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<h2>HL Paper 2</h2><div class="question">
<p>Suppose that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> is the first term of a geometric series with common ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<p>Prove, by mathematical induction, that the sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> terms, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_n}">
<mrow>
<msub>
<mi>s</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> is given by</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_n} = \frac{{{u_1}\left( {1 - {r^n}} \right)}}{{1 - r}}">
<mrow>
<msub>
<mi>s</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>r</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1 \Rightarrow {s_1} = {u_1}">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>, so true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>R1</strong></em></p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span>, ie. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_k} = \frac{{{u_1}\left( {1 - {r^k}} \right)}}{{1 - r}}">
<mrow>
<msub>
<mi>s</mi>
<mi>k</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> for statements such as “let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span>”.</p>
<p><strong>Note:</strong> Subsequent marks after the first <em><strong>M1</strong></em> are independent of this mark and can be awarded.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{k + 1}} = {s_k} + {u_1}{r^k}">
<mrow>
<msub>
<mi>s</mi>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>s</mi>
<mi>k</mi>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{k + 1}} = \frac{{{u_1}\left( {1 - {r^k}} \right)}}{{1 - r}} + {u_1}{r^k}">
<mrow>
<msub>
<mi>s</mi>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{k + 1}} = \frac{{{u_1}\left( {1 - {r^k}} \right)}}{{1 - r}} + \frac{{{u_1}{r^k}\left( {1 - r} \right)}}{{1 - r}}">
<mrow>
<msub>
<mi>s</mi>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{k + 1}} = \frac{{{u_1} - {u_1}{r^k} + {u_1}{r^k} - r{u_1}{r^k}}}{{1 - r}}">
<mrow>
<msub>
<mi>s</mi>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
<mo>−</mo>
<mi>r</mi>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{k + 1}} = \frac{{{u_1}\left( {1 - {r^{k + 1}}} \right)}}{{1 - r}}">
<mrow>
<msub>
<mi>s</mi>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>r</mi>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and if true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span> then true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, the statement is true for any positive integer (or equivalent). <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award the final <em><strong>R1</strong> </em>mark provided at least four of the previous marks are gained.</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="specification">
<p>A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.</p>
<p>The model is of the form</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><msub><mi>X</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>Z</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">M</mi><mfenced><mtable><mtr><mtd><msub><mi>X</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><msub><mi>Z</mi><mi>n</mi></msub></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>n</mi></msub></math> is the probability of the gene being in its normal state after dividing for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mtext>th</mtext></math> time, and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Z</mi><mi>n</mi></msub></math> is the probability of it being in another state after dividing for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mtext>th</mtext></math> time, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math>.</p>
<p>Matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math> is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="specification">
<p>The gene is in its normal state when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn></math>. Calculate the probability of it being in its normal state</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>What does <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> represent in this context?</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math>.</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>Find the eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math>.</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>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>in the long term.</p>
<div class="marks">[2]</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>02</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the probability of mutating from ‘not normal state’ to ‘normal state’ <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>A1</strong> </em>can only be awarded if it is clear that transformation is <strong>from</strong> the mutated state.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo>-</mo><mi>λ</mi><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>02</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to find eigenvalues. Any indication that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mrow><mi mathvariant="bold-italic">M</mi><mo>-</mo><mi>λ</mi><mi mathvariant="bold-italic">I</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> has been used is sufficient for the <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>94</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>.</mo><mn>98</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>-</mo><mn>0</mn><mo>.</mo><mn>0012</mn><mo>=</mo><mn>0</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>92</mn><mi>λ</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>92</mn><mo>=</mo><mn>0</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>92</mn><mo> </mo><mo> </mo><mfenced><mfrac><mn>23</mn><mn>25</mn></mfrac></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>02</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>02</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>92</mn><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> can be awarded for attempting to find either eigenvector.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>02</mn><mi>y</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>02</mn><mi>y</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>02</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Accept any multiple of the given eigenvectors.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>02</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn></mtd></mtr></mtable></mfenced><mn>5</mn></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>744</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>0852</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>256</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>915</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Condone omission of the initial state vector for the <em><strong>M1</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>744</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>744311</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd></mtr></mtable></mfenced></math> seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>25</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></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;">
<p>There was some difficulty in interpreting the meaning of the values in the transition matrix, but most candidates did well with the rest of the question. In part (d) there was frequently evidence of a correct method, but a failure to identify the correct probabilities. It was surprising to see a significant number of candidates diagonalizing the matrix in part (d) and this often led to errors. Clearly this was not necessary.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<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="specification">
<p>A transformation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, of a plane is represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>′</mo><mo>=</mo><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">r</mi><mo>+</mo><mi mathvariant="bold-italic">q</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mo>×</mo><mo> </mo><mn>2</mn></math> matrix, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mo>×</mo><mo> </mo><mn>1</mn></math> vector, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi></math> is the position vector of a point in the plane and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>′</mo></math> the position vector of its image under <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<p>The triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OAB</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>. Under T, these points are transformed to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced></math> respectively.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">S</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math> are matrices.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</mi></math> represents an enlargement with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math>, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math> represents a rotation about <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The transformation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> can also be described by an enlargement scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>)</mo></math>, followed by a rotation about the same centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>)</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>, show that</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</mi></math>.</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>Use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">S</mi></math> to find the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the angle and direction of the rotation represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an equation satisfied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi mathvariant="bold-italic">q</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>3</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>M1</em></strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>M1</em></strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>3</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>M1</em></strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>M1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">S</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2</mn><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mi mathvariant="bold-italic">P</mi><msup><mi mathvariant="bold-italic">S</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong> </em>is for an attempt at rearranging the matrix equation. Award even if the order of the product is reversed.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>2</mn><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">R</mi><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced></math></p>
<p>attempt to solve a system of equations <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>a</mi><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>c</mi><mo>,</mo><mo> </mo><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>d</mi></math> <em><strong>A2</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for two correct equations, <em><strong>A2</strong></em> for all four equations correct.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>866</mn><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>866025</mn><mo>…</mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>866025</mn><mo>…</mo></mtd></mtr></mtable></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The correct answer can be obtained from reversing the matrices, so do not award if incorrect product seen. If the given answer is obtained from the product <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><msup><mi mathvariant="bold-italic">S</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="bold-italic">P</mi></math>, award <em><strong>(A1)(M1)(A0)A0</strong></em>.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>clockwise <em><strong>A1</strong></em></p>
<p>arccosine or arcsine of value in matrix seen <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Both <em><strong>A1</strong></em> marks are dependent on the answer to part (c)(i) and should only be awarded for a valid rotation matrix.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mi mathvariant="bold-italic">q</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi><mo>'</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo>'</mo></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mi>x</mi><mo>-</mo><mi>a</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>-</mo><mi>b</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note</strong>: Accept substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> (and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>'</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo></math>) with particular points given in the question.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mi mathvariant="bold-italic">q</mi></math> using simultaneous equations or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>=</mo><msup><mfenced><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="bold-italic">q</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>651</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>651084</mn><mo>…</mo></mrow></mfenced><mo>,</mo><mo> </mo><mo> </mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>48</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>47662</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>+</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mn>13</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mrow><mn>14</mn><mo>+</mo><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mn>13</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>0</mn><mo>-</mo><mi>a</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>-</mo><mi>b</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> This line, with any of the points substituted, may be seen in part (d)(i) and if so the <em><strong>M1</strong></em> can be awarded there.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>651084</mn><mo>…</mo><mo>,</mo><mo> </mo><mo> </mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>47662</mn><mo>…</mo><mo> </mo></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>+</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mn>13</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mrow><mn>14</mn><mo>+</mo><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mn>13</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></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;">
<p>Part (i) proved to be straightforward for most candidates. A common error in part (ii) was for candidates to begin with the matrix <strong><em>P</em></strong> and to show it successfully transformed the points to their images. This received no marks. For a ‘show that’ question it is expected that the work moves to rather than <em>from</em> the given answer.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(b), (c) These two parts dealt generally with more familiar aspects of matrix transformations and were well done.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(b), (c) These two parts dealt generally with more familiar aspects of matrix transformations and were well done.</p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The trick of recognizing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math> was invariant was generally not seen and as such the question could not be successfully answered.</p>
<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="specification">
<p>A particle moves such that its displacement, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> metres, from a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds is given by the differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>The equation for the motion of the particle is amended to</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>3</mn><mi>t</mi><mo>+</mo><mn>4</mn></math>.</p>
</div>
<div class="specification">
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> the particle is stationary at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> to show that this equation can be written as</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues for the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state the long-term velocity of the particle.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> to write the differential equation as a system of coupled, first order differential equations.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> to find the displacement of the particle when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the long-term velocity of the particle.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Award <strong>M1</strong> for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mtable><mtr><mtd><mo>-</mo><mi>λ</mi><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><em><strong>Note:</strong></em> Award <em><strong>M1</strong> </em>for an attempt to find eigenvalues. Any indication that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mrow><mi mathvariant="bold-italic">M</mi><mo>-</mo><mi>λ</mi><mi mathvariant="bold-italic">I</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> has been used is sufficient for the <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>λ</mi><mfenced><mrow><mo>-</mo><mn>5</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>λ</mi><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(on a phase portrait the particle approaches <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> increases so long term velocity (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>) is)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>Note:</strong></em> Only award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> if both eigenvalues in part (a)(ii) are negative. If at least one is positive accept an answer of ‘<em>no limit</em>’ or ‘<em>infinity</em>’, or in the case of one positive and one negative also accept ‘<em>no limit or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="italic">0</mn></math> (depending on initial conditions)</em>’.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>3</mn><mi>t</mi><mo>+</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> in any recurrence formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>t</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><msub><mi>y</mi><mi>n</mi></msub></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mfenced><mrow><mn>3</mn><msub><mi>t</mi><mi>n</mi></msub><mo>+</mo><mn>4</mn><mo>-</mo><mn>5</mn><msub><mi>y</mi><mi>n</mi></msub><mo>-</mo><mn>6</mn><msub><mi>x</mi><mi>n</mi></msub></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>(when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>,) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>64402</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>644</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A2</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the velocity</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>It was clear that second order differential equations had not been covered by many schools. Fortunately, many were able to successfully answer part (ii) as this was independent of the other two parts. For part (iii) it was expected that candidates would know that two negative eigenvalues mean the system tends to the origin and so the long-term velocity is 0. Some candidates tried to solve the system. It should be noted that when the command term is ‘state’ then no further working out is expected to be seen.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Forming a coupled system from a second order differential equation and solving it using Euler’s method is a technique included in the course guide. Candidates who had learned this technique were successful in this question.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>A change in grazing habits has resulted in two species of herbivore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math>, competing for food on the same grasslands. At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> environmentalists begin to record the sizes of both populations. Let the size of the population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, and the size of the population <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>. The following model is proposed for predicting the change in the sizes of the two populations:</p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mi>y</mi></math></p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi>y</mi></math></p>
<p style="padding-left: 60px;">for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>For this system of coupled differential equations find</p>
</div>
<div class="specification">
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> has a population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn></math>.</p>
</div>
<div class="specification">
<p>It is known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> has an initial population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2900</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the eigenvalues.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the eigenvectors.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down the general solution of the system of equations.</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>Sketch the phase portrait for this system, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math>.</p>
<p>On your sketch show</p>
