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<h2>HL Paper 3</h2><div class="specification">
<p>This question investigates some applications of differential equations to modeling population growth.</p>
<p>One model for population growth is to assume that the rate of change of the population is proportional to the population, i.e. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = kP">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>P</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
<mi>P</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is the time (in years) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> is the population</p>
</div>
<div class="specification">
<p>The initial population is 1000.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.003">
<mi>k</mi>
<mo>=</mo>
<mn>0.003</mn>
</math></span>, use your answer from part (a) to find</p>
</div>
<div class="specification">
<p>Consider now the situation when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is not a constant, but a function of time.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.003 + 0.002t">
<mi>k</mi>
<mo>=</mo>
<mn>0.003</mn>
<mo>+</mo>
<mn>0.002</mn>
<mi>t</mi>
</math></span>, find</p>
</div>
<div class="specification">
<p>Another model for population growth assumes</p>
<ul>
<li>there is a maximum value for the population, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>.</li>
<li>that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is not a constant, but is proportional to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - \frac{P}{L}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mfrac>
<mi>P</mi>
<mi>L</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</li>
</ul>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the general solution of this differential equation is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = A{{\text{e}}^{kt}}"> <mi>P</mi> <mo>=</mo> <mi>A</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mi>k</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A \in \mathbb{R}"> <mi>A</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </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>the population after 10 years</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>the number of years it will take for the population to triple.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{t \to \infty } P"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mi>P</mi> </math></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the solution of the differential equation, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = f\left( t \right)"> <mi>P</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the number of years it will take for the population to triple.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</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="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{R}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span>, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = g\left( t \right)"> <mi>P</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the initial population is 1000, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L = 10000"> <mi>L</mi> <mo>=</mo> <mn>10000</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 0.003"> <mi>m</mi> <mo>=</mo> <mn>0.003</mn> </math></span>, find the number of years it will take for the population to triple.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{P}} {\text{d}}P = \int {k{\text{d}}t} "> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mi>k</mi> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,P = kt + c"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>=</mo> <mi>k</mi> <mi>t</mi> <mo>+</mo> <mi>c</mi> </math></span> <em><strong> A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = {e^{kt + c}}"> <mi>P</mi> <mo>=</mo> <mrow> <msup> <mi>e</mi> <mrow> <mi>k</mi> <mi>t</mi> <mo>+</mo> <mi>c</mi> </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="P = A{e^{kt}}"> <mi>P</mi> <mo>=</mo> <mi>A</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>k</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = {e^c}"> <mi>A</mi> <mo>=</mo> <mrow> <msup> <mi>e</mi> <mi>c</mi> </msup> </mrow> </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>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0{\text{,}}\,\,P = 1000"> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>=</mo> <mn>1000</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow A = 1000"> <mo stretchy="false">⇒</mo> <mi>A</mi> <mo>=</mo> <mn>1000</mn> </math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( {10} \right) = 1000{e^{0.003\left( {10} \right)}} = 1030"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mn>10</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1000</mn> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mrow> <mo>(</mo> <mrow> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mo>=</mo> <mn>1030</mn> </math></span> <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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 = 1000{e^{0.003t}}"> <mn>3000</mn> <mo>=</mo> <mn>1000</mn> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mi>t</mi> </mrow> </msup> </mrow> </math></span> <em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = \frac{{{\text{ln}}\,3}}{{0.003}} = 366"> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> <mrow> <mn>0.003</mn> </mrow> </mfrac> <mo>=</mo> <mn>366</mn> </math></span> years <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{t \to \infty } P = \infty "> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mi>P</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{P}} {\text{d}}P = \int {\left( {0.003 + 0.002t} \right){\text{d}}t} "> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.003</mn> <mo>+</mo> <mn>0.002</mn> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,P = 0.003t + 0.001{t^2} + c"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>=</mo> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>c</mi> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = {e^{0.003t + 0.001{t^2} + c}}"> <mi>P</mi> <mo>=</mo> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>c</mi> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0{\text{,}}\,\,P = 1000"> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>=</mo> <mn>1000</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {e^c} = 1000"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>e</mi> <mi>c</mi> </msup> </mrow> <mo>=</mo> <mn>1000</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = 1000{e^{0.003t + 0.001{t^2}}}"> <mi>P</mi> <mo>=</mo> <mn>1000</mn> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mrow> </msup> </mrow> </math></span></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 = 1000{e^{0.003t + 0.001{t^2}}}"> <mn>3000</mn> <mo>=</mo> <mn>1000</mn> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mrow> </msup> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,3 = 0.003t + 0.001{t^2}"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>=</mo> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>Use of quadratic formula or GDC graph or GDC polysmlt <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 31.7"> <mi>t</mi> <mo>=</mo> <mn>31.7</mn> </math></span> years <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = m\left( {1 - \frac{P}{L}} \right)"> <mi>k</mi> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mi>P</mi> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> , where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m"> <mi>m</mi> </math></span> is the constant of