <em>k </em>for which the following system of equations has no solutions and the value of <em>k </em>for the system to have an infinite number of solutions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x - 3y + z = 3\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x + 5y - 2z = 1\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[16y - 6z = k\]</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that the system of equations can be solved, find the solutions in the form of a vector equation of a line, <strong><em>r </em></strong>= <strong><em>a </em></strong>+ <span style="font: 12.5px Helvetica;">λ</span><strong><em>b </em></strong>, where the components of <strong><em>b </em></strong>are integers.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The plane \( \div \) is parallel to both the line in part (b) and the line \(\frac{{x - 4}}{3} = \frac{{y - 6}}{{ - 2}} = \frac{{z - 2}}{0}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \div \)</span> contains the point (1, 2, 0) , show that the Cartesian equation of ÷ is 16<em>x </em>+ 24<em>y </em>− 11<em>z </em>= 64 .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The <em>z-</em>axis meets the plane </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \div \)</span> at the point P. Find the coordinates of P.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the angle between the line \(\frac{{x - 2}}{3} = \frac{{y + 5}}{4} = \frac{z}{2}\) and the plane </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \div \)</span> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">in augmented matrix form \(\left| {\begin{array}{*{20}{c}}<br> 1&{ - 3}&1&3 \\ <br> 1&5&{ - 2}&1 \\ <br> 0&{16}&{ - 6}&k <br>\end{array}} \right|\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find a line of zeros <strong><em>(M1)<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({r_2} - {r_1}\left| {\begin{array}{*{20}{c}}<br> 1&{ - 3}&1&3 \\ <br> 0&8&{ - 3}&{ - 2} \\ <br> 0&{16}&{ - 6}&k <br>\end{array}} \right|\) <em><strong>(A1)</strong></em><strong><em><br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({r_3} - 2{r_2}\left| {\begin{array}{*{20}{c}}<br> 1&{ - 3}&1&3 \\ <br> 0&8&{ - 3}&{ - 2} \\ <br> 0&{0}&{0}&{k + 4} <br>\end{array}} \right|\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">there is an infinite number of solutions when \(k = - 4\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">there is no solution when</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k \ne - 4,{\text{ }}(k \in \mathbb{R})\) <strong><em>R1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Approaches other than using the augmented matrix are acceptable. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">using \(\left| {\begin{array}{*{20}{c}}<br> 1&{ - 3}&1&3 \\ <br> 0&8&{ - 3}&{ - 2} \\ <br> 0&{0}&{ 0}&{k + 4} <br>\end{array}} \right|\) and letting \(\boldsymbol{z} = \lambda \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(8y - 3\lambda = - 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow y = \frac{{3\lambda - 2}}{8}\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x - 3y + z = 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x - \left( {\frac{{9\lambda - 6}}{8}} \right) + \lambda = 3\) <strong> <em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 8x - 9\lambda + 6 + 8\lambda = 24\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x = \frac{{18 + \lambda }}{8}\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {\frac{{18}}{8}} \\ <br> { - \frac{2}{8}} \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> {\frac{1}{8}} \\ <br> {\frac{3}{8}} \\ <br> 1 <br>\end{array}} \right)\) <strong> <em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \left( {\begin{array}{*{20}{c}}<br> {\frac{9}{4}} \\ <br> { - \frac{1}{4}} \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 3 \\ <br> 8 <br>\end{array}} \right)\) <strong> <em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept equivalent answers. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[7 marks]</span><br></em></strong></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;">recognition that \(\left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> { - 2} \\ <br> 0 <br>\end{array}} \right)\) is parallel to the plane <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">direction normal of the plane is given by \(\left| {\begin{array}{*{20}{c}}<br> \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\ <br> 1&3&8 \\ <br> 3&{ - 2}&0 <br>\end{array}} \right|\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">= 16<em><strong>i</strong></em> + 24<em><strong>j</strong></em> – 11<em><strong>k</strong></em> <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Cartesian equation of the plane is given by 16<em>x</em> + 24<em>y</em> –11<em>z</em> = <em>d</em> and a point which fits this equation is (1, 2, 0) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 16 + 48 = d\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>d</em> = 64 <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">hence Cartesian equation of plane is 16<em style="font-style: italic;">x</em> + 24<em style="font-style: italic;">y</em> –11<em style="font-style: italic;">z</em> = 64 <strong><em>AG</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative methods using dot product.