<ul>
<li>the equation of the line defined by the eigenvector in the first quadrant</li>
<li>at least two trajectories either side of this line using arrows on those trajectories to represent the change in populations as <em>t</em> increases</li>
</ul>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a condition on the size of the initial population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> if it is to avoid its population reducing to zero.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> at which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> at this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>. Give your answer to the nearest <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> herbivores.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>3</mn><mo>-</mo><mi>λ</mi></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>4</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <strong>(M1)(A1)</strong></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Attempt to solve either</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong>or </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></p>
<p>or equivalent <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong>A1</strong><strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> accept equivalent forms</p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <strong>A1</strong><strong>A1</strong><strong>A1</strong></p>
<p> </p>
<p><strong>Note: A1</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> correctly labelled, <strong>A1</strong> for at least two trajectories above <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> and <strong>A1</strong> for at least two trajectories below <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math>, including arrows.</p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>2000</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext mathvariant="italic">e</mtext><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext mathvariant="italic">e</mtext><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo>,</mo><mo> </mo><mn>2900</mn><mo>=</mo><mo>-</mo><mn>2</mn><mi>A</mi><mo>+</mo><mi>B</mi></math> <strong>M1A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for the substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2900</mn></math></p>
<p> </p>
<p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>-</mo><mn>300</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mn>2300</mn></math> <strong>A1</strong><strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mo>-</mo><mn>300</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>2300</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>6</mn><mo>.</mo><mn>79</mn><mo> </mo><mfenced><mrow><mn>6</mn><mo>.</mo><mn>7896</mn><mo>…</mo></mrow></mfenced></math> (years) <strong>A1</strong></p>
<p> </p>
<p><strong>[6 marks]</strong></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>600</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mo>×</mo><mn>6</mn><mo>.</mo><mn>79</mn></mrow></msup><mo>+</mo><mn>2300</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mo>×</mo><mn>6</mn><mo>.</mo><mn>79</mn></mrow></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26827</mn><mo>.</mo><mn>9</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26830</mn></math> (to the nearest <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> animals) <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math></p>
<p>This equation links a variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> with the temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> are positive constants and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The Arrhenius equation predicts that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> is a straight line.</p>
</div>
<div class="specification">
<p>Write down</p>
</div>
<div class="specification">
<p>The following data are found for a particular reaction, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is measured in Kelvin and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>mol</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>:</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Find an estimate of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac></math> is always positive.</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mo>∞</mo></mrow></munder><mi>k</mi><mo>=</mo><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>k</mi><mo>=</mo><mn>0</mn></math>, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</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>(i) the gradient of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>;</p>
<p>(ii) the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</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>Find the equation of the regression line for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use chain rule, including the differentiation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac><mo>=</mo><mi>A</mi><mo>×</mo><mfrac><mi>c</mi><msup><mi>T</mi><mn>2</mn></msup></mfrac><mo>×</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>this is the product of positive quantities so must be positive <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>R1</strong> </em>may be awarded for correct argument from <strong>their</strong> derivative. <em><strong>R1</strong> </em>is not possible if their derivative is not always positive.</p>
<p> </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><img src="data:image/png;base64,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"> <em><strong>A1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for an increasing graph, entirely in first quadrant, becoming concave down for larger values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, <em><strong>A1</strong></em> for tending towards the origin and <em><strong>A1</strong> </em>for asymptote labelled at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi></math> of both sides <strong>OR</strong> substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>T</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>A</mi><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>A</mi></math> <em><strong>(A1)</strong></em></p>
<p><br>(i) so gradient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p><br>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>A</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The implied <em><strong>(M1)</strong></em> and <em><strong>(A1)</strong></em> can only be awarded if <strong>both</strong> correct answers are seen. Award zero if only one value is correct <strong>and</strong> no working is seen.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>an attempt to convert data to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> <em><strong>(M1)</strong></em></p>
<p>e.g. at least one correct row in the following table</p>
<p><img src="data:image/png;base64,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"></p>
<p>line is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi><mo>=</mo><mo>-</mo><mn>13400</mn><mo>×</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mn>15</mn><mo>.</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>13383</mn><mo>.</mo><mn>1</mn><mo>…</mo><mo>×</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mn>15</mn><mo>.</mo><mn>0107</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>13400</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>13383</mn><mo>.</mo><mn>1</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to rearrange or solve graphically <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>A</mi><mo>=</mo><mn>15</mn><mo>.</mo><mn>0107</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn><mo> </mo><mn>300</mn><mo> </mo><mn>000</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>3</mn><mo> </mo><mn>304</mn><mo> </mo><mn>258</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> <strong>Note</strong>: Accept an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3269017</mn></math>… from use of <math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mn>3</mn><mi>sf</mi></math> value.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question caused significant difficulties for many candidates and many did not even attempt the question. Very few candidates were able to differentiate the expression in part (a) resulting in difficulties for part (b). Responses to parts (c) to (e) illustrated a lack of understanding of linearizing a set of data. Those candidates that were able to do part (d) frequently lost a mark as their answer was given in <em>x</em> and <em>y</em>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mn>1.02</mn> </mrow> <mrow> <mn>20</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>P</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1.02</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></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>Given that Phil’s aim is to own the house after 20 years, find the value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P"> <mi>P</mi> </math></span> 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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> years. Show that an expression for the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q"> <mi>Q</mi> </math></span> is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span>.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q"> <mi>Q</mi> </math></span> that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="150000 \times {1.035^{20}}"> <mn>150000</mn> <mo>×</mo> <mrow> <msup> <mn>1.035</mn> <mrow> <mn>20</mn> </mrow> </msup> </mrow> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \$ 298468"> <mo>=</mo> <mi mathvariant="normal">$</mi> <mn>298468</mn> </math></span> <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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P"> <mi>P</mi> </math></span> and common ratio 1.02 <strong><em>(M1)</em></strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P + 1.02P + \ldots + {1.02^{19}}P{\text{ }}\left( { = P(1 + 1.02 + \ldots + {{1.02}^{19}})} \right)"> <mi>P</mi> <mo>+</mo> <mn>1.02</mn> <mi>P</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow> <msup> <mn>1.02</mn> <mrow> <mn>19</mn> </mrow> </msup> </mrow> <mi>P</mi> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1.02</mn> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow> <msup> <mrow> <mn>1.02</mn> </mrow> <mrow> <mn>19</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>explicitly identify <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = P,{\text{ }}r = 1.02"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mi>P</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>r</mi> <mo>=</mo> <mn>1.02</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 20"> <mi>n</mi> <mo>=</mo> <mn>20</mn> </math></span> (may be seen as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_{20}}"> <mrow> <msub> <mi>S</mi> <mrow> <mn>20</mn> </mrow> </msub> </mrow> </math></span>). <strong><em>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{20}} = \frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}"> <mrow> <msub> <mi>s</mi> <mrow> <mn>20</mn> </mrow> </msub> </mrow> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mn>1.02</mn> </mrow> <mrow> <mn>20</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>P</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1.02</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span> <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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24.297 \ldots P = 298468"> <mn>24.297</mn> <mo>…</mo> <mi>P</mi> <mo>=</mo> <mn>298468</mn> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = 12284"> <mi>P</mi> <mo>=</mo> <mn>12284</mn> </math></span> <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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q({1.028^n}) = 5000(1 + 1.028 + {1.028^2} + {1.028^3} + \ldots + {1.028^{n - 1}})"> <mi>Q</mi> <mo stretchy="false">(</mo> <mrow> <msup> <mn>1.028</mn> <mi>n</mi> </msup> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5000</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1.028</mn> <mo>+</mo> <mrow> <msup> <mn>1.028</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>1.028</mn> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow> <msup> <mn>1.028</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q = \frac{{5000\left( {1 + 1.028 + {{1.028}^2} + {{1.028}^3} + ... + {{1.028}^{n - 1}}} \right)}}{{{{1.028}^n}}}"> <mi>Q</mi> <mo>=</mo> <mfrac> <mrow> <mn>5000</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>1.028</mn> <mo>+</mo> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}"> <mo>=</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>the initial value of the first withdrawal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{1.028}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p>the initial value of the second withdrawal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{{{1.028}^2}}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>R1</em></strong></p>
<p>the investment required for these two withdrawals is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q = \frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}"> <mi>Q</mi> <mo>=</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span> <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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\frac{{5000}}{{1.028}}}}{{1 - \frac{1}{{1.028}}}}"> <mfrac> <mrow> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mn>1.028</mn> </mrow> </mfrac> </mrow> </mfrac> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 178571.428 \ldots "> <mo>=</mo> <mn>178571.428</mn> <mo>…</mo> </math></span></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="specification">
<p>A shock absorber on a car contains a spring surrounded by a fluid. When the car travels over uneven ground the spring is compressed and then returns to an equilibrium position.</p>
<p style="text-align: center;"><img 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"></p>
<p>The displacement, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, of the spring is measured, in centimetres, from the equilibrium position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>. The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> can be modelled by the following second order differential equation, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time, measured in seconds, after the initial displacement.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mn>3</mn><mover><mi>x</mi><mo>˙</mo></mover><mo>+</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>The differential equation can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mover><mi>x</mi><mo>˙</mo></mover></mtd></mtr><mtr><mtd><mover><mi>y</mi><mo>˙</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">A</mi><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>2</mn></math> matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mover><mi>x</mi><mo>˙</mo></mover></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi></math>.</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>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues of matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvectors of matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> the shock absorber is displaced <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mtext>cm</mtext></math> and its velocity is zero, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mover><mi>x</mi><mo>˙</mo></mover><mo>⇒</mo><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mover><mi>x</mi><mo>¨</mo></mover></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>+</mo><mn>3</mn><mfenced><mi>y</mi></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong><em>R1</em></strong></p>
<p><br><strong>Note:</strong> If no explicit reference is made to <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mover><mi>x</mi><mo>¨</mo></mover></math>, or equivalent, award <em><strong>A0R1</strong></em> if second line is seen. If <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> used instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>, award <em><strong>A0R0</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi></math> <strong><em>AG</em></strong></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mo>-</mo><mi>λ</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></mtd><mtd><mo>-</mo><mn>3</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mfenced><mrow><mi>λ</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>25</mn><mo>=</mo><mn>0</mn></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mo>;</mo><mo> </mo><mi>λ</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>5</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></mtd><mtd><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>2</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for a valid attempt to find either eigenvector. Accept equivalent forms of the eigenvectors. <br>Do not award <em><strong>FT</strong></em> for eigenvectors that do not satisfy both rows of the matrix.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong><em>M1A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo> </mo><mo>⇒</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>8</mn><mo>,</mo><mo> </mo><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>y</mi><mo>=</mo><mn>0</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>A</mi><mo>-</mo><mn>2</mn><mi>B</mi><mo>=</mo><mn>8</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mn>0</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo> </mo><mo>;</mo><mo> </mo><mi>B</mi><mo>=</mo><mo>-</mo><mn>5</mn></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>10</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup></math> <strong><em>A1</em></strong></p>
<p><strong><br>Note:</strong> Do not award the final <em><strong>A1</strong></em> if the answer is given in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math>.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and a set of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axes.</p>
<p>In each case, the drone moves to a new position represented by the following transformations:</p>
<ul>
<li>a rotation anticlockwise of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> radians about <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math></li>
<li>a reflection in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><msqrt><mn>3</mn></msqrt></mfrac></math></li>
<li>a rotation clockwise of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> radians about <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</li>
</ul>
<p>All the movements are performed in the listed order.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a single matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> that defines a transformation that represents the overall change in position.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">P</mi><mn>2</mn></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state what the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">P</mi><mn>2</mn></msup></math> indicates for the possible movement of the drone.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Three drones are initially positioned at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math>. After performing the movements listed above, the drones are positioned at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>′</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mo>′</mo></math> respectively.</p>
<p>Show that the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math> is equal to the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></math> .</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a single transformation that is equivalent to the three transformations represented by matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>rotation anticlockwise <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr></mtable></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>(M1)A1</em></strong></p>
<p>reflection in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><msqrt><mn>3</mn></msqrt></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> <strong><em>(A1)</em></strong></p>
<p>matrix is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p>rotation clockwise <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>866</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>an attempt to multiply three matrices <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi mathvariant="bold-italic">P</mi><mn>2</mn></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if final answer not resolved into the identity matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>if the overall movement of the drone is repeated <strong><em>A1</em></strong></p>
<p>the drone would return to its original position <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mtext>det</mtext><mo> </mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced><mo>=</mo><mfenced open="|" close="|"><mrow><mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced><mo>-</mo><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <strong><em>A1</em></strong></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext><mo>=</mo></math> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><mfenced open="|" close="|"><mrow><mtext>det</mtext><mo> </mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced></math> <strong><em>R1</em></strong></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext><mo>=</mo></math> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></math> <strong><em>AG</em></strong></p>