proportionality <em><strong>A1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = m\left( {1 - \frac{P}{L}} \right)P"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mi>P</mi> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>P</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{P\left( {L - P} \right)}}} {\text{d}}P = \int {\frac{m}{L}{\text{d}}t} "> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </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}{{P\left( {L - P} \right)}} = \frac{A}{P} + \frac{B}{{L - P}}"> <mfrac> <mn>1</mn> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>A</mi> <mi>P</mi> </mfrac> <mo>+</mo> <mfrac> <mi>B</mi> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \equiv A\left( {L - P} \right) + BP"> <mn>1</mn> <mo>≡</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mi>P</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \frac{1}{L}{\text{,}}\,\,B = \frac{1}{L}"> <mi>A</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{L}\int {\left( {\frac{1}{P} + \frac{1}{{L - P}}} \right){\text{d}}P} = \int {\frac{m}{L}{\text{d}}t} "> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> <mo>∫</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{L}\left( {{\text{ln}}\,P - {\text{ln}}\left( {L - P} \right)} \right) = \frac{m}{L}t + c"> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>t</mi> <mo>+</mo> <mi>c</mi> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {\frac{P}{{L - P}}} \right) = mt + d"> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>P</mi> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>m</mi> <mi>t</mi> <mo>+</mo> <mi>d</mi> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = cL"> <mi>d</mi> <mo>=</mo> <mi>c</mi> <mi>L</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{P}{{L - P}} = C{e^{mt}}"> <mfrac> <mi>P</mi> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> </mfrac> <mo>=</mo> <mi>C</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = {e^d}"> <mi>C</mi> <mo>=</mo> <mrow> <msup> <mi>e</mi> <mi>d</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="P\left( {1 + C{e^{mt}}} \right) = CL{e^{mt}}"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>C</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>C</mi> <mi>L</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = \frac{{CL{e^{mt}}}}{{\left( {1 + C{e^{mt}}} \right)}}{\text{ }}\left( { = \frac{L}{{\left( {D{e^{ - mt}} + 1} \right)}}{\text{,}}\,\,{\text{where}}\;D = \frac{1}{C}} \right)"> <mi>P</mi> <mo>=</mo> <mfrac> <mrow> <mi>C</mi> <mi>L</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>C</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mi>L</mi> <mrow> <mrow> <mo>(</mo> <mrow> <mi>D</mi> <mrow> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>where</mtext> </mrow> <mspace width="thickmathspace"></mspace> <mi>D</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>C</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[10 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1000 = \frac{{10000}}{{D + 1}}"> <mn>1000</mn> <mo>=</mo> <mfrac> <mrow> <mn>10000</mn> </mrow> <mrow> <mi>D</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = 9"> <mi>D</mi> <mo>=</mo> <mn>9</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 = \frac{{10000}}{{9{e^{ - 0.003t}} + 1}}"> <mn>3000</mn> <mo>=</mo> <mfrac> <mrow> <mn>10000</mn> </mrow> <mrow> <mn>9</mn> <mrow> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mn>0.003</mn> <mi>t</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 450"> <mi>t</mi> <mo>=</mo> <mn>450</mn> </math></span> years <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</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;">
[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.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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will explore some of the properties of special functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">g</mi></math> and their relationship with the trigonometric functions, sine and cosine.</strong></p>
<p><br>Functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> are defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>,</mo><mo> </mo><mi>u</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math>, find expressions, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>u</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>u</mi></math>, for</p>
</div>
<div class="specification">
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi></math> are known as circular functions as the general point (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>,</mo><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></math>) defines points on the unit circle with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>.</p>
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> are known as hyperbolic functions, as the general point ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>θ</mi><mo>)</mo><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math> ) defines points on a curve known as a hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>. This hyperbola has two asymptotes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> satisfies the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>u</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find, and simplify, an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>, stating the coordinates of any axis intercepts and the equation of each asymptote.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math> can be rotated to coincide with the curve defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mi>k</mi><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</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>f</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo> </mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p><em><strong><br></strong></em><strong>Note: </strong>Accept combinations of METHODS 1 & 2 that meet at equivalent expressions.</p>
<p><em><strong><br></strong></em><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> into the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> <em><strong>(M1)</strong></em></p>
<p>obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>-i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi><mo>+</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong></em> can be awarded for the use of sine and cosine being odd and even respectively.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><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"><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi><mo>-</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p>substituting and attempt to simplify <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><em><strong>[2 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"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p>substituting expressions found in part (c) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi></mrow></mfenced></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"><mi>f</mi><mfenced><mrow><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></msup></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>u</mi></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept equivalent final answers that have been simplified removing all imaginary parts eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>−</mo><mn>1</mn></math>etc</p>