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]<br></em></strong></span></p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">the plane crosses the <em>z</em>-axis when <em>x</em> = <em>y</em> = 0 <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">coordinates of P are \(\left( {0,\,0,\, - \frac{{64}}{{11}}} \right)\) <strong><em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for stating \(z = - \frac{{64}}{{11}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept. \(\left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 0 \\ <br> { - \frac{{64}}{{11}}} <br>\end{array}} \right)\)<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">recognition that the angle between the line and the direction normal is given by:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 4 \\ <br> 2 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> {16} \\ <br> {24} \\ <br> {-11} <br>\end{array}} \right) = \sqrt {29} \sqrt {953} \cos \theta \) where \(\theta \) is the angle between the line and the normal vector <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 122 = \sqrt {29} \sqrt {953} \cos \theta \) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \theta = 42.8^\circ {\text{ (0.747 radians)}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">hence the angle between the line and the plane is 90° – 42.8° = 47.2° (0.824 radians) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks] </em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept use of the formula <em><strong>a</strong></em>.<em><strong>b</strong></em> = \(\left| {} \right.\)<em><strong>a</strong></em>\(\left. {} \right|\)\(\left| {} \right.\)<em><strong>b</strong></em>\(\left| {\sin \theta } \right.\) .</span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the sum of all the multiples of 3 between 100 and 500.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">102 + 105 + … + 498 <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">so number of terms = 133 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{133}}{2}(2 \times 102 + 132 \times 3)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">= 39900 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (102 + 498) \times \frac{{133}}{2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">= 39900 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sum\limits_{n = 34}^{166} {3n} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">= 39900 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(500 \div 3 = 166.666...{\text{ and }}100 \div 3 = 33.333...\)</span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(102 + 105 + ... + 498 = \sum\limits_{n = 1}^{166} {3n} - \sum\limits_{n = 1}^{33} {3n} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sum\limits_{n = 1}^{166} {3n} = 41583\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sum\limits_{n = 1}^{33} {3n} = 1683\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">the sum is 39900 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates got full marks in this question. Some mistakes were detected when trying to find the number of terms of the arithmetic sequence, namely the use of the incorrect value <em>n</em> = 132 ; a few interpreted the question as the sum of multiples between the 100th and 500th terms. Occasional application of geometric series was attempted.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The complex number \(z = - \sqrt 3 + {\text{i}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the modulus and argument of <em>z</em> , giving the argument in degrees.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the cube root of <em>z</em> which lies in the first quadrant of the Argand diagram, giving your answer in Cartesian form.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the smallest positive integer <em>n</em> for which \({z^n}\) is a positive real number.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bmod (z) = 2,{\text{ }}\arg (z) = 150^\circ \) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({z^{\frac{1}{3}}} = {2^{\frac{1}{3}}}(\cos 50^\circ + {\text{i}}\sin 50^\circ )\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.810 + 0.965{\text{i}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we require to find a multiple of 150 that is also a multiple of 360, so by any method, <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>n</em> = 12 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Only award 1 mark for part (c) if </span><em style="font-family: 'times new roman', times; font-size: medium;">n</em><span style="font-family: 'times new roman', times; font-size: medium;"> = 12 is based on \(\arg (z) = - 30\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="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) (i) Find the sum of all integers, between 10 and 200, which are divisible by 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;"> (ii) Express the above sum using sigma notation.</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;">An arithmetic sequence has first term 1000 and common difference of −6 . The sum of the first <em>n </em>terms of this sequence is negative.</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) Find the least value of <em>n</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) \(n = 27\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_{27}} = \frac{{14 + 196}}{2} \times 27\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2835\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_{27}} = \frac{{27}}{2}(2 \times 14 + 26 \times 7)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2835\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 3</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_{27}}\sum\limits_{n = 1}^{27} {7 + 7n} \) <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;">\( = 2835\) <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) \(\sum\limits_{n = 1}^{27} {(7 + 7n)} \) or equivalent <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \(\sum\limits_{n = 2}^{28} {7n} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[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) \(\frac{n}{2}\left( {2000 - 6(n - 1)} \right) < 0\) <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;">\(n > 334.333\)</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;">\(n = 335\) <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: 20.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept working with equalities.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 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>[2 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;"><strong><em>Total [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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A geometric sequence has a first term of 2 and a common ratio of 1.05. Find the value of the smallest term which is greater than 500.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2 \times {1.05^{n - 1}} > 500\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n - 1 > \frac{{\log {\text{ }}250}}{{\log {\text{ }}1.05}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n - 1 > 113.1675…\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 115\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({u_{115}} = 521\) <strong><em>A1 N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept graphical solution with appropriate sketch.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks] </em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates misread the question and stopped at showing that the required term was the \({115^{{\text{th}}}}\). </span></p>