<p><br><strong>Note:</strong> Award at most <em><strong>A1R0</strong></em> for responses that omit modulus sign.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>statement of fact that rotation leaves area unchanged <strong><em>R1</em></strong></p>
<p>statement of fact that reflection leaves area unchanged <strong><em>R1</em></strong></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext><mo>=</mo></math> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></math> <strong><em>AG</em></strong></p>
<p> </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>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>attempt to find angles associated with values of elements in matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>cos</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced></mtd><mtd><mi>sin</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd><mi>sin</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced></mtd><mtd><mo>-</mo><mi>cos</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></p>
<p>reflection (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>tan</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mi>x</mi></math>) <strong><em>(M1)</em></strong></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <strong><em>A1</em></strong></p>
<p>reflection in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>tan</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></mfenced><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>268</mn><mi>x</mi></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = m{x^3} + n{x^2} + px + q">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>m</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mi>x</mi>
<mo>+</mo>
<mi>q</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> are integers.</p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> passes through the point (0, 0).</p>
</div>
<div class="specification">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> also passes through the point (3, 18).</p>
</div>
<div class="specification">
<p class="indent1" style="margin-top: 12.0pt; tab-stops: 2.0cm 72.0pt 108.0pt 144.0pt 180.0pt 216.0pt 252.0pt 288.0pt 324.0pt 360.0pt 396.0pt 432.0pt 468.0pt 504.0pt;">The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> also passes through the points (1, 0) and (–1, –10).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></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 class="indent1" style="margin-top:12.0pt;">Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="27m + 9n + 3p = 18">
<mn>27</mn>
<mi>m</mi>
<mo>+</mo>
<mn>9</mn>
<mi>n</mi>
<mo>+</mo>
<mn>3</mn>
<mi>p</mi>
<mo>=</mo>
<mn>18</mn>
</math></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 class="indent1" style="margin-top:12.0pt;">Write down the other two linear equations in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></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 class="indent1a" style="margin-top:12.0pt;">Write down these three equations as a matrix equation.</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 class="indent1a" style="margin-top:12.0pt;">Solve this matrix equation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1a" style="margin-top:12.0pt;">The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> can also be written <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = x\left( {x - 1} \right)\left( {rx - s} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>r</mi>
<mi>x</mi>
<mo>−</mo>
<mi>s</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> are integers. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> = 0 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Attempting to substitute (3, 18) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m{3^3} + n{3^2} + {p3} = 18">
<mi>m</mi>
<mrow>
<msup>
<mn>3</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mn>3</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mi>p</mi>
<mn>3</mn>
</mrow>
<mo>=</mo>
<mn>18</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="27m + 9n + 3p = 18">
<mn>27</mn>
<mi>m</mi>
<mo>+</mo>
<mn>9</mn>
<mi>n</mi>
<mo>+</mo>
<mn>3</mn>
<mi>p</mi>
<mo>=</mo>
<mn>18</mn>
</math></span> <em><strong>AG N0</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = 0 <em><strong>A1 N1</strong></em></p>
<p>−<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> − <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = −10 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Evidence of attempting to set up a matrix equation <em><strong> (M1)</strong> </em></p>
<p>Correct <strong>matrix</strong> equation representing the given equations <em><strong>A2 N3 </strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {27}&9&3 \\ 1&1&1 \\ { - 1}&1&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} m \\ n \\ p \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {18} \\ 0 \\ { - 10} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>27</mn>
</mrow>
</mtd>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>m</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ { - 5} \\ 3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1A1A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Factorizing <em><strong> (M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = x\left( {2{x^2} - 5x + 3} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \left( {{x^2} - x} \right)\left( {rx - s} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>r</mi>
<mi>x</mi>
<mo>−</mo>
<mi>s</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 2">
<mi>r</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = 3">
<mi>s</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = x\left( {x - 1} \right)\left( {2x - 3} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>) <em><strong> A1A1 N3</strong></em></p>
<p><em><strong>[3 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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</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>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^5} - 3{x^4} + m{x^3} + n{x^2} + px + q = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>m</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mi>x</mi>
<mo>+</mo>
<mi>q</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{R}">
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>The equation has three distinct real roots which can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,b">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,c">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
</math></span>.</p>
<p>The equation also has two imaginary roots, one of which is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d{\text{i}}">
<mi>d</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}">
<mi>d</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The values <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> are consecutive terms in a geometric sequence.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="abc = 8"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </math></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>Show that one of the real roots is equal to 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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 8{d^2}"> <mi>q</mi> <mo>=</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span>, find the other two real roots.</p>
<div class="marks">[9]</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="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>recognition of the other root <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - d{\text{i}}"> <mo>=</mo> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a + {\text{lo}}{{\text{g}}_2}\,b + {\text{lo}}{{\text{g}}_2}\,c + d{\text{i}} - d{\text{i}} = 3"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>M1A</strong></em><em><strong>1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for sum of the roots, <em><strong>A1</strong> </em>for 3. Award <em><strong>A0M1A0</strong></em> for just <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a + {\text{lo}}{{\text{g}}_2}\,b + {\text{lo}}{{\text{g}}_2}\,c = 3"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo>=</mo> <mn>3</mn> </math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,abc = 3"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow abc = {2^3}"> <mo stretchy="false">⇒</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mn>3</mn> </msup> </mrow> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="abc = 8"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let the geometric series be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}r"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mi>r</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}{r^2}"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{u_1}r} \right)^3} = 8"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mn>8</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}r = 2"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mi>r</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p>hence one of the roots is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}2 = 1"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>2</mn> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{b}{a} = \frac{c}{b}"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> <mo>=</mo> <mfrac> <mi>c</mi> <mi>b</mi> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{b^2} = ac \Rightarrow {b^3} = abc = 8"> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mi>a</mi> <mi>c</mi> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>b</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 2"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p>hence one of the roots is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}2 = 1"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>2</mn> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>product of the roots is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} \times {r_2} \times 1 \times d{\text{i}} \times - d{\text{i}} = - 8{d^2}"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>×</mo> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>×</mo> <mn>1</mn> <mo>×</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>×</mo> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} \times {r_2} = - 8"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>×</mo> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p>sum of the roots is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} + {r_2} + 1 + d{\text{i}} + - d{\text{i}} = 3"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>+</mo> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} + {r_2} = 2"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p>solving simultaneously <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} = - 2"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_2} = 4"> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mn>4</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em><em><strong>A</strong></em><em><strong>1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>product of the roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a \times {\text{lo}}{{\text{g}}_2}\,b \times {\text{lo}}{{\text{g}}_2}\,c \times d{\text{i}} \times - d{\text{i}} = - 8{d^2}"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mo>×</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>×</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo>×</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>×</mo> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em><em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a \times {\text{lo}}{{\text{g}}_2}\,b \times {\text{lo}}{{\text{g}}_2}\,c = - 8"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mo>×</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>×</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{r}"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2"> <mn>2</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2r"> <mn>2</mn> <mi>r</mi> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{lo}}{{\text{g}}_2}\frac{2}{r}} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2r} \right) = - 8"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span></p>
<p>attempt to solve <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {\text{lo}}{{\text{g}}_2}\,r} \right)\left( {1 + {\text{lo}}{{\text{g}}_2}\,r} \right) = - 8"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,r = \pm 3"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>r</mi> <mo>=</mo> <mo>±</mo> <mn>3</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \frac{1}{8}{\text{,}}\,\,8"> <mi>r</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>8</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em><em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2"> <mn>2</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{a}"> <mfrac> <mn>4</mn> <mi>a</mi> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{lo}}{{\text{g}}_2}\,a} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2} \right)\left( {{\text{lo}}{{\text{g}}_2}\,\frac{4}{a}} \right) = - 8"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mn>4</mn> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span></p>
<p>attempt to solve <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{1}{4}{\text{,}}\,\,16"> <mi>a</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>16</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em><em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{4}{\text{,}}\,\,16"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>16</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p>roots are −2, 4 <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p> </p>
<p><em><strong>[9 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="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>1</mn><mo>−</mo><mtext>i</mtext></math>.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>1</mn></msub><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>x</mi></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>2</mn></msub><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mo>(</mo><mi>x</mi><mo>−</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>)</mo></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The current, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>, in an AC circuit can be modelled by the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mi>a</mi><mo> </mo><mi>cos</mi><mo>(</mo><mi>b</mi><mi>t</mi><mo>−</mo><mi>c</mi><mo>)</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> is the frequency and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> is the phase shift.</p>
<p>Two AC voltage sources of the same frequency are independently connected to the same circuit. If connected to the circuit alone they generate currents <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>A</mtext></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>B</mtext></msub></math>. The maximum value and the phase shift of each current is shown in the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">When the two voltage sources are connected to the circuit at the same time, the total current <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>T</mtext></msub></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>A</mtext></msub><mo>+</mo><msub><mi>I</mi><mtext>B</mtext></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plot the position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> on an Argand Diagram.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>b</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, giving the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>x</mi></mrow></msup><mfenced><mrow><mi>c</mi><mo>+</mo><mtext>i</mtext><mi>d</mi></mrow></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mrow><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub></mrow></mfenced></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo> </mo><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>a</mi><mo>≤</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>T</mtext></msub></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the phase shift of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>T</mtext></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOEAAADJCAYAAADVTFUhAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAnDSURBVHhe7d19TFXnHcDxHy8B32gTkgL9w4hC20QkJta1S2arS5d0a2bViv5n0iUtICJvNlFrWrbO+oeJVBdf6/4QUBGsJlPpjBFirLPtJm9yQSeIpvgaq9MEVFQuu8/xidatvMP93cP9fhLivedcTYz5es55nvOcG9LlIwDUhNpfASghQkAZEQLKiBBQRoSAMiIElBEhoIwIAWW9T9Z3ee0LBJKm5mapOFohaWmpEhISYrciIIX0fKwjQhcy/2QffPChHD9+XCorK2T8+PF2DwJSLxFyOupCDQ2NcvjwYWlra5N16wqcKOFeROgyXq9X1qxZ45yCjhkzRsrLy6WlpcXuhRsRoct4PB45efKkzJ49W5KSkmRKcrLs2FFo98KNiNBFzGnnuoIC+b0vwOTkKRIeHi4rli+X4uJiOX/+vP0U3IYIXcScdn7zzQlZkpFht4i8+uo0mTZtmmzf/le7BW5DhC5y4sQ/ZNbMmfLKKy/bLeIcDfPy8qSsrEw6OjrsVrgJEbrI3LlzZMOG9RIa+uw/24wZv5LKykqJjIy0W+AmROgizz//vERFRdl3z4qPn2BfwW2IEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEFBGhIAyIgSUESGgjAgBZUQIKCNCQBkRAsqIEBikzs5OaWhokAcPHtgt/UOEwCDdvHVL3n//DzJ37jypr6+Xrq4uu6dviBAYpJgXXpCvvy6XpKQkmTNnrnya/0e5f/++3du7EF+1PWa7du1a+wqB5NSpU9LS0iILFy60WxAIjh07JjXV1fLSyy9Jfn6+s+2tt37j/NqdXiNMTU21rxBImpua5MyZMxIWFiYzZ82SsWPH2j3QVH/6tFy8eFHi4uJk1aqPnW0pC3r+j7LXCKXLa18gkGzevFl2Fu+UTq9Xpk6dKn/ZsF5GjR5t98Lf2trapKDgCyksLJTZ786WTz/5RKKjox/vDOn5qo9rQheLe/FFKS3dI6d9//tmLs2SzkeP7B74082bN+Vd37XgkSNHZEfhDvmioOBpgH1AhC4XHx8vu3fvkrNnz0pqWvqAh8kxcFFRUbI4PV0OH/67vDFjhoSEhNg9fUOEI8CkSZNk376vJDw8TJavWCEPHz60e+APERERsmBBiowbN85u6R8iHCFiY2Nly5Yt0t7WLhkZS6S9vd3uQaAjwhEkNDRUVq/+s+8asU7y8vLk3r17dg8CGRGOMDExMVJaWip1dadl6dIsecRgTcAjwhHop4M16emLOSIGOCIcocxgTWFRoXg8HsnJyeUaMYAR4QiW4Atxz54SqamtldzcXOno6LB7EEiIcIQzp6alvhA9ngbJzFzqLLtBYCHCIDBx4kTZv3+fhIWFOqOmXkIMKEQYJMwNxeZ+0/v3OyQ1Pb1fS20wvIgwiJh5xM8++5N46j2SnZ3DYE2AIMIgY+6s2VNSIjU1NZKbm8dgTQAgwiAUPzHeWX1hHsXAYI0+IgxSZrBm585iaWxslLS0dLl3lwl9LUQYxBISEqS4uMhZj5iblyd37961e+BPRBjkzJ01e8vKpLq62hmsYWGw/xEhZEL8BOfOGvPszK3bttmt8BcihMMcEc2EfkVFpWzcuIkJfT8iQjzxeEJ/k5SUlEgaqy/8hgjxDBNiUVGh1NXVOfOIDNYMPyLE/zGjpnvLSqWqqkpycnLk4QOeWTOciBA/a4JZfVG6x1l9sSQzk6e4DSMiRLfMYE1RUZGzMNjcWcOp6fAgQvQoMTHBubOmtrZW8vKWMVgzDIgQvTJHxDLfNaKZ0M/Kymb6YogRIfrErNA/ePCA83TpZcs+4iluQ4gI0WdmGdTWrVuk3XdtaB4wzJO+hwYRol+eLAz2eGTJkkwGa4YAEaLfzIT+zuIiZ2GwOTVlsGZwiBADMslM6O8tcyb0zWANC4MHjggxYGawpqRkt7P6YvHiDB6VMUBEiEExt7iZe03NwuCs7GyuEQeACDFoiYmJzndfVFdVOxP6jJr2DxFiSJgJ/fJDh5wAV65cyb2m/UCEGDIxsTGyffuXcvvOHcnMzGSwpo+IEEPq8ReVrnZWX5gJ/QcM1vSKCDHk4mJjHy8MPl0nS7MYrOkNEWJYOIM1u3Y5N33nLfuI777oARFi2DxZfVFVJdnZTOh3hwgxrMyTvs30RX29RxZnZHBE/BlEiGFnTk2LiouktsZ8YzBfQvO/iBB+kZiQIIcOHnTmDz9eterJ802Liovl3+fOSVdXl/1k8Anx/eV7/tt3ee0LBBLzhZ9Hj1Y4D+x1E3NdmJKSIt99973dIhIRESHbvtwmv337bbtlhAnp+VjHkRB+9+PNW/bVY+bo+Pnna+y74EOE8CsT3J3bt+27p5qbmuyr4EOE8KvIyEiJjo62756aPHmyfRV8iBB+ZW5rM9eEPzVq1CjJz8+374IPEcLv0tJSZf369TI/Zb6kpqbKvv375M0337B7gw+joy7l1tHRoMToKBDYiBDqzK1sg32YsLkL5+S33zprGd2GCKHmji+YjZs2yaxZv5bW1la7dWAaGxslZX6KlJeX2y3uQYRQc+nSZfF6vXLt2rVB37aWnJwsBw78Td6bN89ucQ8ihJqkpMnyy9dft+8GJzw8XKZPny6jR4+2W9yDCDEi/Of2bTl46JBcuHDBbnEPIoTrmdNZM2WT+mEqEQIazHdjLFq0SEa58FTUIEIEnOvXr8u5c03d/rS2XrKffCo8LEzCfD9uRIQIODdu/CjNzc3d/ly9etV+cmQgQgScKVOS5J13ftftz2uv/cJ+cmQgQqgyj83v9HoH/dh88/vNn+XGB0kRIdT88EOrNDQ0yoKU+fL9P/8lV65csXv6p629XU6dqpL5782Ty5ev+E5nb9g97sAqCpdiFYWLsIoCCGxECCgjQkAZEQLKiBBQRoSAMiIElBEhoIwIAWVECCgjQkAZEQLKiBBQRoSAMiIElBEhoIwIAWVECCgjQkAZEbpU1HPPSWxcnH0HN+v9QU8AhhVHQkAZEQLKiBBQRoSAMiIElBEhoIwIAWVECCgjQkAZEQKqRP4L8Pc9o3kKR4UAAAAASUVORK5CYII="> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><msup><mtext>e</mtext><mfrac><mrow><mtext>i</mtext><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></msup></math> <strong><em>A1A1</em></strong></p>
<p><br><strong>Note:</strong> Accept an argument of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math>. Do <strong>NOT</strong> accept answers that are not exact.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>x</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></mfenced></mrow></msup></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>x</mi></mrow></msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mrow><mtext>i</mtext><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></mrow></msup></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>x</mi></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mtext>i</mtext></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>x</mi></mrow></msup><mo>×</mo><msqrt><mn>2</mn></msqrt><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mrow><mtext>i</mtext><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></mrow></msup></math> <strong><em>M1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>2</mn></msqrt><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></mrow></msup></math> <strong><em>(A1)</em></strong></p>
<p>attempt extract real part using cis form <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mrow><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>4142</mn><mo>…</mo><mo> </mo><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>785398</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>t</mi></msub><mo>=</mo><mn>12</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi>b</mi><mi>t</mi></mrow></mfenced><mo>+</mo><mn>12</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi>b</mi><mi>t</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>t</mi></msub><mo>=</mo><mn>12</mn><mtext> Re</mtext><mfenced><mrow><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>b</mi><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>b</mi><mi>t</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></mfenced></mrow></msup></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>t</mi></msub><mo>=</mo><mn>12</mn><msqrt><mn>2</mn></msqrt><mo> </mo><mi>cos</mi><mfenced><mrow><mi>b</mi><mi>t</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p>max <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>12</mn><msqrt><mn>2</mn></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><mn>17</mn><mo>.</mo><mn>0</mn></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>phase shift <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>785</mn></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.</p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.</p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.</p>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>On the day of her birth, 1st January 1998, Mary’s grandparents invested <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ x">