<p><em><strong><br></strong></em><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"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>4</mn><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for a value of 1 obtained from either LHS or RHS of given expression.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn></math> (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>) <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Award full marks for showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>z</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>z</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>∀</mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.<br><br><em><strong><br></strong></em><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct curves in the upper quadrants, <em><strong>A1</strong></em> for correct curves in the lower quadrants, <em><strong>A1</strong></em> for correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercepts of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><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>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> (condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>), <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>−</mo><mi>x</mi></math>.</p>
<p><br><em><strong><br></strong></em><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to rotate by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>°</mo></math> in either direction <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Evidence of an attempt to relate to a sketch of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mi>k</mi></math> would be sufficient for this <em><strong>(M1)</strong></em>.</p>
<p><br>attempting to rotate a particular point, eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> rotates to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>,</mo><mo> </mo><mo>±</mo><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></mfrac></mrow></mfenced></math> (or similar) <em><strong>(A1)</strong></em></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">g.</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>
<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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>This question asks you to investigate some properties of the sequence of functions of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right)">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 1 and <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>
<p><strong>Important:</strong> When sketching graphs in this question, you are <strong>not</strong> required to find the coordinates of any axes intercepts or the coordinates of any stationary points unless requested.</p>
</div>
<div class="specification">
<p>For odd values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> > 2, use your graphic display calculator to systematically vary the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>. Hence suggest an expression for odd values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> describing, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, the number of</p>
</div>
<div class="specification">
<p>For even values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> > 2, use your graphic display calculator to systematically vary the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>. Hence suggest an expression for even values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>describing, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, the number of</p>
</div>
<div class="specification">
<p>The sequence of functions, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, defined above can be expressed as a sequence of polynomials of degree <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) = {\text{cos}}\left( {\left( {n + 1} \right)\,{\text{arccos}}\,x} \right)">
<mrow>
<msub>
<mi>f</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the same set of axes, sketch the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_1}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_3}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 1.</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>local maximum points;</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>local minimum points;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On a new set of axes, sketch the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_2}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_4}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>4</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 1.</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>local maximum points;</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>local minimum points.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}^\prime (x) = 0"> <msup> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mi mathvariant="normal">′</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> and hence show that the stationary points on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_n}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> occur at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{cos}}\frac{{k\pi }}{n}"> <mi>x</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in {\mathbb{Z}^ + }"> <mi>k</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> and 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an appropriate trigonometric identity to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_2}(x) = 2{x^2} - 1"> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an appropriate trigonometric identity to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) - {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) + {f_{n - 1}}(x) = 2x{f_n}\left( x \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </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> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_3}(x)"> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> as a cubic polynomial.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_1}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>A1</strong></em></p>
<p>correct graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_3}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>A1</strong></em></p>
<p><img src="data:image/png;base64,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"></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>graphical or tabular evidence that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> has been systematically varied <em><strong>M1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 3, 1 local maximum point and 1 local minimum point</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 5, 2 local maximum points and 2 local minimum points</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 7, 3 local maximum points and 3 local minimum points <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{n - 1}}{2}"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </math></span> local maximum points <em><strong>A1</strong></em></p>
<p><em><strong>[3 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="\frac{{n - 1}}{2}"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </math></span> local minimum points <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Allow follow through from an incorrect local maximum formula expression.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_2}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>A1</strong></em></p>
<p>correct graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_4}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>4</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>A1</strong></em></p>