</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;">Write down the quadratic expression \(2{x^2} + x - 3\) as the product of two linear factors.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, or otherwise, find the coefficient of \(x\) in the expansion of \({\left( {2{x^2} + x - 3} \right)^8}\) .</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><span style="font-family: times new roman,times; font-size: medium;">\(2{x^2} + x - 3 = \left( {2x + 3} \right)\left( {x - 1} \right)\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept \(2\left( {x + \frac{3}{2}} \right)\left( {x - 1} \right)\)</span><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;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Either of these may be seen in (b) and if so <em><strong>A1</strong></em> should be awarded.</span></p>
<p><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;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\left( {2{x^2} + x - 3} \right)^8} = {\left( {2x + 3} \right)^8}{\left( {x - 1} \right)^8}\) <em><strong>M1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = \left( {{3^8} + 8\left( {{3^7}} \right)\left( {2x} \right) + ...} \right)\left( {{{\left( { - 1} \right)}^8} + 8{{\left( { - 1} \right)}^7}\left( x \right) + ...} \right)\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">coefficient of \(x = {3^8} \times 8 \times {\left( { - 1} \right)^7} + {3^7} \times 8 \times 2 \times {\left( { - 1} \right)^8}\) <em><strong>M1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">= −17 496 <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Under ft, final <em><strong>A1</strong></em> can only be achieved for an integer answer.</span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></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;">\({\left( {2{x^2} + x - 3} \right)^8} = {\left( {3 - \left( {x - 2{x^2}} \right)} \right)^8}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = {3^8} + 8\left( { - \left( {x - 2{x^2}} \right)\left( {{3^7}} \right) + ...} \right)\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">coefficient of \(x = 8 \times \left( { - 1} \right) \times {3^7}\) <em><strong>M1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">= −17 496 <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Under ft, final <em><strong>A1</strong></em> can only be achieved for an integer answer.</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;">[4 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 struggled to find an efficient approach to this problem by applying the Binomial Theorem. A disappointing number of candidates attempted the whole expansion which was clearly an unrealistic approach when it is noted that the expansion is to the 8<sup>th</sup> power. The fact that some candidates wrote down Pascal’s Triangle suggested that they had not studied the Binomial Theorem in enough depth or in a sufficient variety of contexts.</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;">Many candidates struggled to find an efficient approach to this problem by applying the Binomial Theorem. A disappointing number of candidates attempted the whole expansion which was clearly an unrealistic approach when it is noted that the expansion is to the 8<sup>th</sup> power. The fact that some candidates wrote down Pascal’s Triangle suggested that they had not studied the Binomial Theorem in enough depth or in a sufficient variety of contexts.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex number \(z = \frac{{2 + 7{\text{i}}}}{{6 + 2{\text{i}}}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express \(z\) in the form \(a + {\text{i}}b\), where \(a,\,b \in \mathbb{Q}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact value of the modulus of \(z\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the argument of \(z\), giving your answer to 4 decimal places.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(z = \frac{{\left( {2 + 7{\text{i}}} \right)}}{{\left( {6 + 2{\text{i}}} \right)}} \times \frac{{\left( {6 - 2{\text{i}}} \right)}}{{\left( {6 - 2{\text{i}}} \right)}}\)Â Â Â <em><strong>(M1)</strong></em></p>
<p>\( = \frac{{26 + 38{\text{i}}}}{{40}} = \left( {\frac{{13 + 19{\text{i}}}}{{20}} = 0.65 + 0.95{\text{i}}} \right)\)Â Â Â <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>attempt to use \(\left| z \right| = \sqrt {{a^2} + {b^2}} \)Â Â Â <em><strong>(M1)</strong></em></p>
<p>\(\left| z \right| = \sqrt {\frac{{53}}{{40}}} \left( { = \frac{{\sqrt {530} }}{{20}}} \right)\) or equivalent    <em><strong>A1</strong></em></p>
<p><strong>Note:<em> A1</em></strong> is only awarded for the correct exact value.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>arg \(z\) = arg(2 + 7i) − arg(6 + 2i)    <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong></p>
<p>arg \(z\) = arctan\(\left( {\frac{{19}}{{13}}} \right)\)      <em><strong>(M1)</strong></em></p>
<p><strong>THEN</strong></p>