<mi mathvariant="normal">$<!-- $ --></mi>
<mi>x</mi>
</math></span> in a savings account. They continued to deposit <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ x">
<mi mathvariant="normal">$<!-- $ --></mi>
<mi>x</mi>
</math></span> on the first day of each month thereafter.</p>
<p>The account paid a fixed rate of 0.4% interest per month. The interest was calculated on the last day of each month and added to the account.</p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ {A_n}">
<mi mathvariant="normal">$<!-- $ --></mi>
<mrow>
<msub>
<mi>A</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> be the amount in Mary’s account on the last day of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n{\text{th}}">
<mi>n</mi>
<mrow>
<mtext>th</mtext>
</mrow>
</math></span> month, immediately after the interest had been added.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_1}">
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_2} = {1.004^2}x + 1.004x">
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1.004</mn>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Write down a similar expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_3}">
<mrow>
<msub>
<mi>A</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_4}">
<mrow>
<msub>
<mi>A</mi>
<mn>4</mn>
</msub>
</mrow>
</math></span>.</p>
<p>(ii) Hence show that the amount in Mary’s account the day before she turned 10 years old is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="251({1.004^{120}} - 1)x">
<mn>251</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>120</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_n}">
<mrow>
<msub>
<mi>A</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> on the day before Mary turned 18 years old showing clearly the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Mary’s grandparents wished for the amount in her account to be at least <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ 20\,000">
<mi mathvariant="normal">$</mi>
<mn>20</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span> the day before she was 18. Determine the minimum value of the monthly deposit <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ x">
<mi mathvariant="normal">$</mi>
<mi>x</mi>
</math></span> 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>As soon as Mary was 18 she decided to invest <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ 15\,000">
<mi mathvariant="normal">$</mi>
<mn>15</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span> of this money in an account of the same type earning 0.4% interest per month. She withdraws <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ 1000">
<mi mathvariant="normal">$</mi>
<mn>1000</mn>
</math></span> 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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_1} = 1.004x">
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1.004</mn>
<mi>x</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_2} = 1.004(1.004x + x)">
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1.004</mn>
<mo stretchy="false">(</mo>
<mn>1.004</mn>
<mi>x</mi>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {1.004^2}x + 1.004x">
<mo>=</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1.004</mn>
<mi>x</mi>
</math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><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> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_3} = 1.004({1.004^2}x + 1.004x + x) = {1.004^3}x + {1.004^2}x + 1.004x">
<mrow>
<msub>
<mi>A</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1.004</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1.004</mn>
<mi>x</mi>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>3</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1.004</mn>
<mi>x</mi>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_4} = {1.004^4}x + {1.004^3}x + {1.004^2}x + 1.004x">
<mrow>
<msub>
<mi>A</mi>
<mn>4</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>4</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>3</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1.004</mn>
<mi>x</mi>
</math></span> <strong><em>A1</em></strong></p>
<p>(ii) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_{120}} = ({1.004^{120}} + {1.004^{119}} + \ldots + 1.004)x">
<mrow>
<msub>
<mi>A</mi>
<mrow>
<mn>120</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>120</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>119</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mn>1.004</mn>
<mo stretchy="false">)</mo>
<mi>x</mi>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{{1.004}^{120}} - 1}}{{1.004 - 1}} \times 1.004x">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mn>1.004</mn>
</mrow>
<mrow>
<mn>120</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>1.004</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mn>1.004</mn>
<mi>x</mi>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 251({1.004^{120}} - 1)x">
<mo>=</mo>
<mn>251</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>120</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>x</mi>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_{216}} = 251({1.004^{216}} - 1)x{\text{ }}\left( { = x\sum\limits_{t = 1}^{216} {{{1.004}^t}} } \right)">
<mrow>
<msub>
<mi>A</mi>
<mrow>
<mn>216</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>251</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>216</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>x</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mi>x</mi>
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>216</mn>
</mrow>
</munderover>
<mrow>
<mrow>
<msup>
<mrow>
<mn>1.004</mn>
</mrow>
<mi>t</mi>
</msup>
</mrow>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="251({1.004^{216}} - 1)x = 20\,000 \Rightarrow x = 58.22 \ldots ">
<mn>251</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>216</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>x</mi>
<mo>=</mo>
<mn>20</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>58.22</mn>
<mo>…</mo>
</math></span> (<strong><em>A1)(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="251({1.004^{216}} - 1)x > 20\,000">
<mn>251</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>216</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>x</mi>
<mo>></mo>
<mn>20</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span>, <strong><em>(M1) </em></strong>for attempting to solve and <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 58.22 \ldots ">
<mi>x</mi>
<mo>></mo>
<mn>58.22</mn>
<mo>…</mo>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 59">
<mi>x</mi>
<mo>=</mo>
<mn>59</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 58">
<mi>x</mi>
<mo>=</mo>
<mn>58</mn>
</math></span>. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \geqslant 59">
<mi>x</mi>
<mo>⩾</mo>
<mn>59</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = {1.004^{12}}{\text{ }}( = 1.049 \ldots )">
<mi>r</mi>
<mo>=</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>12</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>1.049</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15\,000{r^n} - 1000\frac{{{r^n} - 1}}{{r - 1}} = 0 \Rightarrow n = 27.8 \ldots ">
<mn>15</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
<mrow>
<msup>
<mi>r</mi>
<mi>n</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>1000</mn>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mi>n</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>r</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>n</mi>
<mo>=</mo>
<mn>27.8</mn>
<mo>…</mo>
</math></span> (<strong><em>A1)(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for the equation (with their value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>), <strong><em>(M1) </em></strong>for attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> and <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 27.8 \ldots ">
<mi>n</mi>
<mo>=</mo>
<mn>27.8</mn>
<mo>…</mo>
</math></span></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 28">
<mi>n</mi>
<mo>=</mo>
<mn>28</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 27">
<mi>n</mi>
<mo>=</mo>
<mn>27</mn>
</math></span>.</p>
<p> </p>
<p><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>Let <strong><em>A </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 3&1 \\ 4&3 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <strong><em>A</em></strong><sup>2</sup> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span><em><strong>A </strong></em>+ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span><em><strong>I</strong></em> = <strong><em>O</em></strong> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z}">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span> and <strong><em>O </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&0 \\ 0&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> for which the matrix (<strong><em>A</em></strong> − <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><strong><em>I</em></strong>) is singular.</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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> and of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <strong><em>I</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{5}">
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
</math></span><strong><em>A </em></strong>(6<strong><em>I</em></strong> – <strong><em>A</em></strong>).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the result from <strong>part (b) (ii)</strong> to explain why <strong><em>A</em></strong> is non-singular.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the values from <strong>part (b) (i)</strong> to express <strong><em>A</em></strong><sup>4</sup> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span><em><strong>A</strong></em>+ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span><em><strong>I</strong></em> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Z}">
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><em>A</em></strong> − <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><strong><em>I</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {3 - \lambda }&1 \\ 4&{3 - \lambda } \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em><strong> A1</strong></em></p>
<p>If <strong><em>A</em></strong> − <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><strong><em>I</em></strong> is singular then det (<strong><em>A</em></strong> − <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><strong><em>I</em></strong>) = 0<em><strong> (R1)</strong></em></p>
<p>det (<strong><em>A</em></strong> − <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><strong><em>I</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {3 - \lambda } \right)^2} - 4\left( { = {\lambda ^2} - 6\lambda + 5} \right)">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mi>λ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em><strong> (A1)</strong></em></p>
<p>Attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {3 - \lambda } \right)^2} - 4 = 0">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> or equivalent for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> = 1, 5 <em><strong>A1 N2</strong></em></p>
<p><em> <strong>Note:</strong> </em>Candidates need both values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> for the final <em><strong>A1</strong></em>.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\begin{array}{*{20}{c}} 3&1 \\ 4&3 \end{array}} \right)^2} + m\left( {\begin{array}{*{20}{c}} 3&1 \\ 4&3 \end{array}} \right) + n\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&0 \\ 0&0 \end{array}} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>m</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\begin{array}{*{20}{c}} 3&1 \\ 4&3 \end{array}} \right)^2} = \left( {\begin{array}{*{20}{c}} {13}&6 \\ {24}&{13} \end{array}} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>24</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em><strong> (A1)</strong></em></p>
<p>Forming any two independent equations <em><strong> M1</strong></em></p>
<p>(<em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 + m = 0">
<mn>6</mn>
<mo>+</mo>
<mi>m</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="13 + 3m + n = 0">
<mn>13</mn>
<mo>+</mo>
<mn>3</mn>
<mi>m</mi>
<mo>+</mo>
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> or equivalent)</p>
<p><em><strong>Note:</strong> </em>Accept equations in matrix form.</p>
<p>Solving these two equations <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = -6">
<mi>m</mi>
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 5">
<mi>n</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>A</em></strong><sup>2</sup> − 6<strong><em>A</em></strong> + 5<strong><em>I</em></strong> = <strong><em>O</em></strong> <em><strong>(M1) </strong></em></p>
<p>5<strong><em>I</em></strong> = 6<strong><em>A</em></strong> − <strong><em>A</em></strong><sup>2</sup> <em><strong>A1</strong></em></p>
<p>= <strong><em>A</em></strong>(6<strong><em>I</em></strong> − <strong><em>A</em></strong>) <em><strong>A1A1 </strong></em></p>
<p><em><strong>Note:</strong> </em>Award <em><strong>A1</strong></em> for <em><strong>A</strong></em> and <em><strong>A1</strong></em> for (6<em><strong>I</strong></em> − <em><strong>A</strong></em>).</p>
<p><strong><em>I</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{5}">
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
</math></span><strong><em>A</em></strong>(6<strong><em>I</em></strong> − <strong><em>A</em></strong>) <em><strong>AG N0 </strong></em></p>
<p><em><strong>Special Case:</strong></em> Award <em><strong>M1A0A0A0</strong></em> only for candidates following alternative methods.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><strong><em>I</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{5}">
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
</math></span><strong><em>A</em></strong>(6<strong><em>I</em></strong> − <strong><em>A</em></strong>) = <strong><em>A </em></strong><em>×</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{5}">
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
</math></span>(6<strong><em>I</em></strong> − <strong><em>A</em></strong>) <em><strong>M1 </strong></em></p>
<p>Hence by definition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{5}">
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
</math></span>(6<strong><em>I</em></strong> − <strong><em>A</em></strong>) is the inverse of <strong><em>A</em></strong>. <em><strong>R1 </strong></em></p>
<p>Hence <strong><em>A</em></strong><sup>−1</sup> exists and so <strong><em>A</em></strong> is non-singular <em><strong>R1 N0 </strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>As det <strong><em>I</em></strong> = 1 (≠ 0), then <em><strong>R1 </strong></em></p>
<p>det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{5}">
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
</math></span><strong><em>A</em></strong>(6<strong><em>I</em></strong> − <strong><em>A</em></strong>) = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{5}">
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
</math></span> det <strong><em>A</em></strong> <em>×</em> det (6<strong><em>I</em></strong> − <strong><em>A</em></strong>) (≠ 0) <em><strong>M1 </strong></em></p>
<p>⇒ det <strong><em>A</em></strong> ≠ 0 and so <strong><em>A</em></strong> is non-singular. <em><strong> R1 N0 </strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><strong><em>A</em></strong><sup>2</sup> = 6<strong><em>A</em></strong> − 5<strong><em>I</em></strong> <em><strong>(A1) </strong></em></p>
<p><strong><em>A</em></strong><sup>4</sup> = (6<strong><em>A</em></strong> − 5<strong><em>I</em></strong>)<sup>2 </sup><em><strong>M1</strong></em></p>
<p> = 36<strong><em>A</em></strong><sup>2</sup> − 60<strong><em>AI</em></strong> + 25<strong><em>I</em></strong><sup>2 </sup><em><strong>A1</strong></em></p>
<p> = 36(6<strong><em>A</em></strong> − 5<strong><em>I</em></strong>) − 60<strong><em>A</em></strong> + 25<strong><em>I M1</em></strong></p>
<p> = 156<strong><em>A</em></strong> − 155<strong><em>I</em></strong> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = 156, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> = −155) <em><strong>A1 N0 </strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong><em>A</em></strong><sup>2</sup> = 6<strong><em>A</em></strong> − 5<strong><em>I (A1) </em></strong></p>
<p><strong><em>A</em></strong><sup>3</sup> = 6<strong><em>A</em></strong><sup>2</sup> − 5<strong><em>A</em></strong> where <strong><em>A</em></strong><sup>2</sup> = 6<strong><em>A</em></strong> − 5<strong><em>I M1</em></strong></p>
<p> = 31<strong><em>A </em></strong>− 30<strong><em>I A1 </em></strong></p>
<p><strong><em>A</em></strong><sup>4</sup> = 31<strong><em>A</em></strong><sup>2</sup> − 30<strong><em>A</em></strong> where <strong><em>A</em></strong><sup>2</sup> = 6<strong><em>A</em></strong> − 5<strong><em>I M1</em></strong></p>
<p> = 156<strong><em>A</em></strong> − 155<strong><em>I</em></strong> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = 156, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> = −155) <em><strong>A1 N0 </strong></em></p>
<p> </p>
<p><em><strong>Note:</strong></em> Do not accept methods that evaluate A4 directly from <em><strong>A</strong></em>.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</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>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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><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> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1471}}{{32}} + 1">
<mfrac>
<mrow>
<mn>1471</mn>
</mrow>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mn>1</mn>
</math></span></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="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&2 \\ 2&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} p&2 \\ 0&q \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>q</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Find <strong><em>A</em></strong><sup>−1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Find <strong><em>A</em></strong><sup>2</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that 2<strong><em>A</em></strong> + <em><strong>B </strong></em>=<em><strong> </strong></em><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&6 \\ 4&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> and of <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span></span>.</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 class="indent1" style="margin-top:12.0pt;">Hence find <strong><em>A</em></strong><sup>−1</sup><strong><em>B</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Let <strong><em>X</em></strong> be a 2 × 2 matrix such that <strong><em>AX</em></strong> = <strong><em>B</em></strong>. Find <strong><em>X</em></strong>.</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong><em>A</em></strong><sup>−1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&{\frac{1}{2}} \\ {\frac{1}{2}}&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>A</em></strong><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&0 \\ 0&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&4 \\ 4&0 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} p&2 \\ 0&q \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 2&6 \\ 4&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>q</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></span></p>
<p><span style="background-color: #ffffff;"> <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = 2, </span><span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> = 3 <em><strong>A1A1 N3</strong></em></span></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Evidence of attempt to multiply <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <strong><em>A</em></strong><sup>−1</sup><strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&{\frac{1}{2}} \\ {\frac{1}{2}}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&2 \\ 0&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong><em>A</em></strong><sup>−1</sup><strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&{\frac{3}{2}} \\ 1&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{accept}}\left( {\begin{array}{*{20}{c}} 0&{\frac{1}{2}q} \\ {\frac{1}{2}p}&1 \end{array}} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>accept</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>q</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>p</mi>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Evidence of correct approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <strong><em>X</em></strong> = <strong><em>A</em></strong><sup>−1</sup><strong><em>B</em></strong>, setting up a system of equations</p>
<p><strong><em>X</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&{\frac{3}{2}} \\ 1&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{accept}}\left( {\begin{array}{*{20}{c}} 0&{\frac{1}{2}q} \\ {\frac{1}{2}p}&1 \end{array}} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>accept</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>q</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>p</mi>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question, give all answers to two decimal places.</strong></p>