<p><img src="data:image/png;base64,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"></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>graphical or tabular evidence that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> has been systematically varied <em><strong>M1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 2, 0 local maximum point and 1 local minimum point</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 4, 1 local maximum points and 2 local minimum points</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 6, 2 local maximum points and 3 local minimum points <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{n - 2}}{2}"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </math></span> local maximum points <em><strong>A1</strong></em></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="\frac{n}{2}"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </math></span> local minimum points <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\left( x \right)} \right)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}^\prime (x) = \frac{{n\,{\text{sin}}\left( {n\,{\text{arccos}}\left( x \right)} \right)}}{{\sqrt {1 - {x^2}} }}"> <msup> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mi mathvariant="normal">′</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to use the chain rule.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}^\prime (x) = 0 \Rightarrow n\,{\text{sin}}\left( {n\,{\text{arccos}}\left( x \right)} \right) = 0"> <msup> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mi mathvariant="normal">′</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <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="n\,{\text{arccos}}\left( x \right) = k\pi \,\,\,\left( {k \in {\mathbb{Z}^ + }} \right)"> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> <mi>π</mi> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>leading to</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{cos}}\frac{{k\pi }}{n}"> <mi>x</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in {\mathbb{Z}^ + }"> <mi>k</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> and 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>) <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_2}(x) = {\text{cos}}\left( {2\,{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2{\left( {{\text{cos}}\left( {{\text{arccos}}\,x} \right)} \right)^2} - 1"> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>M1</strong></em></p>
<p>stating that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{cos}}\left( {{\text{arccos}}\,x} \right)} \right) = x"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_2}(x) = 2{x^2} - 1"> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) = {\text{cos}}\left( {\left( {n + 1} \right)\,{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{cos}}\left( {n\,{\text{arccos}}\,x + {\text{arccos}}\,x} \right)"> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>use of cos(<em>A</em> + <em>B</em>) = cos <em>A </em>cos <em>B</em> − sin <em>A </em>sin <em>B</em> leading to <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) - {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)"> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n - 1}}(x) = {\text{cos}}\left( {\left( {n - 1} \right)\,{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </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=" = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) + {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)"> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </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="{f_{n + 1}}(x) + {f_{n - 1}}(x) = 2\,{\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </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=" = 2x{f_n}\left( x \right)"> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_3}(x) = 2x{f_2}\left( x \right) - {f_1}(x)"> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2x\left( {2{x^2} - 1} \right) - x"> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4{x^3} - 3x"> <mo>=</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 1 + \frac{y}{x}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 0">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the family of curves which satisfy the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 1 + \frac{y}{x}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 0">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</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="y\left( 1 \right) = 1">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span>, use Euler’s method with step length <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> = 0.25 to find an approximation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( 2 \right)">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</math></span>. Give your answer to two significant figures.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 1 + \frac{y}{x}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( 1 \right) = 1">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</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>Find the percentage error when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( 2 \right)">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</math></span> is approximated by the final rounded value found in part (a). Give your answer to two significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the isocline corresponding to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = k">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \ne 0">
<mi>k</mi>
<mo>≠</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</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>Show that such an isocline can never be a normal to any of the family of curves that satisfy the differential equation.</p>
<div class="marks">[4]</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>attempt to apply Euler’s method <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x_{n + 1}} = {x_n} + 0.25{\text{;}}\,\,{y_{n + 1}} = {y_n} + 0.25 \times \left( {1 + \frac{{{y_n}}}{{{x_n}}}} \right)">
<mrow>
<msub>
<mi>x</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>x</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>+</mo>
<mn>0.25</mn>
<mrow>
<mtext>;</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<msub>
<mi>y</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>+</mo>
<mn>0.25</mn>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>x</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><img 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"> <em><strong>(A1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> for correct <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> values, <strong>A1</strong> for first three correct <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> values.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 3.3 <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="I\left( x \right) = {{\text{e}}^{\int { - \frac{1}{x}{\text{d}}x} }}">
<mi>I</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^{ - {\text{ln}}\,x}}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{x}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{x}\frac{{{\text{d}}y}}{{{\text{d}}x}} - \frac{y}{{{x^2}}} = \frac{1}{x}">
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mi>y</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{y}{x}} \right) = \frac{1}{x}">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{y}{x} = {\text{ln}}\left| x \right| + C">
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( 1 \right) = 1 \Rightarrow C = 1">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>C</mi>