<p>arg \(z\) = 0.9707 (radians) (= 55.6197 degrees)   <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Only award the last <em><strong>A1</strong> </em>if 4 decimal places are given.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>The coefficient of \({x^2}\) in the expansion of \({\left( {\frac{1}{x} + 5x} \right)^8}\) is equal to the coefficient of \({x^4}\) in the expansion of \({\left( {a + 5x} \right)^7},{\text{ }}a \in \mathbb{R}\). Find the value of \(a\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>\(^8{C_r}{\left( {\frac{1}{x}} \right)^{8 - r}}{(5x)^r} = {}^8Cr{\left( 5 \right)^r}{x^{2r - 8}}\) <strong><em>(M1)</em></strong></p>
<p>\(r = 5\) <strong><em>(A1)</em></strong></p>
<p>\(^8{C_5} \times {5^5}{ = ^7}{C_4}{a^3} \times {5^4}\) <strong><em>M1A1</em></strong></p>
<p>\(56 \times 5 = 35{a^3}\)</p>
<p>\({a^3} = 8\) <strong><em>(A1)</em></strong></p>
<p>\(a = 2\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempt to expand both binomials <strong><em>M1</em></strong></p>
<p>\(175000{x^2}\) <strong><em>A1</em></strong></p>
<p>\(21875{a^3}{x^4}\) <strong><em>A1</em></strong></p>
<p>\(175000 = 21875{a^3}\) <strong><em>M1</em></strong></p>
<p>\({a^3} = 8\) <strong><em>(A1)</em></strong></p>
<p>\(a = 2\) <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In an arithmetic sequence the first term is 8 and the common difference is \(\frac{1}{4}\). If the sum of the first 2<em>n</em> terms is equal to the sum of the next <em>n</em> terms, find <em>n</em>.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If \({a_1},{\text{ }}{a_2},{\text{ }}{a_3},{\text{ }} \ldots \) are terms of a geometric sequence with common ratio \(r \ne 1\), show that \({({a_1} - {a_2})^2} + {({a_2} - {a_3})^2} + {({a_3} - {a_4})^2} + \ldots + {({a_n} - {a_{n + 1}})^2} = \frac{{a_1^2(1 - r)(1 - {r^{2n}})}}{{1 + r}}\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_{2n}} = \frac{{2n}}{2}\left( {2(8) + (2n - 1)\frac{1}{4}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = n\left( {16 + \frac{{2n - 1}}{4}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_{3n}} = \frac{{3n}}{2}\left( {2 \times 8 + (3n - 1)\frac{1}{4}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{3n}}{2}\left( {16 + \frac{{3n - 1}}{4}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_{2n}} = {S_{3n}} - {S_{2n}} \Rightarrow 2{S_{2n}} = {S_{3n}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solve \(2{S_{2n}} = {S_{3n}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 2n\left( {16 + \frac{{2n - 1}}{4}} \right) = \frac{{3n}}{2}\left( {16 + \frac{{3n - 1}}{4}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( { \Rightarrow 2\left( {16 + \frac{{2n - 1}}{4}} \right) = \frac{3}{2}\left( {16 + \frac{{3n - 1}}{4}} \right)} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(gdc or algebraic solution) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>n</em> = 63 <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({({a_1} - {a_2})^2} + {({a_2} - {a_3})^2} + {({a_3} - {a_4})^2} + \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {({a_1} - {a_1}r)^2} + {({a_1}r - {a_1}{r^2})^2} + ({a_1}{r^2} - {a_1}{r^{^3}}) + \ldots \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\left[ {{a_1}(1 - r)} \right]^2} + {\left[ {{a_1}r(1 - r)} \right]^2} + {\left[ {{a_1}{r^2}(1 - r)} \right]^2} + \ldots + {\left[ {{a_1}{r^{n - 1}}(1 - r)} \right]^2}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> This </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> is for the expression for the last term.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = a_1^2{(1 - r)^2} + a_1^2{r^2}{(1 - r)^2} + a_1^2{r^4}{(1 - r)^2} + \ldots + a_1^2{r^{2n - 2}}{(1 - r)^2}\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = a_1^2{(1 - r)^2}(1 + {r^2} + {r^4} + \ldots + {r^{2n - 2}})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = a_1^2{(1 - r)^2}\left( {\frac{{1 - {r^{2n}}}}{{1 - {r^2}}}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{a_1^2(1 - r)(1 - {r^{2n}})}}{{1 + r}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to solve (a) successfully. A few candidates failed to understand the relationship between \({S_{2n}}\) and \({S_{3n}}\), and hence did not obtain the correct equation. (b) was answered poorly by a large number of candidates. There was significant difficulty in forming correct general statements, and a general lack of rigor in providing justification.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to solve (a) successfully. A few candidates failed to understand the relationship between \({S_{2n}}\) and \({S_{3n}}\), and hence did not obtain the correct equation. (b) was answered poorly by a large number of candidates. There was significant difficulty in forming correct general statements, and a general lack of rigor in providing justification.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The sum of the first 16 terms of an arithmetic sequence is 212 and the fifth term is 8.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the first term and the common difference.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the smallest value of <em>n </em>such that the sum of the first <em>n </em>terms is greater than 600.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_n} = \frac{n}{2}[2a + (n - 1)d]\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(212 = \frac{{16}}{2}(2a + 15d)\,\,\,\,\,( = 16a + 120d)\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({n^{th}}{\text{ term is }}a + (n - 1)d\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(8 = a + 4d\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneously: <strong><em>(M1) </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(d = 1.5,{\text{ }}a = 2\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{n}{2}[4 + 1.5(n - 1)] > 600\) <strong> <em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 3{n^2} + 5n - 2400 > 0\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow n > 27.4...,{\text{ }}(n < - 29.1...)\)</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 11px;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not penalize improper use of inequalities. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow n = 28\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This proved to be a good start to the paper for most candidates. The vast majority made a meaningful attempt at this question with many gaining the correct answers. Candidates who lost marks usually did so because of mistakes in the working. In part (b) the most efficient way of gaining the answer was to use the calculator once the initial inequality was set up. A small number of candidates spent valuable time unnecessarily manipulating the algebra before moving to the calculator.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This proved to be a good start to the paper for most candidates. The vast majority made a meaningful attempt at this question with many gaining the correct answers. Candidates who lost marks usually did so because of mistakes in the working. In part (b) the most efficient way of gaining the answer was to use the calculator once the initial inequality was set up. A small number of candidates spent valuable time unnecessarily manipulating the algebra before moving to the calculator.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The first term and the common ratio of a geometric series are denoted, respectively, by <em>a</em> and <em>r</em> where <em>a</em> , \(r \in \mathbb{Q}\). Given that the third term is 9 and the sum to infinity is 64, find the value of <em>a</em> and the value of <em>r</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we are given that \(a{r^2} = 9\) and \(\frac{a}{{1 - r}} = 64\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">dividing, \({r^2}(1 - r) = \frac{9}{{64}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(64{r^3} - 64{r^2} + 9 = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = 0.75,{\text{ }}a = 16\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coefficient of \({x^{ - 2}}\) in the expansion of \({(x - 1)^3}{\left( {\frac{1}{x} + 2x} \right)^6}\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">expanding \({(x - 1)^3} = {x^3} - 3{x^2} + 3x - 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">expanding \({\left( {\frac{1}{2} + 2x} \right)^6}\) gives</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(64{x^6} + 192{x^4} + 240{x^2} + \frac{{60}}{{{x^2}}} + \frac{{12}}{{{x^4}}} + \frac{1}{{{x^6}}} + 160\) <strong><em>(M1)A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for an attempt at expanding using binomial.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> Award <strong><em>A1 </em></strong>for \(\frac{{60}}{{{x^2}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> Award <strong><em>A1 </em></strong>for \(\frac{{12}}{{{x^4}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{60}}{{{x^2}}} \times - 1 + \frac{{12}}{{{x^4}}} \times - 3{x^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1) </em></strong>only if both terms are considered.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">therefore coefficient \({x^{ - 2}}\) is \( - 96\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \( - 96{x^{ - 2}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award full marks if working with the required terms only without giving the entire expansion.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[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="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A bank offers loans of $<em>P</em> at the beginning of a particular month at a monthly interest rate of <em>I</em> . The interest is calculated at the end of each month and added to the amount outstanding. A repayment of $<em>R</em> is required at the end of each month. Let \({\text{\$}}{S_n}\) denote the amount outstanding immediately after the \({n^{{\text{th}}}}\) monthly repayment.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find an expression for \({S_1}\) and show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{S_2} = P{\left( {1 + \frac{I}{{100}}} \right)^2} - R\left( {1 + \left( {1 + \frac{I}{{100}}} \right)} \right).\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine a similar expression for \({S_n}\) . Hence show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{S_n} = P{\left( {1 + \frac{I}{{100}}} \right)^n} - \frac{{100R}}{I}\left( {{{\left( {1 + \frac{I}{{100}}} \right)}^n} - 1} \right)\]</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sue borrows $5000 at a monthly interest rate of 1 % and plans to repay the loan in 5 years (<em>i.e.</em> 60 months).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate the required monthly repayment, giving your answer correct to two decimal places.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) After 20 months, she inherits some money and she decides to repay the loan completely at that time. How much will she have to repay, giving your answer correct to the nearest $?</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({S_1} = P\left( {1 + \frac{I}{{100}}} \right) - R\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_2} = P{\left( {1 + \frac{I}{{100}}} \right)^2} - R\left( {1 + \frac{I}{{100}}} \right) - R\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = P{\left( {1 + \frac{I}{{100}}} \right)^2} - R\left( {1 + \left( {1 + \frac{I}{{100}}} \right)} \right)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) extending this,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_n} = P{\left( {1 + \frac{I}{{100}}} \right)^n} - R\left( {1 + \left( {1 + \frac{I}{{100}}} \right) + \ldots + {{\left( {1 + \frac{I}{{100}}} \right)}^{n - 1}}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = P{\left( {1 + \frac{I}{{100}}} \right)^n} - \frac{{R\left( {{{\left( {1 + \frac{I}{{100}}} \right)}^n} - 1} \right)}}{{\frac{I}{{100}}}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = P{\left( {1 + \frac{I}{{100}}} \right)^n} - \frac{{100R}}{I}\left( {{{\left( {1 + \frac{I}{{100}}} \right)}^n} - 1} \right)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) putting \({S_{60}} = 0,{\text{ }}P = 5000,{\text{ }}I = 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5000 \times {1.01^{60}} = 100R{\text{ }}({1.01^{60}} - 1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(R = (\$ )111.22\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(n = 20,{\text{ }}P = 5000,{\text{ }}I = 1,{\text{ }}R = 111.22\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_{20}} = 5000 \times {1.01^{20}} - 100 \times 111.22({1.01^{20}} - 1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (\$ )3652\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">which is the outstanding amount</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In the arithmetic series with \({n^{{\text{th}}}}\) term \({u_n}\) , it is given that \({u_4} = 7\) and \({u_9} = 22\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the minimum value of <em>n</em> so that \({u_1} + {u_2} + {u_3} + ... + {u_n} > 10\,000\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({u_4} = {u_1} + 3d = 7\) , \({u_9} = {u_1} + 8d = 22\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> 5</span><em style="font-family: 'times new roman', times; font-size: medium;">d</em><span style="font-family: 'times new roman', times; font-size: medium;"> = 15 gains both above marks</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({u_1} = - 2\) , \(d = 3\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_n} = \frac{n}{2}\left( { - 4 + (n - 1)3} \right) > 10\,000\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 83\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was well answered by most candidates. A few did not realise that the answer had to be an integer.</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">Solve the simultaneous equations</p>
<p class="p1">\[\ln \frac{y}{x} = 2\]</p>
<p class="p1">\[\ln {x^2} + \ln {y^3} = 7.\]</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p2"><span class="Apple-converted-space">\(\ln \frac{y}{x} = 2 \Rightarrow \ln x + \ln y = 2\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(\ln {x^2} + \ln {y^3} = 7 \Rightarrow 2\ln x + 3\ln y = 7\) Â Â </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1">attempting to solve for \(x\) and \(y\) \(\left( {{\text{to obtain }}\ln x = \frac{1}{5}{\text{ and }}\ln y = \frac{{11}}{5}} \right)\)<span class="s2">Â <span class="Apple-converted-space">Â Â </span></span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\(x = {{\text{e}}^{\frac{1}{5}}}{\text{ }}( = 1.22)\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(y = {{\text{e}}^{\frac{{11}}{5}}}{\text{ }}( = 9.03)\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p2"><span class="Apple-converted-space">\(\ln \frac{y}{x} = 2 \Rightarrow y = {{\text{e}}^2}x\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(\ln {x^2} + \ln {{\text{e}}^6}{x^3} = 7\) Â Â </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1">attempting to solve for \(x\) <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p3"><span class="Apple-converted-space">\(x = {{\text{e}}^{\frac{1}{5}}}{\text{ }}( = 1.22)\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><span class="Apple-converted-space">\(y = {{\text{e}}^{\frac{{11}}{5}}}{\text{ }}( = 9.03)\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>METHOD 3</strong></p>
<p class="p2"><span class="Apple-converted-space">\(\ln \frac{y}{x} = 2 \Rightarrow y = {{\text{e}}^2}x\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(\ln {x^2} + \ln {y^3} = 7 \Rightarrow \ln ({x^2}{y^3}) = 7\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\({x^2}{y^3} = {{\text{e}}^7}\) Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">substituting \(y = {e^2}x\) into \({x^2}{y^3} = {{\text{e}}^7}\)Â <span class="s2">(to obtain \({{\text{e}}^6}{x^5} = {{\text{e}}^7}\)</span>) <span class="Apple-converted-space">Â Â </span><strong><em>M1</em></strong></p>
<p class="p3"><span class="Apple-converted-space">\(x = {{\text{e}}^{\frac{1}{5}}}{\text{ }}( = 1.22)\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><span class="Apple-converted-space">\(y = {{\text{e}}^{\frac{{11}}{5}}}{\text{ }}( = 9.03)\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Reasonably well done. Candidates who did not obtain the correct solution generally made an error when attempting to apply logarithmic or exponential laws and hence made erroneous substitutions.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \ln x\) . The graph of <em>f </em>is transformed into the graph of the function <em>g </em>by a translation of \(\left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> { - 2} <br>\end{array}} \right)\), followed by a reflection in the <em>x</em>-axis. Find an expression for \(g(x)\), giving your answer as a single logarithm.</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;">\(h(x) = f(x - 3) - 2 = \ln (x - 3) - 2\) <strong><em>(M1)(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(g(x) = -h(x) = 2 - \ln (x - 3)\) <strong><em>M1</em></strong></span></p>
<p> <strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Note<em>: </em></strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">M1 </strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">only if it is clear the effect of the reflection in the </span><em style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">x</em><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">-axis:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">the expression is correct <strong><em>OR<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">there is a change of signs of the previous expression <strong><em>OR<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">there’s a graph or an explanation making it explicit</span></p>