<p>Bryan decides to purchase a new car with a price of €14 000, but cannot afford the full amount. The car dealership offers two options to finance a loan.</p>
<p><strong>Finance option A:</strong></p>
<p>A 6 year loan at a nominal annual interest rate of 14 % <strong>compounded quarterly</strong>. No deposit required and repayments are made each quarter.</p>
</div>
<div class="specification">
<p><strong>Finance option B:</strong></p>
<p>A 6 year loan at a nominal annual interest rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> % <strong>compounded monthly</strong>. Terms of the loan require a 10 % deposit and monthly repayments of €250.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the repayment made each quarter.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total amount paid for the car.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the interest paid on the loan.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount to be borrowed for this option.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the annual interest rate, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State which option Bryan should choose. Justify your answer.</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>Bryan chooses option B. The car dealership invests the money Bryan pays as soon as they receive it.</p>
<p>If they invest it in an account paying 0.4 % interest per month and inflation is 0.1 % per month, calculate the real amount of money the car dealership has received by the end of the 6 year period.</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>N = 24<br>I % = 14<br>PV = −14000<br>FV = 0<br>P/Y = 4<br>C/Y = 4 <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for an attempt to use a financial app in their technology, award <em><strong>A1</strong> </em>for all entries correct. Accept PV = 14000.</p>
<p>(€)871.82 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>4 × 6 × 871.82 <em><strong>(M1)</strong></em></p>
<p>(€) 20923.68 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>20923.68 − 14000 <em><strong>(M1)</strong></em></p>
<p>(€) 6923.68 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.9 × 14000 (= 14000 − 0.10 × 14000) <em><strong>M1</strong></em></p>
<p>(€) 12600.00 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>N = 72</p>
<p>PV = 12600</p>
<p>PMT = −250</p>
<p>FV = 0</p>
<p>P/Y = 12</p>
<p>C/Y = 12 <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to use a financial app in their technology, award <em><strong>A1</strong> </em>for all entries correct. Accept PV = −12600 provided PMT = 250.</p>
<p>12.56(%) <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>EITHER</strong></em></p>
<p>Bryan should choose Option A <em><strong>A1</strong></em></p>
<p>no deposit is required <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong></em> for stating that no deposit is required. Award <em><strong>A1</strong> </em>for the correct choice from that fact. Do not award <em><strong>R0A1</strong></em>.</p>
<p><em><strong>OR</strong></em></p>
<p>Bryan should choose Option B <em><strong>A1</strong></em></p>
<p>cost of Option A (6923.69) > cost of Option B (72 × 250 − 12600 = 5400) <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for a correct comparison of costs. Award <em><strong>A1</strong></em> for the correct choice from that comparison. Do not award <em><strong>R0A1</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>real interest rate is 0.4 − 0.1 = 0.3% <em><strong>(M1)</strong></em></p>
<p>value of other payments 250 + 250 × 1.003 + … + 250 × 1.003<sup>71</sup></p>
<p>use of sum of geometric sequence formula or financial app on a GDC <em><strong>(M1)</strong></em></p>
<p>= 20 058.43</p>
<p>value of deposit at the end of 6 years</p>
<p>1400 × (1.003)<sup>72</sup> = 1736.98 <em><strong>(A1)</strong></em></p>
<p>Total value is (€) 21 795.41 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Both <em><strong>M</strong></em> marks can awarded for a correct use of the GDC’s financial app:</p>
<p style="padding-left:120px;">N = 72 (6 × 12)<br>I % = 3.6 (0.3 × 12)<br>PV = 0<br>PMT = −250<br>FV =<br>P/Y = 12<br>C/Y = 12</p>
<p style="padding-left:120px;"><em><strong>OR</strong></em></p>
<p style="padding-left:120px;">N = 72 (6 × 12)<br>I % = 0.3<br>PV = 0<br>PMT = −250<br>FV =<br>P/Y = 1<br>C/Y = 1</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Matrices <strong><em>A</em></strong>, <strong><em>B</em></strong> and <strong><em>C</em></strong> are defined by</p>
<p><strong><em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5&1 \\ 7&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&4 \\ { - 3}&{15} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>15</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>C</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 9&{ - 7} \\ 8&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>7</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Let <strong><em>X</em></strong> be an unknown 2 × 2 matrix satisfying the equation</p>
<p style="text-align: center;"><strong><em>AX</em> </strong>+<strong> <em>B</em> </strong>=<strong> <em>C</em></strong>.</p>
<p>This equation may be solved for <strong><em>X</em></strong> by rewriting it in the form</p>
<p style="text-align: center;"><strong><em>X</em> </strong>=<strong> <em>A</em></strong><sup>−1</sup><strong> <em>D</em></strong>.</p>
<p>where <strong><em>D</em></strong> is a 2 × 2 matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <strong><em>A</em></strong><sup>−1</sup>.</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 <strong><em>D</em></strong>.</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>Find <strong><em>X</em></strong>.</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="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong><em>A</em></strong><sup>−1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{2}{3}}&{ - \frac{1}{3}} \\ { - \frac{7}{3}}&{\frac{5}{3}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3}\left( {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 7}&5 \end{array}} \right)">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>7</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {0.667}&{ - 0.333} \\ { - 2.33}&{ - 1.67} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.667</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.333</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2.33</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1.67</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)(A1)(N2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>AX</em></strong> = <strong><em>C</em></strong> − <strong><em>B</em></strong> (may be implied) <em><strong>(A1)</strong></em></p>
<p><strong><em>X</em></strong> = <strong><em>A</em></strong><sup>−1 </sup><strong>(<em>C</em></strong><em>−</em><strong><em>B</em>)</strong> <em><strong>(A1)</strong></em></p>
<p><strong><em>D</em> </strong>=<strong> <em>C </em></strong><em>−</em><strong> <em>B</em></strong> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 7&{ - 11} \\ {11}&{ - 13} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1) (N3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>X</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&{ - 3} \\ 2&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A2) (N2)</strong></em></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="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
<mi>S</mi>
</math></span><sub><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sub> be the sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> terms of the arithmetic series 2 + 4 + 6 + ….</p>
</div>
<div class="specification">
<p>Let <strong><em>M </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>It may now be assumed that <strong><em>M</em></strong><sup><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&{2n} \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>n</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> ≥ 4. The sum <strong><em>T</em></strong><sub><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sub> is defined by</p>
<p style="text-align: center;"><strong><em>T</em></strong><sub><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sub> = <strong><em>M</em></strong><sup>1</sup> + <strong><em>M</em></strong><sup>2</sup> + <strong><em>M</em></strong><sup>3</sup> + ... + <strong><em>M</em></strong><sup><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
<mi>S</mi>
</math></span><sub>4</sub>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
<mi>S</mi>
</math></span><sub>100</sub>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>M</em></strong><sup>2</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <strong><em>M</em></strong><sup>3</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&6 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <strong><em>M</em></strong><sup>4</sup>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>T</em></strong><sub>4</sub>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using your results from part (a) (ii), find <strong><em>T</em></strong><sub>100</sub>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
<mi>S</mi>
</math></span><sub>4</sub> = 20 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span><sub>1</sub> = 2, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> = 2 <em><strong>(A1) </strong></em></p>
<p>Attempting to use formula for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
<mi>S</mi>
</math></span><sub><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sub> <em><strong>M1 </strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
<mi>S</mi>
</math></span><sub>100</sub> = 10100 <em><strong> A1 N2 </strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>M</em></strong><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&4 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A2 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>For writing <strong><em>M</em></strong><sup>3</sup> as <strong><em>M</em></strong><sup>2</sup> × <strong><em>M</em></strong> or <strong><em>M</em></strong> × <strong><em>M</em></strong><sup>2</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{or}}\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&4 \\ 0&1 \end{array}} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>or</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong><em>M</em></strong><sup>3</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {1 + 0}&{4 + 2} \\ {0 + 0}&{0 + 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>0</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>4</mn>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0</mn>
<mo>+</mo>
<mn>0</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2</strong></em></p>
<p><strong><em>M</em></strong><sup>3</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&6 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> AG N0</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>M</em></strong><sup>4</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&8 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>T</em></strong><sub>4</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&4 \\ 0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&6 \\ 0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&8 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&{20} \\ 0&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mrow>
<mn>20</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>T</em></strong><sub>100</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&4 \\ 0&1 \end{array}} \right) + \ldots + \left( {\begin{array}{*{20}{c}} 1&{200} \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mn>200</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {100}&{10100} \\ 0&{100} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>100</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>10100</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mn>100</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&1 \\ 2&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Write down the determinant of <strong><em>M</em></strong>.</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="indent1" style="margin-top:12.0pt;"> Write down <strong><em>M</em></strong><sup>−1</sup>.</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="indent1" style="margin-top:12.0pt;"><strong>Hence</strong> solve <strong><em>M</em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 4 \\ 8 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p class="indent1" style="margin-top:12.0pt;"> </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="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>det <strong><em>M</em></strong> = −4 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>M</em></strong><sup>−1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{4}\left( {\begin{array}{*{20}{c}} { - 1}&{ - 1} \\ { - 2}&2 \end{array}} \right)\,\,\,\left( { = \left( {\begin{array}{*{20}{c}} {\frac{1}{4}}&{\frac{1}{4}} \\ {\frac{1}{2}}&{ - \frac{1}{2}} \end{array}} \right)} \right)">
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1 N2</strong></em></p>
<p><strong>Note:</strong><em> </em>Award <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{4}">
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> and <em><strong>A1</strong> </em>for the correct matrix. </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><em>X </em></strong>=<strong><em> M</em></strong><sup>−1</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4 \\ 8 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{X}} = - \frac{1}{4}\left( {\begin{array}{*{20}{c}} { - 1}&{ - 1} \\ { - 2}&2 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 4 \\ 8 \end{array}} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>X</mtext>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong><em>X </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ { - 2} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3">
<mi>x</mi>
<mo>=</mo>
<mn>3</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 2">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span>) <em><strong>A1A1 N0</strong></em></p>
<p> </p>
<p><strong>Note:</strong><em> </em>Award no marks for an <strong>algebraic</strong> solution of the system <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + y = 4">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>=</mo>
<mn>4</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y = 8">
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>=</mo>
<mn>8</mn>
</math></span>. </p>
<p><em><strong>[3 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="specification">
<p>Let <strong><em>A</em> </strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em> </strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0 \\ d&e \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>d</mi>
</mtd>
<mtd>
<mi>e</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>. Giving your answers in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="e">
<mi>e</mi>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>write down <strong><em>A</em> </strong>+<strong> <em>B</em></strong>.</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 <strong><em>AB</em></strong>.</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="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong><em>A</em></strong> + <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&0 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&0 \\ d&e \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>d</mi>
</mtd>
<mtd>
<mi>e</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {a + 1}&b \\ {c + d}&e \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>c</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
</mtd>
<mtd>
<mi>e</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>A</em></strong><strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&0 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&0 \\ d&e \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>d</mi>
</mtd>
<mtd>
<mi>e</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1A1A1</strong></em></p>
<p><strong>Note</strong>: Award <em><strong>N2</strong></em> for finding <strong>BA</strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ {ad + ce}&{bd} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mi>d</mi>
<mo>+</mo>
<mi>c</mi>
<mi>e</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>b</mi>
<mi>d</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = a + b{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{b}} \in {\mathbb{R}^ + }">
<mrow>
<mtext>b</mtext>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span> and let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arg}}\,z = \theta ">
<mrow>
<mtext>arg</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>=</mo>
<mi>θ<!-- θ --></mi>
</math></span>.</p>
</div>
<div class="question">
<p>Show the points represented by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z - 2a">
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</math></span> on the following Argand diagram.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><img src="data:image/png;base64,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"> A1</p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in first quadrant and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z - 2a">
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</math></span> its reflection in the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
<p><em><strong>[1 mark]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider a geometric sequence with a first term of 4 and a fourth term of −2.916.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio of this sequence.</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 sum to infinity of this sequence.</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="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_4} = {u_1}{r^3} \Rightarrow - 2.916 = 4{r^3}"> <mrow> <msub> <mi>u</mi> <mn>4</mn> </msub> </mrow> <mo>=</mo> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <mi>r</mi> <mn>3</mn> </msup> </mrow> <mo stretchy="false">⇒</mo> <mo>−</mo> <mn>2.916</mn> <mo>=</mo> <mn>4</mn> <mrow> <msup> <mi>r</mi> <mn>3</mn> </msup> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p>solving, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = - 0.9"> <mi>r</mi> <mo>=</mo> <mo>−</mo> <mn>0.9</mn> </math></span> <em><strong> (M1)A1</strong></em></p>
<p> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } = \frac{4}{{1 - \left( { - 9} \right)}}"> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">∞</mi> </msub> </mrow> <mo>=</mo> <mfrac> <mn>4</mn> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>9</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{40}}{{19}}\,\left( { = 2.11} \right)"> <mo>=</mo> <mfrac> <mrow> <mn>40</mn> </mrow> <mrow> <mn>19</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2.11</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[2 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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M = \left( {\begin{array}{*{20}{c}} a&2 \\ 2&{ - 1} \end{array}} \right)">
<mi>M</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in \mathbb{Z}">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^2}">
<mrow>
<msup>
<mi>M</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></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>If <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^2}">
<mrow>
<msup>
<mi>M</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5&{ - 4} \\ { - 4}&5 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></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>Using this value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^{ - 1}}">
<mrow>
<msup>
<mi>M</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and <strong>hence </strong>solve the system of equations:</p>
<p style="padding-left:180px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - x + 2y = - 3">
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span></p>
<p style="padding-left:180px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y = 3">
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^2} = \left( {\begin{array}{*{20}{c}} a&2 \\ 2&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} a&2 \\ 2&{ - 1} \end{array}} \right)">
<mrow>
<msup>
<mi>M</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {{a^2} + 4}&{2a - 2} \\ {2a - 2}&5 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)(A1)(A1)(A1)</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2a - 2 = - 4">
<mn>2</mn>
<mi>a</mi>
<mo>−</mo>
<mn>2</mn>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a = - 1">
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p>Substituting: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^2} + 4 = {\left( { - 1} \right)^2} + 4 = 5">
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mo>=</mo>
<mn>5</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Candidates may solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^2} + 4 = 5">
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mo>=</mo>
<mn>5</mn>
</math></span> to give <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \pm 1">
<mi>a</mi>
<mo>=</mo>
<mo>±</mo>
<mn>1</mn>
</math></span>, and then show that only <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 1">
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> satisfies <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2a - 2 = - 4">
<mn>2</mn>
<mi>a</mi>
<mo>−</mo>
<mn>2</mn>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span>.</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M = \left( {\begin{array}{*{20}{c}} { - 1}&2 \\ 2&{ - 1} \end{array}} \right)">
<mi>M</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^{ - 1}} = - \frac{1}{3}\left( {\begin{array}{*{20}{c}} { - 1}&{ - 2} \\ { - 2}&{ - 1} \end{array}} \right)">