<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="y = x\,{\text{ln}}\left| x \right| + x">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mi>x</mi>
</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="v = \frac{y}{x}">
<mi>v</mi>
<mo>=</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}x}} = \frac{1}{x}\frac{{{\text{d}}y}}{{{\text{d}}x}} - \frac{1}{{{x^2}}}y">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mi>y</mi>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v + x\frac{{{\text{d}}v}}{{{\text{d}}x}} = 1 + v">
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int 1 \,{\text{d}}v = \int {\frac{1}{x}} \,{\text{d}}x">
<mo>∫</mo>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = {\text{ln}}\left| x \right| + C">
<mi>v</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{y}{x} = {\text{ln}}\left| x \right| + C">
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( 1 \right) = 1 \Rightarrow C = 1">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>C</mi>
<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="y = x\,{\text{ln}}\left| x \right| + x">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 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="y\left( 2 \right) = 2\,{\text{ln}}\,2 + 2 = 3.38629 \ldots ">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mn>3.38629</mn>
<mo>…</mo>
</math></span></p>
<p>percentage error <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{3.38629 \ldots - 3.3}}{{3.38629 \ldots }} \times 100{\text{% }}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>3.38629</mn>
<mo>…</mo>
<mo>−</mo>
<mn>3.3</mn>
</mrow>
<mrow>
<mn>3.38629</mn>
<mo>…</mo>
</mrow>
</mfrac>
<mo>×</mo>
<mn>100</mn>
<mrow>
<mtext>% </mtext>
</mrow>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p>= 2.5% <em><strong> A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = k \Rightarrow 1 + \frac{y}{x} = k">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
<mo stretchy="false">⇒</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
<mo>=</mo>
<mi>k</mi>
</math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left( {k - 1} \right)x">
<mi>y</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
</math></span></p>
<p> </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>gradient of isocline equals gradient of normal <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k - 1 = - \frac{1}{k}">
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>k</mi>
</mfrac>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k\left( {k - 1} \right) = - 1">
<mi>k</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k + 1 = 0">
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<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="\Delta = 1 - 4 < 0">
<mi mathvariant="normal">Δ</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>4</mn>
<mo><</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore ">
<mo>∴</mo>
</math></span> no solution <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Accept alternative reasons for no solutions.</p>
<p> </p>
<p><em><strong>[4 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;">
[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>This question will investigate power series, as an extension to the Binomial Theorem for negative and fractional indices.</p>
<p>A power series in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is defined as a function of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> where the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_i} \in \mathbb{R}">
<mrow>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>It can be considered as an infinite polynomial.</p>
</div>
<div class="specification">
<p>This is an example of a power series, but is only a finite power series, since only a finite number of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_i}">
<mrow>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
</math></span> are non-zero.</p>
</div>
<div class="specification">
<p>We will now attempt to generalise further.</p>
<p>Suppose <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span> can be written as the power series <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...">
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^5}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</math></span> using the Binomial Theorem.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the power series <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - x + {x^2} - {x^3} + {x^4} - ...">
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span></p>
<p>By considering the ratio of consecutive terms, explain why this series is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 1}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which this equality is true.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 2}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</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>Repeat this process to find the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 3}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, by recognising the pattern, deduce the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - n}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mi>n</mi>
</mrow>
</msup>
</mrow>
</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>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</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>By substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_0}">
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating both sides of the expression and then substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, find the value of <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>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Repeat this procedure to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_2}">
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> and <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>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the first four terms in what is called the Extended Binomial Theorem for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{1 + {x^2}}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, using integration, find the power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, giving the first four non-zero terms.</p>
<div class="marks">[4]</div>
<div class="question_part_label">k.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + 5x + 10{x^2} + 10{x^3} + 5{x^4} + {x^5}">
<mn>1</mn>
<mo>+</mo>
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</math></span> <em><strong>M1A1</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>It is an infinite GP with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 1{\text{,}}\,\,r = - x">
<mi>a</mi>
<mo>=</mo>
<mn>1</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
</math></span> <em><strong>R1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } = \frac{1}{{1 - \left( { - x} \right)}} = \frac{1}{{1 + x}} = {\left( {1 + x} \right)^{ - 1}}">
<mrow>
<msub>
<mi>S</mi>
<mi mathvariant="normal">∞</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>M1A1AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 1}} = 1 - x + {x^2} - {x^3} + {x^4} - ...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1{\left( {1 + x} \right)^{ - 2}} = - 1 + 2x - 3{x^2} + 4{x^3} - ...">
<mo>−</mo>
<mn>1</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 2}} = 1 - 2x + 3{x^2} - 4{x^3} + ...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2{\left( {1 + x} \right)^{ - 3}} = - 2 + 6x - 12{x^2} + 20{x^3}...">