<p> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = \ln {{\text{e}}^2} - \ln (x - 3)\) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = \ln \left( {\frac{{{{\text{e}}^2}}}{{x - 3}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was well attempted but many candidates could have scored better had they written down all the steps to obtain the final expression. In some cases, as the final expression was incorrect and the middle steps were missing, candidates scored just 1 mark. That could be a consequence of a small mistake, but the lack of working prevented them from scoring at least all method marks. Some candidates performed the transformations well but were not able to use logarithms properties to transform the answer and give it as a single logarithm.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>k</em> such that the following system of equations does not have a unique solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[kx + y + 2z = 4\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[ - y + 4z = 5\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[3x + 4y + 2z = 1\]</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">determinant = 0 <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k( - 2 - 16) - (0 - 12) + 2(0 + 3) = 0\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 18k + 18 = 0\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">writes in the form</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> k&1&2&4 \\ <br> 0&{ - 1}&4&5 \\ <br> 3&4&2&1 <br>\end{array}} \right)\)</span></span><span style="font-family: 'times new roman', times; font-size: medium;"> (or attempts to solve simultaneous equations) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Having two 0’s in first column (obtaining two equations in the same two variables) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> k&1&2&4 \\ <br> 0&{ - 1}&4&5 \\ <br> 3&4&{18k - 18}&{21k - 27} <br>\end{array}} \right)\) (or isolating one variable) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> The <strong><em>A1</em></strong> is to be awarded for the 18<em>k</em> – 18. The final column may not be seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>k</em> = 1 <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></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;">Candidates who used the determinant method usually obtained full marks. Few students used row reduction and of those the success was varied. However, many candidates attempted long algebraic methods, which frequently went wrong at some stage. Of those who did work through to correctly isolate one variable, few were able to interpret the resultant value of \(k\) .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove by mathematical induction that, for \(n \in {\mathbb{Z}^ + }\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + n{\left( {\frac{1}{2}} \right)^{n - 1}} = 4 - \frac{{n + 2}}{{{2^{n - 1}}}}.\]</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">A.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Using integration by parts, show that \(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{1}{5}{{\text{e}}^{2x}}} (2\sin x - \cos x) + C\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Solve the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \sqrt {1 - {y^2}} {{\text{e}}^{2x}}\sin x\), given that <em>y</em> = 0 when <em>x</em> = 0,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">writing your answer in the form \(y = f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Sketch the graph of \(y = f(x)\) , found in part (b), for \(0 \leqslant x \leqslant 1.5\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the coordinates of the point P, the first positive intercept on the <em>x</em>-axis, and mark it on your sketch.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) The region bounded by the graph of \(y = f(x)\) and the <em>x</em>-axis, between the origin and P, is rotated 360° about the <em>x</em>-axis to form a solid of revolution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the volume of this solid.</span></p>
<div class="marks">[17]</div>
<div class="question_part_label">B.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">prove that \(1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + n{\left( {\frac{1}{2}} \right)^{n - 1}} = 4 - \frac{{n + 2}}{{{2^{n - 1}}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for <em>n</em> = 1</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{LHS}} = 1,{\text{ RHS}} = 4 - \frac{{1 + 2}}{{{2^0}}} = 4 - 3 = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so true for <em>n</em> = 1 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">assume true for <em>n</em> = <em>k</em> <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + k{\left( {\frac{1}{2}} \right)^{k - 1}} = 4 - \frac{{k + 2}}{{{2^{k - 1}}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">now for <em>n</em> = <em>k</em> +1</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">LHS: \(1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + k{\left( {\frac{1}{2}} \right)^{k - 1}} + (k + 1){\left( {\frac{1}{2}} \right)^k}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 4 - \frac{{k + 2}}{{{2^{k - 1}}}} + (k + 1){\left( {\frac{1}{2}} \right)^k}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 4 - \frac{{2(k + 2)}}{{{2^k}}} + \frac{{k + 1}}{{{2^k}}}\,\,\,\,\,\)(or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 4 - \frac{{(k + 1) + 2}}{{{2^{(k + 1) - 1}}}}\,\,\,\,\,\)(accept \(4 - \frac{{k + 3}}{{{2^k}}}\)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore if it is true for <em>n</em> = <em>k</em> it is true for <em>n</em> = <em>k</em> + 1. It has been shown to be true for <em>n</em> = 1 so it is true for all \(n{\text{ }}( \in {\mathbb{Z}^ + })\). <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> To obtain the final <strong><em>R</em></strong> mark, a reasonable attempt at induction must have been made.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">A.