<mrow>
<msup>
<mi>M</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}\left( {\begin{array}{*{20}{c}} 1&2 \\ 2&1 \end{array}} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{1}{3}}&{\frac{2}{3}} \\ {\frac{2}{3}}&{\frac{1}{3}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - x + 2y = - 3">
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y = 3">
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} { - 1}&2 \\ 2&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 3} \\ 3 \end{array}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em><em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} {\frac{1}{3}}&{\frac{2}{3}} \\ {\frac{2}{3}}&{\frac{1}{3}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} { - 1}&2 \\ 2&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\frac{1}{3}}&{\frac{2}{3}} \\ {\frac{2}{3}}&{\frac{1}{3}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} { - 3} \\ 3 \end{array}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p>ie <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p class="accept"><strong>Note: </strong>The solution must use matrices<span style="font-style: normal;">.</span> Award no marks for solutions using other methods<span style="font-style: normal;">.</span></p>
<p><em><strong>[6 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="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}">
<mi>γ<!-- γ --></mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>The matrix <strong><em>A</em> </strong>is defined by <strong><em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} \gamma &1 \\ 0&{\frac{1}{\gamma }} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>γ<!-- γ --></mi>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mi>γ<!-- γ --></mi>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Deduce that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\gamma ^2} = \gamma - 1">
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>γ</mi>
<mo>−</mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - \gamma } \right)^6}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>γ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>6</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"><strong><em>A</em></strong><sup>3</sup><strong> </strong>= –<strong><em>I</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"><strong><em>A</em></strong><sup>–1</sup> = <strong><em>I</em></strong> – <strong><em>A</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma ">
<mi>γ</mi>
</math></span> is a root of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^2} - z + 1 = 0">
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>z</mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\gamma ^2} - \gamma + 1 = 0">
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>γ</mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore {\gamma ^2} = \gamma - 1">
<mo>∴</mo>
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>γ</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for the use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^2} - z + 1 = 0">
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>z</mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> in any way.</p>
<p>Award <em><strong>R1</strong></em> for a correct reasoned approach.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\gamma ^2} = \frac{{ - 1 + {\text{i}}\sqrt 3 }}{2}">
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma - 1 = \frac{{1 + {\text{i}}\sqrt 3 }}{2} - 1 = \frac{{ - 1 + {\text{i}}\sqrt 3 }}{2}">
<mi>γ</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - \gamma } \right)^6} = {\left( { - {\gamma ^2}} \right)^6}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>γ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>6</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>6</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( \gamma \right)^{12}}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mi>γ</mi>
<mo>)</mo>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {{\gamma ^3}} \right)^4}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>γ</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( { - 1} \right)^4}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1">
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - \gamma } \right)^6}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>γ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>6</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 - 6\gamma + 15{\gamma ^2} - 20{\gamma ^3} + 15{\gamma ^4} - 6{\gamma ^5} + {\gamma ^6}">
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>6</mn>
<mi>γ</mi>
<mo>+</mo>
<mn>15</mn>
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>20</mn>
<mrow>
<msup>
<mi>γ</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mrow>
<msup>
<mi>γ</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mrow>
<msup>
<mi>γ</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>γ</mi>
<mn>6</mn>
</msup>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt at binomial expansion.</p>
<p>use of any previous result e.g. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 - 6\gamma + 15{\gamma ^2} + 20 - 15\gamma + 6{\gamma ^2} + 1">
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>6</mn>
<mi>γ</mi>
<mo>+</mo>
<mn>15</mn>
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>20</mn>
<mo>−</mo>
<mn>15</mn>
<mi>γ</mi>
<mo>+</mo>
<mn>6</mn>
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1">
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note: </strong>As the question uses the word ‘hence’, other methods that do not use previous results are awarded no marks.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>A</em></strong><sup>2</sup> = <strong><em>A</em></strong> <em>–</em> <strong><em>I</em></strong> </p>
<p>⇒<strong><em> A</em></strong><sup>3</sup> = <strong><em>A</em></strong><sup>2</sup><em> – <strong>A</strong></em> <em><strong>M1A1</strong></em> </p>
<p> = <strong><em>A</em></strong><em> – <strong>I</strong> – <strong>A</strong></em> <em><strong> A1</strong></em> </p>
<p> = <em>–<strong>I</strong></em> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Allow other valid methods.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>I</em></strong> = <strong><em>A</em></strong> <em>–</em> <strong><em>A</em></strong><sup>2 </sup></p>
<p><strong><em>A</em></strong><sup>–1</sup> = <strong><em>A</em></strong><sup>–1</sup><strong><em>A</em></strong> – <strong><em>A</em></strong><sup>–1</sup><strong><em>A</em></strong><sup>2</sup> <em><strong>M1A1 </strong></em></p>
<p>⇒ <strong><em>A</em></strong><sup>–1</sup> = <strong><em>I</em></strong> <em>–</em> <strong><em>A</em></strong> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Allow other valid methods.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Long term experience shows that if it is sunny on a particular day in Vokram, then the probability that it will be sunny the following day is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math>. If it is not sunny, then the probability that it will be sunny the following day is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>3</mn></math>.</p>
<p>The transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> is used to model this information, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> can be written as a product of three matrices, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">D</mi><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math> is a diagonal matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is sunny today. Find the probability that it will be sunny in three days’ time.</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 eigenvalues and eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math>.</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>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the long-term percentage of sunny days in Vokram.</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>finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mn>3</mn></msup></math> <strong>OR </strong> use of tree diagram <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mn>3</mn></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>65</mn></mtd><mtd><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>525</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>35</mn></mtd><mtd><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>475</mn></mtd></mtr></mtable></mfenced></math></p>
<p>the probability of sunny in three days’ time is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>65</mn></math> <em><strong>A1</strong></em></p>
<p> </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 find eigenvalues <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong><br>Note:</strong> Any indication that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mrow><mi mathvariant="bold-italic">T</mi><mo>-</mo><mi>λ</mi><mi mathvariant="bold-italic">I</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> has been used is sufficient for the <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn><mo>-</mo><mi>λ</mi></mtd><mtd><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>7</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>8</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>.</mo><mn>7</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>λ</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p>attempt to find either eigenvector <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>y</mi><mo>=</mo><mi>x</mi><mo>⇒</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math> so an eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>⇒</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math> so an eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept multiples of the stated eigenvectors.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo> </mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mo> </mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo> </mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo> </mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Examiners should be aware that different, correct, matrices <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> may be seen.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math> must be consistent with each other.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><msup><mn>5</mn><mi>n</mi></msup><mo>→</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>only if their <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup></math> corresponds to their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>6</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>4</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo>%</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The matrix <em><strong>M</strong></em> is given by <em><strong>M</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} 1&2&2 \\ 3&1&1 \\ 2&3&1 \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em><strong>M</strong></em><sup>3</sup> can be written as a quadratic expression in <em><strong>M</strong></em> in the form <em>a<strong>M</strong></em><sup>2</sup> + <em>b<strong>M</strong></em> + <em>c<strong>I</strong></em> , determine the values of the constants <em>a</em>, <em>b</em> and <em>c</em>.</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>Show that <em><strong>M</strong></em><sup>4</sup> = 19<em><strong>M</strong></em><sup>2</sup> + 40<em><strong>M</strong></em> + 30<em><strong>I</strong></em>.</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>Using mathematical induction, prove that <em><strong>M</strong></em><sup>n</sup> can be written as a quadratic expression in <em><strong>M</strong></em> for all positive integers <em>n </em>≥ 3.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a quadratic expression in <em><strong>M</strong></em> for the inverse matrix <em><strong>M</strong></em><sup>–1</sup>.</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><em><strong>M</strong></em><sup>2</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} {11}&{10}&6 \\ 8&{10}&8 \\ {13}&{10}&8 \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>; <em><strong>M</strong></em><sup>3</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} {53}&{50}&{38} \\ {54}&{50}&{34} \\ {59}&{60}&{44} \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>53</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>50</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>38</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>54</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>50</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>34</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>59</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>60</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>44</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <em><strong>(A1)(A1)</strong></em></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} {53}&{50}&{38} \\ {54}&{50}&{34} \\ {59}&{60}&{44} \end{array}} \right] = a\left[ {\begin{array}{*{20}{c}} {11}&{10}&6 \\ 8&{10}&8 \\ {13}&{10}&8 \end{array}} \right] + b\left[ {\begin{array}{*{20}{c}} 1&2&2 \\ 3&1&1 \\ 2&3&1 \end{array}} \right] + c\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>53</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>50</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>38</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>54</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>50</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>34</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>59</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>60</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>44</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>then, for example</p>
<p>11<em>a</em> + <em>b</em> + <em>c</em> = 53</p>
<p>10<em>a</em> + 2<em>b</em> = 50 <em><strong> M1A1</strong></em></p>
<p>6<em>a</em> + 2<em>b</em> = 38</p>
<p>the solution is <em>a</em> = 3, <em>b</em> =10, <em>c</em> =10 <em><strong>(M1)A1</strong></em></p>
<p>(<em><strong>M</strong></em><sup>3</sup> = 3<em><strong>M</strong></em><sup>2</sup> +10<em><strong>M</strong></em> +10<em><strong>I</strong></em>)</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>det(<em><strong>M</strong></em> − <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><em><strong>I</strong></em>) = 0 <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {1 - \lambda } \right)\left( {{{\left( {1 - \lambda } \right)}^2} - 3} \right) - 2\left( {3\left( {1 - \lambda } \right) - 2} \right) + 2\left( {9 - 2\left( {1 - \lambda } \right)} \right) = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>9</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\lambda ^3} + 3{\lambda ^2} + 10\lambda + 10 = 0">
<mo>−</mo>
<mrow>
<msup>
<mi>λ</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>10</mn>
<mi>λ</mi>
<mo>+</mo>
<mn>10</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong> M1A1</strong></em></p>
<p>applying the Cayley – Hamilton theorem <em><strong>(M1)</strong></em></p>
<p><em><strong>M</strong></em><sup>3</sup> = 3<em><strong>M</strong></em><sup>2</sup> +10<em><strong>M</strong></em> +10<em><strong>I</strong></em> and so <em>a</em> = 3, <em>b</em> =10, <em>c</em> =10 <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>M</strong></em><sup>4</sup> = 3<em><strong>M</strong></em><sup>3</sup> + 10<em><strong>M</strong></em><sup>2</sup> + 10<em><strong>M M1</strong></em></p>
<p>= 3(3<em><strong>M</strong></em><sup>2</sup> + 10<em><strong>M</strong></em> + 10<strong><em>I</em></strong>) + 10<em><strong>M</strong></em><sup>2</sup> + 10<em><strong>M M1</strong></em></p>
<p>=19<em><strong>M</strong></em><sup>2</sup> + 40<em><strong>M</strong></em> +30<strong><em>I AG</em></strong> </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>the statement is true for <em>n</em> = 3 as shown in part (a) <em><strong>A1</strong></em></p>
<p>assume true for <em>n</em> = <em>k</em>, <em>ie</em> <em><strong>M</strong></em><sup>k</sup> = <em>p<strong>M</strong></em><sup>2</sup> + <em>q<strong>M</strong></em> + <em>r<strong>I</strong></em> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Subsequent marks after this <em><strong>M1</strong> </em>are independent and can be awarded.</p>
<p><em><strong>M</strong><sup>k</sup></em><sup>+1</sup> = <em>p<strong>M</strong><span style="font-size: 11.66px;"><sup>3</sup></span></em> + <em>q<strong>M</strong></em><sup>2</sup> + <em>r</em><strong><em>M </em></strong> <em><strong>M1</strong></em></p>
<p>= <em>p</em>(3<em><strong>M</strong></em><sup>2</sup> +10<em><strong>M</strong></em> +10<em><strong>I</strong></em>) + <em>q<strong>M</strong><sup>2</sup></em> + <em>r<strong>M M1</strong></em></p>
<p>= (3<em>p</em> +<em>q</em>)<em><strong>M</strong></em><sup>2</sup> + (10<em>p</em> + <em>r</em>)<span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: bold;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">M</span> + 10<em>p<strong>I</strong></em> <em><strong>A1</strong></em></p>
<p>hence true for <em>n</em> = <em>k</em> ⇒ true for <em>n</em> = <em>k</em> +1 and since true for <em>n</em> = 3, the statement is proved by induction <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong></em> provided at least four of the previous marks are gained.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>M</strong></em><sup>2</sup> = 3<strong><em>M</em></strong> + 10<em><strong>I</strong></em> + 10<em><strong>M</strong></em><sup>–1</sup> <em><strong>M1</strong></em></p>
<p><em><strong>M</strong></em><sup>–1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{10}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>10</mn>
</mrow>
</mfrac>
</math></span>(<em><strong>M</strong></em><sup>2</sup> − 3<strong><em>M</em></strong> − 10<em><strong>I</strong></em>) <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1a" style="margin-top:12.0pt;">Write down the inverse of the matrix <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&{ - 3}&0 \\ 2&0&1 \\ 4&1&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></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="indent1a" style="margin-top:12.0pt;">Hence or otherwise solve</p>
<p class="indent1a" style="margin-top:12.0pt;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="x - 3y = 1">
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p class="indent1a" style="margin-top:12.0pt;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="2x + z = 2">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
</math></span></p>
<p class="indent1a" style="margin-top:12.0pt;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="4x + y + 3z = - 1">
<mn>4</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></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="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong><em>A</em></strong><sup>−1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 0.2}&{1.8}&{ - 0.6} \\ { - 0.4}&{0.6}&{ - 0.2} \\ {0.4}&{ - 2.6}&{1.2} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1.8</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.6</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0.4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2.6</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1.2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A2 N2</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>For recognizing that the equations may be written as <strong><em>A</em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p>For attempting to calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = {{\text{A}}^{ - 1}}\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 4, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 1, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> = −6 <em><strong> A2 N4</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&1&1 \\ 0&1&1 \\ 0&0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 1&1&0 \\ 1&1&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Given that <strong><em>X</em> </strong>= <strong><em>B</em> </strong>– <strong><em>A</em></strong><sup>–1</sup> and <strong><em>Y</em> </strong>= <strong><em>B</em></strong><sup>–1</sup> – <strong><em>A</em></strong>,</p>
</div>
<div class="specification">
<p>You are told that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A^n} = \left( {\begin{array}{*{20}{c}} 1&n&{\frac{{n\left( {n + 1} \right)}}{2}} \\ 0&1&n \\ 0&0&1 \end{array}} \right)">
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>n</mi>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{A^n}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} 1&x&y \\ 0&1&x \\ 0&0&1 \end{array}} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">find<strong><em> X </em></strong>and<strong> <em>Y</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">does <strong><em>X</em></strong><sup>–1</sup> + <strong><em>Y</em></strong><sup>–1</sup> have an inverse? Justify your conclusion.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">and hence find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A^n} + {\left( {{A^n}} \right)^{ - 1}}">
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><em>X</em></strong> = <strong><em>B</em></strong> – <strong><em>A</em></strong><sup>–1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 1&1&0 \\ 1&1&1 \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 1&{ - 1}&0 \\ 0&1&{ - 1} \\ 0&0&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1&0 \\ 1&0&1 \\ 1&1&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong><em>Y </em></strong>=<strong><em> B</em></strong><sup>–1</sup><strong><em> </em></strong><em>–</em><strong><em> A </em></strong><em>= </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0&0 \\ { - 1}&1&0 \\ 1&{ - 1}&1 \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 1&1&1 \\ 0&1&1 \\ 0&0&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&{ - 1}&{ - 1} \\ { - 1}&0&{ - 1} \\ 0&{ - 1}&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em> <strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>X</em></strong><sup>–1</sup> <em>+</em> <strong><em>Y</em></strong><sup>–1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&{ - 1}&0 \\ 1&0&{ - 1} \\ 0&1&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong><em>X</em></strong><sup>–1</sup> <em>+</em> <strong><em>Y</em></strong><sup>–1</sup> has no inverse <em><strong> A1 </strong></em></p>