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mn>12</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>20</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 3}} = 1 - 3x + 6{x^2} - 10{x^3}...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - n}} = 1 - nx + \frac{{n\left( {n + 1} \right)}}{{2{\text{!}}}}{x^2} - \frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{{3{\text{!}}}}{x^3}...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<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>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1A1A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{1^q} = {a_0} \Rightarrow {a_0} = 1">
<mrow>
<msup>
<mn>1</mn>
<mi>q</mi>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q{\left( {1 + x} \right)^{q - 1}} = {a_1} + 2{a_2}x + 3{a_3}{x^2} + ...">
<mi>q</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_1} = q">
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mi>q</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q\left( {q - 1} \right){\left( {1 + x} \right)^{q - 2}} = 1 \times 2{a_2} + 2 \times 3{a_3}x + ...">
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mn>2</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_2} = \frac{{q\left( {q - 1} \right)}}{{2{\text{!}}}}">
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q\left( {q - 1} \right)\left( {q - 2} \right){\left( {1 + x} \right)^{q - 3}} = 1 \times 2 \times 3{a_3} + ...">
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_3} = \frac{{q\left( {q - 1} \right)\left( {q - 2} \right)}}{{3{\text{!}}}}">
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q} = 1 + qx + \frac{{q\left( {q - 1} \right)}}{{2{\text{!}}}}{x^2} + \frac{{q\left( {q - 1} \right)\left( {q - 2} \right)}}{{3{\text{!}}}}{x^3}...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>q</mi>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{1 + {x^2}}} = 1 - {x^2} + {x^4} - {x^6} + ...">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>6</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x + c = x - \frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} - \frac{{{x^7}}}{7} + ...">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>7</mn>
</msup>
</mrow>
</mrow>
<mn>7</mn>
</mfrac>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>M1A1</strong></em></p>
<p>Putting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0 \Rightarrow c = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x = x - \frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} - \frac{{{x^7}}}{7} + ...">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>7</mn>
</msup>
</mrow>
</mrow>
<mn>7</mn>
</mfrac>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">k.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">k.</div>
</div>
<br><hr><br><div class="specification">
<p style="text-align: left;">The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x){\text{ }}={\text{ }}{(\arcsin{\text{ }}x)^2},{\text{ }} - 1 \leqslant x \leqslant 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>=</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>1</mn>
</math></span>.</p>
<p> </p>
</div>
<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> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {x^2}} \right)f''\left( x \right) - xf'\left( x \right) - 2 = 0">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<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="f'\left( 0 \right) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</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>By differentiating the above equation twice, show that</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\left( {1 - {x^2}} \right){f^{\left( 4 \right)}}\left( x \right) - 5x{f^{\left( 3 \right)}}\left( x \right) - 4f''\left( x \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 3 \right)}}\left( x \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 4 \right)}}\left( x \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> denote the 3rd and 4th derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> respectively.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> up to and including the term in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \frac{1}{3}{x^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</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>Use this series approximation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> to find an approximate value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\pi ^2}">
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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="f'\left( x \right) = \frac{{2\,{\text{arcsin}}\,\left( x \right)}}{{\sqrt {1 - {x^2}} }}">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for an attempt at chain rule differentiation.<br>Award <em><strong>M0A0</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 2\,{\text{arcsin}}\,\left( x \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 0 \right) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>AG</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>differentiating gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {x^2}} \right){f^{\left( 3 \right)}}\left( x \right) - 2xf''\left( x \right) - f'\left( x \right) - xf''\left( x \right)\left( { = 0} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>x</mi>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>differentiating again gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {x^2}} \right){f^{\left( 4 \right)}}\left( x \right) - 2x{f^{\left( 3 \right)}}\left( x \right) - 3f''\left( x \right) - 3x{f^{\left( 3 \right)}}\left( x \right) - f''\left( x \right)\left( { = 0} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>3</mn>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note</strong>: Award <em><strong>M1</strong> </em>for an attempt at product rule differentiation of at least one product in each of the above two lines.<br>Do not penalise candidates who use poor notation.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {x^2}} \right){f^{\left( 4 \right)}}\left( x \right) - 5x{f^{\left( 3 \right)}}\left( x \right) - 4f''\left( x \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find <strong>one of</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( 0 \right)">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 3 \right)}}\left( 0 \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 4 \right)}}\left( 0 \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span> by substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> into relevant differential equation(s) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Condone <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( 0 \right)">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span> found by calculating <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{{2\,{\text{arcsin}}\,\left( x \right)}}{{\sqrt {1 - {x^2}} }}} \right)">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {f\left( 0 \right) = 0,\,f'\left( 0 \right) = 0} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( 0 \right) = 2">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 4 \right)}}\left( 0 \right) - 4f''\left( 0 \right) = 0 \Rightarrow {f^{\left( 4 \right)}}\left( 0 \right) = 8">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>8</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 3 \right)}}\left( 0 \right) = 0">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{{2{\text{!}}}}{x^2} + \frac{8}{{4{\text{!}}}}{x^4}">
<mfrac>
<mn>2</mn>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>8</mn>
<mrow>