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = - \cos x{{\text{e}}^{2x}} + \int {2{{\text{e}}^{2x}}\cos x{\text{d}}x} } \) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = - \cos x{{\text{e}}^{2x}} + 2{{\text{e}}^{2x}}\sin x - \int {4{{\text{e}}^{2x}}\sin x{\text{d}}x} \) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = - \cos x{{\text{e}}^{2x}} + 2{{\text{e}}^{2x}}\sin x} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{1}{5}{{\text{e}}^{2x}}(2\sin x - \cos x) + C} \) <strong><em>AG</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\sin x{{\text{e}}^{2x}}{\text{d}}x = \frac{{\sin x{{\text{e}}^{2x}}}}{2} - \int {\cos x\frac{{{{\text{e}}^{2x}}}}{2}{\text{d}}x} } \) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin x{{\text{e}}^{2x}}}}{2} - \cos x\frac{{{{\text{e}}^{2x}}}}{4} - \int {\sin x\frac{{{{\text{e}}^{2x}}}}{4}{\text{d}}x} \) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{5}{4}\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{{{{\text{e}}^{2x}}\sin x}}{2} - \frac{{\cos x{{\text{e}}^{2x}}}}{4}} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{1}{5}{{\text{e}}^{2x}}(2\sin x - \cos x) + C} \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\frac{{{\text{d}}y}}{{\sqrt {1 - {y^2}} }} = \int {{{\text{e}}^{2x}}\sin x{\text{d}}x} } \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arcsin y = \frac{1}{5}{{\text{e}}^{2x}}(2\sin x - \cos x)( + C)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">when \(x = 0,{\text{ }}y = 0 \Rightarrow C = \frac{1}{5}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = \sin \left( {\frac{1}{5}{{\text{e}}^{2x}}(2\sin x - \cos x) + \frac{1}{5}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) </span><img 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" alt><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px; display: inline; float: none;"><span class="Apple-style-span" style="font-family: 'times new roman', times; font-size: medium; display: inline; float: none; line-height: normal;"> <strong><em>A1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P is (1.16, 0) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for 1.16 seen anywhere, <strong><em>A1</em></strong> for complete sketch.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow FT on their answer from (b)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(V = \int_0^{1.162...} {\pi {y^2}{\text{d}}x} \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1.05\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow FT on their answers from (b) and (c)(i).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">B.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part A: Given that this question is at the easier end of the ‘proof by induction’ spectrum, it was disappointing that so many candidates failed to score full marks. The <em>n</em> = 1 case was generally well done. The whole point of the method is that it involves logic, so ‘let n = k’ or ‘put n = k’, instead of ‘assume ... to be true for n = k’, gains no marks. The algebraic steps need to be more convincing than some candidates were able to show. It is astonishing that the R1 mark for the final statement was so often not awarded.</span></p>
<div class="question_part_label">A.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part B: Part (a) was often well done, although some faltered after the first integration. Part (b) was also generally well done, although there were some errors with the constant of integration. In (c) the graph was often attempted, but errors in (b) usually led to manifestly incorrect plots. Many attempted the volume of integration and some obtained the correct value.</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;">Consider the equation \({z^3} + a{z^2} + bz + c = 0\) , where \(a\) , \(b\), \(c \in \mathbb{R}\) . The points in the</span> <span style="font-family: times new roman,times; font-size: medium;">Argand diagram representing the three roots of the equation form the vertices of a</span> <span style="font-family: times new roman,times; font-size: medium;">triangle whose area is \(9\). Given that one root is \( - 1 + 3{\text{i}}\) , find</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) the other two roots;<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) \(a\) , \(b\) and \(c\) .</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;">(a) one root is \(- 1 - 3{\text{i}}\) <em><strong> A1</strong></em></span></p>
<p><br><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">distance between roots is \(6\), implies height is \(3\) <em><strong>(M1)A1</strong></em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 1 + 3 = 2 \Rightarrow \) third root is \(2\) <em><strong>A1</strong></em></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;">\( - 1 - 3 = - 4 \Rightarrow \) third root is \(4\) <em><strong>A1</strong></em></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) <strong>EITHER</strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(z - \left( { - 1 + 3{\text{i}}} \right)\) \(z - \left( { - 1 - 3{\text{i}}} \right)\) \(\left( {z - 2} \right) = 0\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow \left( {{z^2} + 2z + 10} \right)\left( {z - 2} \right) = 0\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({{z^2} + 6z - 20 = 0}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(a = 0\)</span>, \(b = 6\) and \(c = 20\)</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;">\(z - \left( { - 1 + 3{\text{i}}} \right)\) \(z - \left( { - 1 - 3{\text{i}}} \right)\) \(\left( {z + 4} \right) = 0\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow \left( {{z^2} + 2z + 10} \right)\left( {z + 4} \right) = 0\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({{z^2} + 6{z^2} + 18z + 40 = 0}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 6\)</span><span style="font-family: times new roman,times; font-size: medium;">, \(b = 18\) and \(c = 40\)</span></p>
<p><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;">[7 marks]</span></strong></em></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;">Most students were able to state the conjugate root, but many were unable to take the</span> <span style="font-family: times new roman,times; font-size: medium;">question further. Of those that then recognised the method, the question was well answered.</span></p>
</div>
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