<p>as det(<strong><em>X</em></strong><sup>–1</sup> <em>+</em> <strong><em>Y</em></strong><sup>–1</sup>) = 0<em> <strong>R1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A^n}{\left( {{A^n}} \right)^{ - 1}} = I \Rightarrow \left( {\begin{array}{*{20}{c}} 1&n&{\frac{{n\left( {n + 1} \right)}}{2}} \\ 0&1&n \\ 0&0&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&x&y \\ 0&1&x \\ 0&0&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right)">
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mi>I</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>n</mi>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} 1&{x + n}&{y + nx + \frac{{n\left( {n + 1} \right)}}{2}} \\ 0&1&{x + n} \\ 0&0&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>n</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>y</mi>
<mo>+</mo>
<mi>n</mi>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>n</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p>solve simultaneous equations to obtain</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + n = 0">
<mi>x</mi>
<mo>+</mo>
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y + nx + \frac{{n\left( {n + 1} \right)}}{2} = 0">
<mi>y</mi>
<mo>+</mo>
<mi>n</mi>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span><em> <strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - n">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mi>n</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{n\left( {n - 1} \right)}}{2}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em><em><strong>A1N2</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A^n} + {\left( {{A^n}} \right)^{ - 1}} = \, \Rightarrow \left( {\begin{array}{*{20}{c}} 1&n&{\frac{{n\left( {n + 1} \right)}}{2}} \\ 0&1&n \\ 0&0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&{ - n}&{\frac{{n\left( {n - 1} \right)}}{2}} \\ 0&1&{ - n} \\ 0&0&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 2&0&{{n^2}} \\ 0&2&0 \\ 0&0&2 \end{array}} \right)">
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>n</mi>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mi>n</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mi>n</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> measures the concentration of mercury in micrograms per litre.</p>
<p>The situation is modelled using the second order differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math> is the time measured in days since the leak started. It is known that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>If the mercury levels are greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> micrograms per litre, fishing in the river is considered unsafe and is stopped.</p>
</div>
<div class="specification">
<p>The river authority decides to stop people from fishing in the river for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> longer than the time found from the model.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the system of coupled first order equations:</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math></p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi></math></p>
<p style="text-align:left;">can be written as the given second order differential equation.</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 eigenvalues of the system of coupled first order equations given in part (a).</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>Hence find the exact solution of the second order differential equation.</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>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, labelling the maximum point of the graph with its coordinates.</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>Use the model to calculate the total amount of time when fishing should be stopped.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down one reason, with reference to the context, to support this decision.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>differentiating first equation. <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>substituting in for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong><em>AG</em></strong></p>
<p><br><strong>Note:</strong> The <strong>AG</strong> line must be seen to award the final <em><strong>M1</strong></em> mark.</p>
<p> </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>the relevant matrix is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> is also possible.</p>
<p><br>(this has characteristic equation) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>λ</mi><mfenced><mrow><mo>-</mo><mn>3</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 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>the general solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Must have constants, but condone sign error for the <em><strong>M1</strong></em>.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>M1A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to find eigenvectors <em><strong>(M1)</strong></em></p>
<p>respective eigenvectors are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> (or any multiple)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>the initial conditions become:</p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><mo>-</mo><mi>A</mi><mo>-</mo><mn>2</mn><mi>B</mi></math> <em><strong>M1</strong></em></p>
<p>this is solved by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></p>
<p>so the solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img src="data:image/png;base64,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"> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct shape (needs to go through origin, have asymptote at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> and a single maximum; condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>). Award <em><strong>A1</strong></em> for correct coordinates of maximum.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>intersecting graph with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p style="padding-left:60px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUUAAACFCAYAAADM6+pdAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAByLSURBVHhe7Z0JmBTFGYYLEBGNJ3gSSBQRBVGIoogElGCIQYxHMIhEREUUr4iK9wVIMKBGFEVFMIoEiMaDgAESroBJOEQEBDlMUCOHiKyA4AGdfmur195hZnfuaXa/93n6menq2Z2e7q6v/v+vv6qqeD5GCCGEpap7FUII4SNRFEKIEBJFIYQIIVEUQogQlVoUX375ZbN27Vq3J4QQlVgU33nnHdO5c2fTq1cvVyKEEJVUFHfs2GF69OhhatWqZebNm2dee+01d0QIUdmplKL4yiuvmLp165o2bdqY3/72t+aGG24wRUVF7qgQojJT6URx586dZsiQIWbo0KF2v2nTpubiiy82t99+u90XQlRuKp0o9uvXz3To0MEcfvjhrsSYvn37msWLF5tx48a5EiFEZaVSieKmTZvM5MmTTe/evV1JMTVq1DBjxowxzz77rCsRQlRWKpUoHnDAAWb27NlWBGOpU6eOmTJlitsTQlRWKmVHixBCJEKiKIQQISSKQggRQqIohBAhJIpCCBFitxbFTz/91AwePNgmYM+YMcOVCiFE+uzWovj555/bVJqFCxe6EiGEyIzdWhSPOeYY07x5c7cnhBCZs9vHFKtWVVhUCJE9pChCCBFCoiiEECF2+3WfP/jgA1O/fn0zffp0Oz9iLMyKk2jJgUmTJplBgwaZenXruhIhRJQ4vkmTuHMV5JIKL4rLli0z27dvd3ul6dOnj502rGHDhq4kOsyeNcuc3qqV24sOGzduNOv8Rua4Ro1cSXRY+t575tDDDjMHHXSQK4kOUb2fW7ZsMatWrjQnNm3qSqLDRx9+aGruvbdp0KCBK8kTiOLuzKpVqxB1zxdFV5I8nTp18hYtWuT2osXECRPcu2ixbt06b97cuW4vWnBenF8Uier93LRpk+cLttuLFsuXL7dbvtntY4qBFbh161b7KoQQmbBbi+J7vrs0duxY41t8ZurUqWbu3LnuiBBCpMduLYqNGjUyDzzwgF1GgOF+SuQWQmSKUnKEECKERFEIIUJIFIUQIoREUQghQkgUhRAihERRCCFCSBSFECKERFEIIUJIFIUQIoREUQghQkgUhRAihERRCCFCSBSFECKERFEIIUJIFIUQIoREUQghQkgUhRAihERRCCFCSBSFECKERFEIIUJIFIUQIoREUQghQkgUhRAihERRCCFCSBSFECKERFEIIUJERhS//fZbM2zYMHPRRReZwYMHm+3bt7sjiXnxxRft54Nt2rRp7ogQQqRHJERx586d5oILLjDLly83o0ePNtu2bTMdO3a0QpmILVu2mCeffNLtGVOlShXTrFkztyeEEOkRCVH84x//aKZOnWr69+9v9thjD3PHHXeYd99914wYMcJ9YlcQxAEDBphx48bZbezYseaAAw5wR4UQIj0iIYrPPPOMOemkk8zee+9t9xHG008/3ZbHA9d66NChZvz48XKZhRBZpeCiiOv873//2xx11FGupJjvf//71lqM50IvWLDAHH300eYPf/iDadu2rfnxj39s1qxZ444KIUT6FFwUP//8c/PVV1+ZWrVquZJisBq/+eYbezyW0047zfz9738369atsy72woULzdlnn11mDFIIIZKhiufj3hcERO+ggw4yt956q/nd737nSo3p3bu3efTRR83GjRvNgQce6ErjgwuNxThlyhTTrl07V1rMypUrrejGg+/o27evOfbYY11JdPjHP/5hLeCowf1Y61vljRo3diXR4b0lS8xhhx9un6eoEdX7SYflyhUrTNMIdlJ+uHq12atmTdOgQQNXkh8KLoqAlXjuueeakSNHuhJjunXrZiZOnGg+/fRTV1I2WI9XX321/bswCN/HH3/s9krzz3/+06b/1PVd9ahBOOBwv4JHje1+A7Ptyy/LbagKAQ1sTd/D2KtGDVcSHaJ6P/HGioqKTO3atV1JdNjsCzZhtXyLokEUC81FF13ktWjRwu0Vc8opp3idOnVye+XTsWNHb968eW4vOfj/ixYtcnvRYuKECe5dtFi3bp03b+5ctxctOC/OL4pE9X5u2rTJmz1rltuLFsuXL7dbvolE73OfPn3M/PnzzUcffWT3aVWJE+JSxwPr8euvv3Z7xvzvf/+zrTA92EIIkQmREEXEbODAgaZz5842Z5HRKY888ohp3ry5PU7co0ePHmbWrFl2/7zzzjPHHXec+f3vf2+ee+45M2rUKPPYY4/ZY0IIkQmREEUg9keKDTmKL7/8sunVq5c78h2MWoE333zTPPzww+aHP/yhjUXedtttZq+99rLHhBAiEyIjikDuYadOncyhhx7qSor53ve+Z5599lmb0A377beftRbZDj74YFsmhBDZIFKiKIQQhUaiKIQQISSKQggRQqIohBAhJIpCCBFCoiiEECEkikIIESJlUZw7d65Nrg7DMLs//elP5oMPPnAl5fPFF1/Y+RLj8dlnn9nv+Nvf/pbUWi3xSOecWA4hdgIKZvhmph0hosSmTZvss8lrwL/+9S9bb5KdRAVY+iPRXKSUU4f++te/2lFlsTABx5///GdbT8PDblMhW9rBTFlvvPGG28uMlERx8uTJ9st/+ctfuhJjk6oZlsdchgzFY+hdWaxevdr069fPHHbYYXFn1uYG/OAHP7BJ3GeddZadIWPx4sXuaHKkek6zZ8+2v6lhw4bmvffec6XFMPPOoEGDzNtvv+1KhCgsGA033nijufzyy0uW4LjhhhvsTFHUmzp16pRavygeQT1kFpoZM2e40u+gzlAnvvzyS7Ns2TJzyimnlKoDTIXG4AnmM2VUGQMr4glnWWRTO84880wr8OHpB9PGTQxRLr6S25lsduzY4Uo8b/78+V716tXtMVi/fr1Xs2ZNb1YSs274AuRde+21bq+YrVu3eu3atfMWLlxo91999VVv3333tTPmJEsq5xSeJYfv5HJMnz7d7ofZuHGj16xZM6+oqMiV5B7NkpM6lWWWnLZt25aa3WmC/7979uzp+RaUt2LFCltfqlWr5q1atcp9IjHUw+uvv87tFcPzzt/7lqAr8TzfMPDat29v3/viY/cDqDPUneeff96VlE8y9TTRLDnxtCOAGbemTJni9tIjaUvx9ttvN+eff76pWvW7P2H8MRO0HnnkkXafIXennnqqtazKg6F7sbAsweOPP25OOOEEu09LxBjopUuX2v1kSPec4p1PAHMHtmzZ0i6sJUQhwT3GfTz++ONdibHzhWIZ+gaEHSrLe994seGg8oj33DO/In/vC6wrKYbhtcA8A1imAW3atDG+oKU0X2S2tSMAvcC7y2QW/qREkSm98PsxUcPgTjNbTZhjjjnG+C2H20sN/n/sLNj77LNPSkuXZvucArjxw4cPTzvGKUQ2YDb62Hp41VVXlTJWqDNMnnLiiSe6ktSoV6+edZcxSuhD+M9//mNeeeUVu3pmwJ577uneGRvf+9WvfmV++tOfupLyyVU9JYTw4Ycf2jBcuiQlisxK41uVpX4ErdWGDRt2aR2wqmhpCMJmg5kzZ5qbbrrJ7ZVNLs+J387fB9OXFQruA7Eb1qehsaIT6P1ly8w777yTcOMzbFgUa9euNZs3b047MC4KB7FEZouPFZNYqDMXX3xx2jN9I7B0WmC9ESu89NJL7Sz4WKFhWHSOz7FGOzNWJWud5bKeItacJyt9pktSokhLgGkeNluDdU9ip+ziQkEm5msAwdz999/ftljJkMtzCmbuQWRyCWJFS/fJJ5+YSZMm2fkln3rqKTvf5J133ml6+w3EPXffbS0GAtW4U6+9/roV63gb15DPsPF5whP33nOPua1PH3PzzTebe++91zz00EPW5XrxxRfNa6+9ZtfgXuj/TkQ31eC5yB10ONIoliV2dIwwxyjzkWYCAsW6Rz//+c/tc8RE0LH1B/echpY6ev/999vPJEOutYO6mkk9TWqNFkxjzFqskwBOnBX3rrzyylI9XT179rTzIlKZmBsxESeffLJp0aKFeeKJJ1xJaUg1uOaaa8zTTz9dEssoj1TPiZ4vRIH4DCkB9evXt78TVzkW/paGgYWu7vFFJRsQt0H8iN3govB+o28N7Ov/Xs6TnnfWr+EB3c//7v38h49zCNyjTCAMQG8dlWiLbzlu2brVWpC04mxYJbTYRf59qO63viwGxcb1rVq1imnatJm1JCirVq2a+6+FZf68eaau7/odcsghriQ6vOlbWmf7ApMJNGz0Lid6RoHYP5ZbMM1eeVAPW7Y8zQwZ8rgrKYYeZXq3MUjoFWa+09/85jdxxZZ61759eyvaYY1IRLL1NIhpxq7RUp52IOSIdaKUv/JIylLEQoxdEY+TJjCKmIShcjOTdlmCWB5YS0wcy81IVhAhl+cUtGLh2E06cLNYnvUp/2G4w3+Ahw0bZi1D4ildfJfnwQEDrPA29288LhBxGmYgb3jssdZC4F5kKohAK43Ykr7B/+b6nHHGGbYydO3a1aZ80PoPGjzY3gsqR5MmTUx1/xqu+WSNmTBhgg2K3+Jbm5wvD/fYsWPtb6OV5jchuCJ7BJ5aotUpsfJbtWqVtCAmgqVAEF6eBUAM8VR4VuNZcdQrBC5ZcllPgbqaST1N6i/x0fH1YzsZuBBz5swpKUfMWGuFwG+68D+4CbR4gcvKj+T/JkMuzgmwnCA2rlIenDtuBh1V9913nxk6dKgVjCYnnGB6+4Ly4IMPmssuu8wuf1nf/9+0oFGCBxRrlQ4wgthn+i4V4hksSfuAL4i//vWvrXDjRrH86YwZM6wLd5fv7t/tu/o0bi+NGmXjUmQYLH//fXs9sZRF8gTPXjxrjGfsv//9rznnnHNcSfFAi3RAdBE/6k4AjWJZQkN9C393eeSqngLPVqr1NExS7jMJzVgJ/AiUPIAKT9c8gknFfuGFF2wFwoLApcI9w8T92c9+Zv8+APOYfTZ6tapXr27LaSmoYGTkh+MmJG2OGzfO9kIjKrhsl1xySdyblMw58T8IHl933XUl7jMtI716JKN2797d/bfvoLMJ14Vz4f+VBe4n14xUIjL26xxxhBU83HN+VzKtWDbcrVywfv1685Ev6if5lmx5ULEIg7AFbvnm4BVX3d8Mj59v+QYxa4SV8AAeAvvBK8fLu24V3X0GepTPPvtsa7kF0NmBq0sHTBDK2Lp1q/nJT35iPxfUQxo1LMmAoB6SAodrHtRDGitEi/AJIkWdGj16tOnQoYOtd7i11157ra0v/E/q7ft+Q3fXXXfZ+wTUF7yiK664Iu16igfC/w8sVkikHQEs2YoxxZpNaEk6JL3uM3FFxCNePI2ezQULFpjWrVuXWkqAH0DPMT86MOmpEHQgBHABLrjgAvuelo3WLpYaNWrYC0Orcv3119vhR7h0ZS0uXtY50cGABRN0NHDTlixZ4j5hbE9asGhWADcQl7OsjHlaa3r+lrpG5NQWLewomWREMJaKIIrJQgVBOFlon4rIRmtPheF1k7+/039McfeDDfGkEvHKqA7KlixeXOFFEfFCLMLxMobaxbO6seiJ+wb1kFhjIDBBPSTEwf1s1LhxST0MoKONoYMIHWIauO9AL3jQyUL6TjCyJgARJu0GjyLdetrDF2bOuYsvxFCWdgTQsYhRwyicdN3wpEURC+nCCy80b731VqkcpULAUB8uVtOmTV1JeoQ7WsqCG8RDgeDFxji5fPTUTp02zVbqlrTG/kOQSiw0HpVJFJMhEEiuMe95RUht+YYNtsGkkuBF1Kpdu0Qs2cLCGdvjmS+yeT+JM1MHiMtlCtePxqRlhnHIeGD9YYWmW08TdbSURbdu3WwsHu80XZIWRSCIjpvJDSkUtIpYk/SGZUqyonjHHXfYcaDh0AFwHmPHjDHb/ArZrl0722Jmq8GQKKYGMSkabEQPy9wKJ9amb10E7xHOvWrWLLEw6dE/0BdR3tt9vyFDOBmdkW2yeT+xpBm5kUpmRiJyJYp4TXT8JJumE49URZGJKRifncl3QkqiCHwp8QPUON8QZyD5OBgalCnJiCK5faQ/0DscgMtBxvzb8+eb9n6LRNwj0x6zWCSKqVNeTJF4U2Bt4orZlCNfFNiC/W3+veVe7u9bloFI8hpsgYCypdIAZvt+cr7PP/+8jY3Huq6pkCtRJD2nPGOjPFIRRdxm7l84/pguKYtiRSIZUcS6oDeVeArjn+k9Hf/GG6aem8kHdy0XSBRTJxsdLQgnghOIZPA+EM9AQL/+6iuzj+sYCotmIJy8Eovjlc6AdO4nngjpTbjJZSVsZ0KuRDEbpOM+ZwOJYhmiiGVKkDhIB2rhP5w82AR3M20Fy0OimDr57H3m2QgszvAWiKbtXfff0wMfuPRH1KlTIpS8Bu8RUay9sNtOtgX5oog0DTKuYTZiiLFIFHel0otiEJyPB51LJCmHoZufhznXkIsW7o2LCuSwIQiZuGy5go4XhIVshajAtUIgN278zGzftt3s2LnT9rbv3LnDZiOFwW0PxJL4KA1QAHmiDLvLNoju5s1f+HWg7DSzQkBjQGqQRDGPIIqkLLDFg1xDkq7DkGKQjwRrrDGsnqhBJac3PtE1KyTkt1KRctFRkimp3k+Gu4XT04hp05OLwMYbKcQopxo19vSFtboVV6xTrgUbeZ+JGgobZ92wwRyWI/c8E4jZEqKSKOaRstznRYsWmVGjRtnEWHKegERWevzygdzn1KlIydt4KKTAYZnjYpP5wZhfwNKkYUIgg3Hr7AeuO+/ZSODmOC48QkkMNBDJYNvD95SKvigyTZqcUKqc70wnvzabyH0uAPFEkSRYph0iabUzCetNmtjOFR6UTPMiU0GimDoVbUQLydOrVq2yz2ft2rVdaeogrFYgA6F0r2x0JjE0k1QlyrFCt/qvCG8N39rEKwpvgSUe3ijDMuU9Fin72UCiWABiRZGWd+SIETbv8PLu3c3BBaxcEsXUqWiimA8SdbQgmFYg3Sthk/B+UEZ2RlDG+2/ceGkEFZFELDEoEMs9q1e31iq98exznJg+x3H/g9egnDkCSIlKRRSJK3MerOOSLhJFJ4oMWXpu+HC7aBbDigo9akeimDoSxdTJdu8zFmYgmHbjvS9SvCd+yTEs1+B9EApAhhBXCMq/9T9zRYodLQw/ZKAFw4CZ2yCtEACiWFkJFq5i0ao+t96a8YI32UQLV6VOZVm4Kpv4lpU3O4mF5grB8gQLV5WHb2F6HTt2tAt4vfvuu640eQobSY0A/gNhp7W6pGtXO1RPCLF7U7duXdtBSqYIM/swew8udbJUePeZC8L0XfFgHHfNmnuZ+vWPziiQnQuUp5g6UcxTDIjq/SzOU9ycs5FZmYBbXbVKFbP3Pvu4ktThtzEkl9gkM+w0btzYHSkDay9WUnCfFy9e7PaiRaNGjdy7aDFt2jSvV69ebi9acF6cXxSJ6v3k+aceRJEnnnjCbumyY8cOb8iQId4RRxzhPfbYY6XWrC+LSu8+CyEqHsyPyhyuWIek1zEfarKdLhJFIUSFgdghE0gzOznLmvzlL3+xMcZUkCgKISoM06ZNs/FbZhditYB0qLCiSOcK45ZjVwwTQlRcsBCZAzWTjqMKKYpcFBKzGfPZo0cPu5qcEEIkQ4UTRWYGZ6UxJnNgdvAxY8aYO++808yePdt9QgghElPtflY8r0DccsstdogQQgiMp2SCTiaK7dy5sy0L8DzP5i3xmSjCmi9RhBzF8PIMUYLzimIOJUT1fjLRQ1L5ewWgTp06dssnFS55m3n+mIxz7NixrsSYnj172n2Se4UQoiwqlPvMXHIbNmzYZT0LstkZ+M40SUIIURYVShQZggZMVxQGdxroeIkSzAYyceJEs3DhQldSNgx7okc92AgLVEZSiQ/HXrNgvZ3KBMu/8qywkD6zkycDEyuHr1sw0XK+wIBhOWPOm6GIeQX3uaLgXzyvevXq3jXXXONKirnqqqu8GjVq2ONRYcmSJd6JJ57oPf74417Xrl29Rx991B1JzEMPPUSoo2Tr37+/O5IffEHxbrzxRu+4445zJfnlmWee8Ro2bOgdcsghrqR8Yq/ZCy+84I7kh0mTJtlz5rsbNGiQ9+9fv36916RJk5LfT/0YOnSoO5qYk08+ueRvqlSp4i1dutQdyT0zZ870Wrdu7XXo0MGrWbOmd+mll7oj+aHCjX1u1aqV1759e7dXzFlnneW1bNnS7RUexLlx48beoEGD7L7fKtqbP6uMKZw2b95c8DGq8+bNsw9oKqKUbe6+++6kv7/Q12zFihW24Xv66ae94cOH2/eIzPjx490ncg8GwujRo+0zx/kwnVa1atW8VatWuU/syltvvZWUcOaCbdu2ldQLmD59ule1alWvqKjIleSeCpeSw+pfc+bMKZkqCNcBl4n1VaICQ48Ym3nOOefYfXpLWb7SfxjsfjyGDRtmfAutoJ1FJ510kl1VrpCkMtV9oa8Z93nKlCn22bviiivsrEzEt1nuIl+w2BWpaazRcvTRR5snn3zSLrmxfPly94ldIY2N6xaEnfIJoS/GKQe0adPGzmAVGxLLJRVOFFmCtEuXLjZpm8V+eO3WrZtdQzcqUFm4yeG0lkaNGtn4CQ9sLAg7cTSS0uldJ2sf4S8EhV7MKFmicM14DsOrHtL4MQ73qKOOciW55+qrr3bvikEcaViaN2/uSkrDdfNdZXPeeedZMUKgPvvsM3c0P4RnvacBeeSRR/I7E76zGCscuArjxo3z1q5d60qiQ9u2bb169eq5vWJuu+0261oRAyoLXNhmzZrZ2NCCBQtcaf4YOHBgQd3ndL4/fM14LgoFzyLTWBEuKRQ333yz9/DDD7u9xOBujxgxwtt3332t25/veDwxd2LtxDN9jyqv31/hLMUAXIVOnTpFcmJPrK1Yi4sWGmjJywIXFjcMi+Opp55ypaIswtfs9ddfd6X558EHH7RWT6GSy1euXGnXMg+7p4ngOezevbu9XmRHcP3yCZ4T1+qee+6xGRojR450R3JPhRXFKFO/fv1dUiNwUXBXiDmVB2vyMpQx0YziYleCa0YeayFg9pb9998/7ZlbMoXUJET5pZdeKrfhDXPmmWeaFi1aFORZI/TwwAMPmPPPP9/OiZgvJIoFgLVgWLmMdX0DyAPjAUwWHph4i/iLxHDNjjzySLeXP7DOWNy+X79+riS/4IX06dPH9O/fP6lGN5ZCP2s/+tGPMlqyNFUkigWAIHbDhg3Nq6++avexEnFRWGgnHvSkxwa7cWduuukmtydiSXTNCKnkE3p56QUfOHCgKyle5H7NmjVuL7cgiLjLvXv3LhlDTG98opmjuG6M/grAQmSEGCGIQrF69Wrbe583XGxR5BkSoUnmJSGZXLpwXhj5dVdeeaXni6bdJ7GbPEYSp8k569u3b16TacPcf//9Nvie7HoX2eayyy6zHSYsYxlmzpw5XpcuXUrKo3DN+D46Vn7xi1/Ye8x24YUX2vuej44Dcv5IgKaax25B4n+8Z80XQftKbiUdW/yffEG9YKAF+bB0lN577715T3iv1IvhFxrGak+dOtXmhGE5BhD/wQrs2LGjOffcc20ZMSniYXy2UK7M5MmT7cQarJDWsmVLGx+LHWeeS/j+sBXTqlWrku+fO3euGTJkiBkwYEDJ9POFvmZYaGvXrnV739G6dWu7UHuuwRqdNWuW2yvNGWecYd3ieM8a15khqKxxwmfyDQvaf/zxxzYGy8xC+e6YkigKIUQIxRSFECKERFEIIUJIFIUQIoREUQghQkgUhRAihERRCCFCSBSFECKERFEIIUow5v914flHhzz0OwAAAABJRU5ErkJggg=="></p>
<p>so the time fishing is stopped between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>1830</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>11957</mn><mo>…</mo></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>.</mo><mn>06</mn><mo> </mo><mfenced><mrow><mn>343</mn><mo>…</mo></mrow></mfenced></math> days <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>Any reasonable answer. For example:</em></p>
<p>There are greater downsides to allowing fishing when the levels may be dangerous than preventing fishing when the levels are safe.</p>
<p>The concentration of mercury may not be uniform across the river due to natural variation / randomness.</p>
<p>The situation at the power plant might get worse.</p>
<p>Mercury levels are low in water but still may be high in fish. <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for a reasonable answer that refers to this specific context (and not a generic response that could apply to <em>any</em> model).</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many candidates did not get this far, but the attempts at the question that were seen were generally good. The greater difficulties were seen in parts (e) and (f), but this could be a problem with time running out.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question">
<p>The matrices <strong><em>A</em></strong>, <strong><em>B</em></strong>, <strong><em>X</em></strong> are given by</p>
<p><em><strong>A</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&1 \\ { - 5}&6 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <em><strong>B</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&8 \\ 0&{ - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <em><strong>X</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{where}}">
<mrow>
<mtext>where</mtext>
</mrow>
</math></span></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{Q}">
<mi>d</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<p>Given that <strong><em>AX</em></strong> + <strong><em>X</em></strong> = <strong><em>Β</em></strong>, find the <strong>exact</strong> values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span><em>,</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span><em>,</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span><em>.</em></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&1 \\ { - 5}&6 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><strong><em>X</em></strong> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><strong><em>X</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&8 \\ 0&{ - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&1 \\ { - 5}&7 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><strong><em>X</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&8 \\ 0&{ - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></span> <em><strong>(M1)</strong></em></p>
<p>Pre-multiply by inverse of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&1 \\ { - 5}&7 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong><em>X</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{33}}\left( {\begin{array}{*{20}{c}} 7&{ - 1} \\ 5&4 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 4&8 \\ 0&{ - 3} \end{array}} \right)">
<mfrac>
<mn>1</mn>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)(A1)</strong></em></p>
<p><strong>Note:</strong><em> </em>Award <em><strong>(A1)</strong></em> for determinant, <em><strong>(A1)</strong></em> for matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 7&{ - 1} \\ 5&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{33}}\left( {\begin{array}{*{20}{c}} {28}&{59} \\ {20}&{28} \end{array}} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>28</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>59</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>20</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>28</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)(A1)</strong></em><em><strong>(A1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { \Rightarrow a = \frac{{28}}{{33}}{\text{,}}\,\,b = \frac{{59}}{{33}}{\text{,}}\,\,c = \frac{{20}}{{33}}{\text{,}}\,\,d = \frac{{28}}{{33}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>28</mn>
</mrow>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>59</mn>
</mrow>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>20</mn>
</mrow>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>28</mn>
</mrow>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&1 \\ { - 5}&6 \end{array}} \right)\left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right) + \left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 4&8 \\ 0&{ - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {3a + c}&{3b + d} \\ { - 5a + 6c}&{ - 5b + 6d} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 4&8 \\ 0&{ - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mi>a</mi>
<mo>+</mo>
<mi>c</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>3</mn>
<mi>b</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>a</mi>
<mo>+</mo>
<mn>6</mn>
<mi>c</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>b</mi>
<mo>+</mo>
<mn>6</mn>
<mi>d</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4a + c = 4">
<mn>4</mn>
<mi>a</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mn>4</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 5a + 7c = 0">
<mo>−</mo>
<mn>5</mn>
<mi>a</mi>
<mo>+</mo>
<mn>7</mn>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4b + d = 8">
<mn>4</mn>