<mn>4</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Only award the above <em><strong>A1</strong></em>, for correct first differentiation in part (b) leading to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 3 \right)}}\left( 0 \right) = 0">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> stated or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 3 \right)}}\left( 0 \right) = 0">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> seen from use of the general Maclaurin series.<br><strong>Special Case:</strong> Award <em><strong>(M1)A0A1</strong></em> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 4 \right)}}\left( 0 \right) = 8">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>8</mn>
</math></span> is stated without justification or found by working backwards from the general Maclaurin series.</p>
<p>so the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> up to and including the term in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \frac{1}{3}{x^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span> <em><strong>AG</strong></em></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>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \frac{1}{3}{x^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p>the series approximation gives a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{13}}{{48}}">
<mfrac>
<mrow>
<mn>13</mn>
</mrow>
<mrow>
<mn>48</mn>
</mrow>
</mfrac>
</math></span></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\pi ^2} \simeq \frac{{13}}{{48}} \times 36">
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>≃</mo>
<mfrac>
<mrow>
<mn>13</mn>
</mrow>
<mrow>
<mn>48</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mn>36</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \simeq 9.75\,\,\left( { \simeq \frac{{39}}{4}} \right)">
<mo>≃</mo>
<mn>9.75</mn>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>≃</mo>
<mfrac>
<mrow>
<mn>39</mn>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept 9.76.</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="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4{x^2} + {y^2} - xy}}{{{x^2}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>, with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> when <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>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method, with step length <span class="mjpage"><math alttext="h = 0.1" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>h</mi> <mo>=</mo> <mn>0.1</mn> </math></span>, to find an approximate value of <span class="mjpage"><math alttext="y" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> </math></span> when <span class="mjpage"><math alttext="x = 1.4" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>=</mo> <mn>1.4</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>Sketch the isoclines for <span class="mjpage"><math alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>4</mn> </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>Express <span class="mjpage"><math alttext="{m^2} - 2m + 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>m</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>4</mn> </math></span> in the form <span class="mjpage"><math alttext="{\left( {m - a} \right)^2} + b" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>m</mi> <mo>−</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>b</mi> </math></span> , where <span class="mjpage"><math alttext="a{\text{, }}b \in \mathbb{Z}" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</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>Solve the differential equation, for <span class="mjpage"><math alttext="x > 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>></mo> <mn>0</mn> </math></span>, giving your answer in the form <span class="mjpage"><math alttext="y = f\left( x \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math alttext="y = f\left( x \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> for <span class="mjpage"><math alttext="1 \leqslant x \leqslant 1.4" xmlns="http://www.w3.org/1998/Math/MathML"> <mn>1</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>1.4</mn> </math></span> .</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to the curvature of your sketch in part (c)(iii), and without further calculation, explain whether you conjecture <span class="mjpage"><math alttext="f\left( {1.4} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mn>1.4</mn> </mrow> <mo>)</mo> </mrow> </math></span> will be less than, equal to, or greater than your answer in part (a).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iv.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>(M1)(A1)(A1)(A1)A1</strong></em></p>
<p><span class="mjpage"><math alttext="y\left( {1.4} \right) \approx 5.34" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mrow> <mo>(</mo> <mrow> <mn>1.4</mn> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>5.34</mn> </math></span></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct <span class="mjpage"><math alttext="y" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> </math></span> value.<br>For the intermediate <span class="mjpage"><math alttext="y" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> </math></span> values, accept answers that are accurate to 2 significant figures.<br>The final <span class="mjpage"><math alttext="y" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> </math></span> value must be accurate to 3 significant figures or better.</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>attempt to solve <span class="mjpage"><math alttext="\frac{{4{x^2} + {y^2} - xy}}{{{x^2}}} = 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mfrac> <mrow> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>4</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math alttext=" \Rightarrow {y^2} - xy = 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math alttext="y\left( {y - x} \right) = 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math alttext="y = 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span> or <span class="mjpage"><math alttext="y = x" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mi>x</mi> </math></span></p>
<p><img 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"> <em><strong>A1</strong></em><em><strong>A1</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><span class="mjpage"><math alttext="{m^2} - 2m + 4 = {\left( {m - 1} \right)^2} + 3\,\,\,\,\,\left( {a = 1,\,b = 3} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>m</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>=</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</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>recognition of homogeneous equation,<br>let <span class="mjpage"><math alttext="y = vx" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mi>v</mi> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p>the equation can be written as</p>
<p><span class="mjpage"><math alttext="v + x\frac{{{\text{d}}v}}{{{\text{d}}x}} = 4 + {v^2} - v" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>v</mi> <mo>+</mo> <mi>x</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>4</mn> <mo>+</mo> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>v</mi> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math alttext="x\frac{{{\text{d}}v}}{{{\text{d}}x}} = {v^2} - 2v + 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>v</mi> <mo>+</mo> <mn>4</mn> </math></span></p>