<mi>b</mi>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>8</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 5b + 7d = - 3">
<mo>−</mo>
<mn>5</mn>
<mi>b</mi>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Notes:</strong> Award <em><strong>(A1)</strong></em> for each pair of equations. <br>Allow <strong>ft</strong> from their equations.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a = \frac{{28}}{{33}}{\text{,}}\,\,b = \frac{{59}}{{33}}{\text{,}}\,\,c = \frac{{20}}{{33}}{\text{,}}\,\,d = \frac{{28}}{{33}}}">
<mrow>
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>28</mn>
</mrow>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>59</mn>
</mrow>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>20</mn>
</mrow>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>28</mn>
</mrow>
<mrow>
<mn>33</mn>
</mrow>
</mfrac>
</mrow>
</math></span> <em><strong>(A1)(A1)</strong></em><em><strong>(A1)(A1)</strong></em></p>
<p><strong>Note:</strong><em> </em>Award <em><strong>(A0)(A0)(A1)(A1)</strong></em> if the final answers are given as decimals <em>ie</em> 0.848, 1.79, 0.606, 0.848.</p>
<p><em><strong>[8 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>M</em></strong><sup>2</sup> = <strong><em>M</em></strong> where <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right){\text{,}}\,\,bc \ne 0">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mi>c</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>. </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + d = 1">
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="bc">
<mi>b</mi>
<mi>c</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;"><strong>Hence</strong> show that <strong><em>M</em></strong> is a singular matrix.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">If all of the elements of <strong><em>M</em></strong> are positive, find the range of possible values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Show that (<strong><em>I</em></strong> − <strong><em>M</em></strong>)<sup>2</sup> = <strong><em>I</em></strong> − <strong><em>M</em></strong> where <strong><em>I</em></strong> is the identity matrix.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Attempting to find <strong><em>M</em></strong><sup>2</sup> <em><strong>M1</strong></em></p>
<p><strong><em>M</em></strong><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {{a^2} + bc}&{ab + bd} \\ {ac + cd}&{bc + {d^2}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mi>c</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>a</mi>
<mi>b</mi>
<mo>+</mo>
<mi>b</mi>
<mi>d</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mi>c</mi>
<mo>+</mo>
<mi>c</mi>
<mi>d</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>b</mi>
<mi>c</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>d</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b\left( {a + d} \right) = b">
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>b</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c\left( {a + d} \right) = c">
<mi>c</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>c</mi>
</math></span><em><strong> A1</strong></em></p>
<p>Hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + d = 1">
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> (as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \ne 0">
<mi>b</mi>
<mo>≠</mo>
<mn>0</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \ne 0">
<mi>c</mi>
<mo>≠</mo>
<mn>0</mn>
</math></span>) <em><strong>AG N0</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^2} + bc = a">
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mi>a</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow bc = a - {a^2}">
<mo stretchy="false">⇒</mo>
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mi>a</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong> A1</strong></em><em><strong> N1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>Using det <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ad - bc">
<mi>a</mi>
<mi>d</mi>
<mo>−</mo>
<mi>b</mi>
<mi>c</mi>
</math></span> <em><strong>M1</strong></em></p>
<p>det <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ad - a\left( {1 - a} \right)">
<mi>a</mi>
<mi>d</mi>
<mo>−</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or det <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {1 - a} \right) - a\left( {1 - a} \right)">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>(or equivalent) <em><strong> A1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="= 0">
<mo>=</mo>
<mn>0</mn>
</math></span> using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + d = 1">
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 1 - a">
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</math></span> to simplify their expression <em><strong> R1</strong></em></p>
<p>Hence <strong><em>M</em></strong> is a singular matrix <em><strong> AG</strong></em><em><strong> N0</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>Using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="bc = a\left( {1 - a} \right)">
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + d = 1">
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> to obtain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="bc = ad">
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mi>a</mi>
<mi>d</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p>det <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ad - bc">
<mi>a</mi>
<mi>d</mi>
<mo>−</mo>
<mi>b</mi>
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ad - bc = 0">
<mi>a</mi>
<mi>d</mi>
<mo>−</mo>
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="bc = ad">
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mi>a</mi>
<mi>d</mi>
</math></span> <em><strong> R1</strong></em></p>
<p>Hence <strong><em>M</em></strong> is a singular matrix <em><strong> AG</strong></em><em><strong> N0</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {1 - a} \right) > 0">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>></mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p>0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> < 1 <em><strong>A1</strong></em><em><strong>A1 N3</strong></em></p>
<p class="indent2"><strong>Note: </strong>Award <em><strong>A1</strong> </em>for correct endpoints and <em><strong>A1</strong> </em>for correct inequality signs.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>Attempting to expand (<strong><em>I</em></strong> − <strong><em>M</em></strong>)<sup>2</sup> <em><strong>M1 </strong></em></p>
<p>(<strong><em>I</em></strong> − <strong><em>M</em></strong>)<sup>2</sup> = <strong><em>I</em></strong> − 2<strong><em>M</em></strong> + <strong><em>M</em></strong><sup>2</sup> <em><strong>A1 </strong></em></p>
<p> = <strong><em>I</em></strong> − 2<strong><em>M</em></strong> + <strong><em>M</em></strong> <em><strong>A1 </strong></em></p>
<p> = <strong><em>I</em></strong> − <strong><em>M</em></strong> <em><strong>AG N0 </strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>Attempting to expand (<strong><em>I</em></strong> − <strong><em>M</em></strong>)<sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\begin{array}{*{20}{c}} {1 - a}&{ - b} \\ { - c}&{1 - d} \end{array}} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>c</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>d</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> (or equivalent) <em><strong>M1</strong></em></p>
<p>(<strong><em>I</em></strong> − <strong><em>M</em></strong>)<sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {{{\left( {1 - a} \right)}^2} + bc}&{ - b\left( {1 - a} \right) - b\left( {1 - d} \right)} \\ { - c\left( {1 - a} \right) - c\left( {1 - d} \right)}&{bc + {{\left( {1 - d} \right)}^2}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mi>c</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>c</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>c</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>b</mi>
<mi>c</mi>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>(or equivalent) <em><strong>A1 </strong></em></p>
<p>Use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + d = 1">
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="bc = a - {a^2}">
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mi>a</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> to show desired result. <em><strong>M1</strong></em></p>
<p>Hence (<strong><em>I</em></strong> − <strong><em>M</em></strong>)<sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {1 - a}&{ - b} \\ { - c}&{1 - d} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>c</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>d</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>AG N0</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><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> for correct expression. Can be shown in recursive form.</p>
<p><strong>THEN</strong></p>
<p>(= 2644.75) = 2645 <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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2500 \times {0.93^{24}} + 250\frac{{\left( {{{0.93}^{24}} - 1} \right)}}{{\left( {0.93 - 1} \right)}}">
<mo>=</mo>
<mn>2500</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>0.93</mn>
<mrow>
<mn>24</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>250</mn>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mn>0.93</mn>
</mrow>
<mrow>
<mn>24</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.93</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p>=3384 <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>Consider the following system of coupled differential equations.</p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mi>x</mi></math></p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi></math></p>
</div>
<div class="specification">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues and corresponding eigenvectors of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math>.</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>Hence, write down the general solution of the system.</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>Determine, with justification, whether the equilibrium point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> is stable or unstable.</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>(i) at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p>(ii) at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</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>Sketch a phase portrait for the general solution to the system of coupled differential equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>6</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>6</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>6</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>6</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mo>-</mo><mn>4</mn><mo>-</mo><mi>λ</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <strong> <em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>4</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn><mi>x</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn><mi>y</mi></mtd></mtr></mtable></mfenced></math> <strong> <em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> This <em><strong>M1</strong></em> can be awarded for attempting to find either eigenvector.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>4</mn><mi>y</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>y</mi></math></p>
<p>possible eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> (or any real multiple)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn><mi>x</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mi>y</mi></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn></math></p>
<p>possible eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> (or any real multiple)<strong> <em>A1</em></strong></p>
<p><em><strong><br>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>4</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong> <em>(M1)A1</em></strong></p>
<p><strong><br>Note:</strong> Award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>4</mn><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>3</mn><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>4</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math>, <em><strong>M1A0</strong></em> if LHS is missing or incorrect.</p>
<p> </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>two (distinct) real negative eigenvalues <strong><em>R1</em></strong></p>
<p>(or equivalent (<em>eg</em> both <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mn>4</mn><mi>t</mi></mrow></msup><mo>→</mo><mn>0</mn><mo>,</mo><mo> </mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mo>→</mo><mn>0</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>→</mo><mo>∞</mo></math>))</p>
<p>⇒ stable equilibrium point <strong><em>A1</em></strong></p>
<p><strong><br>Note:</strong> Do not award <em><strong>R0A1</strong></em>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi></mrow><mrow><mo>-</mo><mn>4</mn><mi>x</mi></mrow></mfrac></math> <strong><em>(M1)</em></strong></p>
<p>(i) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math> <strong><em>A1</em></strong></p>
<p>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <strong><em>A1</em></strong><strong><em>A1</em></strong><strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for a phase plane, with correct axes (condone omission of labels) and at least three non-overlapping trajectories. Award <em><strong>A1</strong></em> for all trajectories leading to a stable node at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>. Award <em><strong>A1</strong></em> for showing gradient is negative at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math>. Award <em><strong>A1</strong></em> for both eigenvectors on diagram.</p>
<p> </p>
<p><em><strong>[4 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>In a small village there are two doctors’ clinics, one owned by Doctor Black and the other owned by Doctor Green. It was noted after each year that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn><mo>%</mo></math> of Doctor Black’s patients moved to Doctor Green’s clinic and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> of Doctor Green’s patients moved to Doctor Black’s clinic. All additional losses and gains of patients by the clinics may be ignored.</p>
<p>At the start of a particular year, it was noted that Doctor Black had <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2100</mn></math> patients on their register, compared to Doctor Green’s <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3500</mn></math> patients.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> indicating the annual population movement between clinics.</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 a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math>, with integer elements, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">D</mi><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math> is a diagonal matrix.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that the long-term transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mo>∞</mo></msup></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mo>∞</mo></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>965</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>05</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>035</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>95</mn></mtd></mtr></mtable></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>95</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>035</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>05</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>965</mn></mtd></mtr></mtable></mfenced></math>.<br>Award the <em><strong>A1</strong></em> for a transposed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> if used correctly in part (b) i.e. preceded by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>×</mo><mn>2</mn></math> matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2100</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>3500</mn></mrow></mfenced></math> rather than followed by a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>1</mn></math> matrix.</p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>965</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>05</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>035</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>95</mn></mtd></mtr></mtable></mfenced><mn>2</mn></msup><mfenced><mtable><mtr><mtd><mn>2100</mn></mtd></mtr><mtr><mtd><mn>3500</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2294</mn></mtd></mtr><mtr><mtd><mn>3306</mn></mtd></mtr></mtable></mfenced></math></p>
<p>so ratio is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2294</mn><mo>:</mo><mn>3306</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>1147</mn><mo>:</mo><mn>1653</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>693889</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mi>x</mi><mo>=</mo><mi>λ</mi><mi>x</mi><mo> </mo><mo>:</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>965</mn><mo>-</mo><mi>λ</mi></mtd><mtd><mn>0</mn><mo>.</mo><mn>05</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>035</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>95</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>965</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>.</mo><mn>95</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>-</mo><mn>0</mn><mo>.</mo><mn>05</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>035</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>915</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn></math> <em><strong>(A1)</strong></em></p>
<p>attempt to find eigenvectors for at least one eigenvalue <em><strong>(M1)</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>915</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> (or any real multiple) <em><strong>(A1)</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></math> (or any real multiple) <em><strong>(A1)</strong></em></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></math> (accept integer valued multiples of their eigenvectors and columns in either order) <em><strong>A1</strong></em></p>
<p><br><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mn>17</mn></mfrac><mfenced><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> This mark is independent, and may be seen anywhere in part (d).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>915</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mi>n</mi></msup><mo>=</mo><mi mathvariant="bold-italic">P</mi><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><msup><mn>915</mn><mi>n</mi></msup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msup><mn>1</mn><mi>n</mi></msup></mtd></mtr></mtable></mfenced><mfrac><mn>1</mn><mn>17</mn></mfrac><mfenced><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)A0</strong></em> for finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mi mathvariant="bold-italic">P</mi></math> correctly.</p>
<p><br>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><msup><mn>915</mn><mi>n</mi></msup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msup><mn>1</mn><mi>n</mi></msup></mtd></mtr></mtable></mfenced><mo>→</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mi>n</mi></msup><mo>→</mo><mfrac><mn>1</mn><mn>17</mn></mfrac><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>AG</strong></em> line must be seen for the final <em><strong>A1</strong></em> to be awarded.</p>
<p><br><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD ONE</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>2100</mn></mtd></mtr><mtr><mtd><mn>3500</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3294</mn></mtd></mtr><mtr><mtd><mn>2306</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>so ratio is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3294</mn><mo>:</mo><mn>2306</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>1647</mn><mo>:</mo><mn>1153</mn><mo>,</mo><mo> </mo><mo> </mo><mn>1</mn><mo>.</mo><mn>42844</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>700060</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD TWO</strong></p>
<p>long term ratio is the eigenvector associated with the largest eigenvalue <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>:</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 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="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 $3.00.</p>
<p>Find the cost of each type of fruit.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> be the cost of one banana, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> the cost of one kiwifruit, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> the cost of one melon</p>
<p>attempt to set up three linear equations <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2b + 3k + 4m = 658">
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mn>3</mn>
<mi>k</mi>
<mo>+</mo>
<mn>4</mn>
<mi>m</mi>
<mo>=</mo>
<mn>658</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5b + 2k + 8m = 1232">
<mn>5</mn>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
<mo>+</mo>
<mn>8</mn>
<mi>m</mi>
<mo>=</mo>
<mn>1232</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5b + 4k = 300">
<mn>5</mn>
<mi>b</mi>
<mo>+</mo>
<mn>4</mn>
<mi>k</mi>
<mo>=</mo>
<mn>300</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve three simultaneous equations <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 36,{\text{ }}k = 30,{\text{ }}m = 124">
<mi>b</mi>
<mo>=</mo>
<mn>36</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>k</mi>
<mo>=</mo>
<mn>30</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>m</mi>
<mo>=</mo>
<mn>124</mn>
</math></span></p>
<p>banana costs ($)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>