<p><span class="mjpage"><math alttext="\int {\frac{1}{{{v^2} - 2v + 4}}} {\text{d}}v = \int {\frac{1}{x}} {\text{d}}x" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>v</mi> <mo>+</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt to separate the variables.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{{{\left( {v - 1} \right)}^2} + 4}}} {\text{d}}v = \int {\frac{1}{x}} {\text{d}}x"><mo>∫</mo><mfrac><mn>1</mn><mrow><msup><mrow><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></mfrac><mtext>d</mtext><mi>v</mi><mo>=</mo><mo>∫</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mtext>d</mtext><mi>x</mi></math></span> from part (c)(i) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math alttext="\frac{1}{{\sqrt 3 }}{\text{arctan}}\left( {\frac{{v - 1}}{{\sqrt 3 }}} \right) = {\text{ln}}\,x\,\,\left( { + c} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math alttext="x = 1{\text{,}}\,y = 2 \Rightarrow v = 2" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo>=</mo> <mn>2</mn> </math></span></p>
<p><span class="mjpage"><math alttext="\frac{1}{{\sqrt 3 }}{\text{arctan}}\left( {\frac{1}{{\sqrt 3 }}} \right) = {\text{ln}}\,1 + c" xmlns="http://www.w3.org/1998/Math/MathML"> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>1</mn> <mo>+</mo> <mi>c</mi> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for using initial conditions to find <span class="mjpage"><math alttext="c" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>c</mi> </math></span>.</p>
<p><span class="mjpage"><math alttext=" \Rightarrow c = \frac{\pi }{{6\sqrt 3 }}\,\,\left( { = 0.302} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mo stretchy="false">⇒</mo> <mi>c</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mrow> <mn>6</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>0.302</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math alttext="{\text{arctan}}\left( {\frac{{v - 1}}{{\sqrt 3 }}} \right) = \sqrt 3 \,{\text{ln}}\,x + \frac{\pi }{6}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </math></span></p>
<p>substituting <span class="mjpage"><math alttext="v = \frac{y}{x}" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>v</mi> <mo>=</mo> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> This <em><strong>M1</strong></em> may be awarded earlier.</p>
<p><span class="mjpage"><math alttext="y = x\left( {\sqrt 3 {\text{tan}}\left( {\sqrt 3 \,{\text{ln}}\,x + \frac{\pi }{6}} \right) + 1} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[10 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYoAAADICAYAAAD2mabDAAAb0UlEQVR4Ae2deZQX1ZXHc87MZDL/5sw5OXM8DEEQAVlkkRCNSFjiJANBFkVkGzAKkriAIiiCKyqKEVSCgCyihmijbFEMjCigjgp6ZGklgdBssnRQEOhma7lz7pNbVLdN92+p3+9Xy6fOKV79qt57de/nVr8v771avicsEIAABCAAgRoIfK+GYxyCAAQgAAEICELBRQABCEAAAjUSQChqxMNBCEAAAhBAKLgGIAABCECgRgIIRY14OAgBCEAAAggF1wAEIBA6AqdPnxZbQ2dcAg1CKBIYdFyGQBQIvP3227J8+V+cYETB3jjbiFDEObr4BoEIEtCexNdffy2XXNJa+vbtE0EP4mcyQhG/mOIRBCJLwIabRo++Uxo2bCAlJSWR9SVOhiMUcYomvkAg4gRUKN5ds0bq1DlPZs+exbBTSOKJUIQkEJgBgSQTUIHQRYec2rVrK7179ZRvvvkGoQjJRYFQhCQQmAGBJBOwIaeRI0d4Q062L8lcwuI7QhGWSGAHBBJKwAThzTeXuSGn55+f6/UkrKeRUDShcRuhCE0oMAQCySSgYlBaWiotW7aQ/v2vc0NOySQRXq8RivDGBssgkAgCOhcxePAgadq0iezdu9f5TE8iXKFHKMIVD6yBQGII2JCTDjXpXU6vv/5nN+Rk+xMDIgKOIhQRCBImQiBuBEwMNm/eLBc0qC+jRt3hzUvEzdc4+INQxCGK+ACBCBIoLy+XTp06Svv2P5OysjKEIsQxRChCHBxMg0DcCNjcg6b69HX9+vVk06aNcXMzdv4gFLELKQ5BILwEbMhpyZLFbl5i7pw59CTCGy7PMoTCQ8EGBCCQawIqFFu3bpHGjS+UoUNvcKezXkauz039mRNAKDJnR0kIQCAFAtaL0KzHysvlF106y2WX/VQOHz7s3eWUQjVkKSABhKKA8Dk1BJJCwHoNt95ys1xwQX3ZsGEDQ04RCj5CEaFgYSoEokpAhcKel5g//4/ODetpmIhE1bck2I1QJCHK+AiBAhJQIfh43Tp3h5Pe6YQwFDAYGZ4aocgQHMUgAIHUCOhrOVq1aindunWV48ePMy+RGrZQ5UIoQhUOjIFAPAjYsJIKQ/fuv3Yv/NuzZ088nEugFwhFAoOOyxDIBwEVixEjbnNDTmvXrmXIKR/Qc3QOhCJHYKkWAkkn8Oyz09xDdTp5bT2MpDOJqv8IRVQjh90QCCkBFYXly5dL3bp15MEHHwiplZiVDgGEIh1a5IUABGokoCKxadMmadSooQwcOEAqKipqzM/BaBBAKKIRJ6yEQCQI6B1Obdu2kS5dOrsnryNhNEbWSgChqBURGSAAgZoI2PzDkSNH5Moru0jr1i1l9+7dzEvUBC1ixxCKiAUMcyEQNgIqFKdOnZJ+/a5zQ076eg5dTEDCZi/2pE8AoUifGSUgAAEfAf3m9ciRI6RevbqycuVKd0RFgiU+BBCK+MQSTyCQVwLWY5g48VF3G+yf5s/3ehEIRV5DkfOTIRQ5R8wJIBBPAioGs2fPciIxZcrkSiKBUMQr5ghFvOKJNxDIKQHrRWj66qsL3LMS4+65xxOJnJ6cygtGAKEoGHpODIHoETChWLFihXs1x/DhN4nOUdj+6HmExakQQChSoUQeCCScgAqBLpp++OGHTiQGDOjn7nZCJOJ/cSAU8Y8xHkIgKwImBCYS+tT1tdf2kfLycnoSWZGNTmGEIjqxwlIIFISACcVnxcXSvHlT6dr1V/L1oUOIREGiUZiTIhSF4c5ZIRBqAiYOmupSXEUkdJ8dC7UjGBcIAYQiEIxUAoF4EfALhYnEzztcIQcPHkQg4hXqlLxBKFLCRCYIJIuACYUNN6lI7N+/n+GmZF0GnrcIhYeCDQhAwIaTNF216h03J+EXCQglkwBCkcy44zUEqiVgPYmZM2e4J647nOlJaGYTkWoLsjPWBBCKWIcX5yBQOwETAEunTJ7sROL664dUmpOw47XXSI64EUAo4hZR/IFAmgRMADS1F/xpqr/tWJpVkj1mBBCKmAUUdyCQLgEVA30Nx113jXE9iWeeedoTCYQiXZrxzI9QxDOueAWBWgmYCOhHh26++XfuBX+znnuu1nJkSB4BhCJ5McdjCHg9Bn0Nx8CBA9xHh4qKirz9IIKAnwBC4afBNgQSRkDf/lqnznny3MyZnkhYTyNhKHC3BgIIRQ1wOASBOBFQATAR0DmJ8ePGOZF4eMIEN0dhvloe+00KAYSCawACCSKgIqDDTcOGDXVzEvqFOhMQBCJBF0KariIUaQIjOwSiTEDf1dSzZw+pf349WbJkcSWRQCiiHNnc2o5Q5JYvtUOg4ASsx7B9e4no6ziaNbtIPvjg/zyRKLiBGBB6AghF6EOEgRDIjoAKxcfr1rn3Nl16aTvZsmWLJxL0IrJjm5TSCEVSIo2fiSNgIrBw4Wvu06Xdu3eT0tJSJxIKw44nDgwOp00AoUgbGQUgEA0CemfTpMcfd3c23XTTMDlWXh4Nw7EydAQQitCFBIMgkD2BsrIyGTr0RicSkyY9Xun21+xrp4akEUAokhZx/I0dgapDSDt37pQrr+wiDRs2kCVLlnjzEbFzHIfyRgChyBtqTgSB3BBQobD1/fffd5PW7dq1lfXr1zMPkRvkiasVoUhcyHE4bgRMJKZPf9a9s0mfk9BJa12q9jbi5jv+5IcAQpEfzpwFAoESMHHQVOcjdLJa39k0duzdom+DZYFAkAQQiiBpUhcE8khARWLr1i3SuXNHNx+xYMECbwgqj2ZwqgQQQCgSEGRcjB8BFQmdqG7cuJFcdtmlUlxc7ImEHmOBQJAEEIogaVIXBHJMQEXgxIkTMm7cPW6o6Te/uV4OHTrkiUSOT0/1CSWAUCQ08LgdPQIqEtu2bZOuXX/lnrSeOXOGez5C99OLiF48o2QxQhGlaGFrogiYAFi6eNEiadz4QtH3NX3yySeeOCASibosCuIsQlEQ7JwUArUTMIHQu5pGjhzhhppuvOE3oq8KZ4FAPgkgFPmkzbkgkAYBFQp9aK59+8ulQYP68vzzc5mLSIMfWYMjgFAEx5KaIJA1ARtGqqiokKnPPOPmIjp36iibN2/2hpqyPgkVQCBNAghFmsDIDoFcElCh2LVrl/Tu1dMNNd13371y/PhxRCKX0Km7VgIIRa2IyACB4AlYz0FT29azzJ//RzdhfcklrWX16lXeUFPVfMFbRI0QODcBhOLcbDgCgZwRsIbf0r1798qgQQNdL+K3vx3uTVj7RSRnxlAxBGohgFDUAojDEMgVAROJxYsXuTe+Nm/eTJb6Xgtux3N1fuqFQKoEEIpUSZEPAgEQsMZf031798qQIYO/ve31xhu8N74GcBqqgECgBBCKQHFSGQRqJqACoZ8offnlP8lFFzV2PQntUeh+FgiElQBCEdbIYFcsCZSUlEjfvn1cL+KmYUO9XoT1NGLpNE5FngBCEfkQ4kAYCVjDbz0F/UbEH/4w1T04p3c0vfnmm94dTWG0H5sg4CeAUPhpsA2BgAj4heLjdeukS5fOUrduHbn7rjFy+PBhTyRMSAI6LdVAICcEEIqcYKXSJBMwkdB3Mo0efacTCBUKFQyEIclXRnR9RyiiGzssDyEBFQKdrC4qekUuvri5NGrUUKZN+4OcPHkyhNZiEgRSI4BQpMaJXBColYCKxIYNG6RHj+7eLa+7d+/2hplqrYAMEAgpAYQipIHBrPATsGEkTQ8cOCBjxox2w0xXXHG5vPPOO54Dls/bwQYEIkYAoYhYwDA3HAS08ddVh5Rmz54lzZpd5IaZpk59xn2qNBxWYgUEgiGAUATDkVoSSGDlW29Jx44d3DDTiBG3yb59+xhmSuB1kASXEYokRBkfsyJgvQdLP//8MxnQv58TiO7du8knn3yMQGRFmMJhJ4BQhD1C2FdwAioQumiPYdSoO9w8RLt2P5GlS5e4O5z0mIlIwY3FAAjkgABCkQOoVBl9AiYOmuoDco89NlEaNmzg3s80depU72NCfoGwMtH3Hg8gUJkAQlGZB78STMAafUuPHTsmM2ZMdy/uq1+/nujX5r766ivXe0gwJlxPIAGEIoFBx+XqCZhAqBiMHz/O3cWkr9249dZbZMeOHZ5A0HOonh9740sAoYhvbPGsFgJVG/xDhw7JI4887ASi0YUN3XyETlzbYvkttf2kEIg7AYQi7hHGv2oJaGNvq85BTJky2T0L0aBBfdeb2L9/v9eDqLYCdkIgQQQQigQFG1fPElCRUIGYPPlJJxD16tV1PQj/sxAmJGdLsQWBZBJAKJIZ90R5bQ2+pYcOHfyOQOzcsSNRTHAWAukQQCjSoUXeSBIwgdBJapuDaNDgfNeDsElqyxNJBzEaAjkmgFDkGDDV54+ANva2+Ld37drl5h10/sHmIKoOMWk5fxmrhxQCEBBBKLgKYkPAegWWfv7553LbbbeKzj80adLI9Sa+/PJLBCE2EceRfBFAKPJFmvPkhYCKhD4opw/H1alznlsHDRrgXgNuApIXQzgJBGJEAKGIUTCT5Io2+rbYtn56VF/z3aZNKycQ/fpdJ6tXrfJ6EJbPUitPCgEI1EwAoaiZD0dDSkAbe1tLSrbJPWPvdg/K6TCTDjdt3LjROx5SFzALApEhgFBEJlQY6ieg36Ves2aNDBo00L3NVT8c9PCECbJ3zx5PIExI/OXYhgAE0ieAUKTPjBJ5IGCNvD/V0x49ekTmzJnjfTCoQ4f2Mm/ePCkrK8uDVZwCAskkgFAkM+6h99ovELq9efNmGTfuHu9FfYMHD3LfpdaeheUNvVMYCIGIEkAoIhq4uJutjf/x48dl4cKFcs01vd3kdIsWzeShhx6U7du3e+5rPlv827aPFAIQyJ4AQpE9Q2rIkoA18JrqunXrVrn//vtEhUFvce3Ro7ssWFAkx8rLszwTxSEAgUwIIBSZUKNMYARMHHSOQcWgd6+eThz0AbmxY++WTZs2eUNLJiiBnZyKIACBlAggFClhIlNQBKyxt3Tt2rVy552j3JPT2nvo2bOHEwwVDhMRyxuUDdQDAQikRwChSI8XudMk4G/sbXv37t3y1FNT5Ior2rvegz4g9/DDE9yQk+VJ8zRkhwAEckgAocghXKr+loA2/keOHJb58/8offpc45570G9QDx9+k6x86y2pqKig98DFAoEQE0AoQhycKJhWUw9A71patuwNGTZsqKgw6NBS79495aWXXpKvv/7auaflbbG6/PvsGCkEIFA4AghF4djH4sz+xl23T5065Z5vGDlyhDfv0KnTz91Q086dO72eQyycxwkIJIQAQpGQQOfKTb84jBp1uzRr1tT1HH72s0vl0UcekeLiYk8c/KKSK3uoFwIQCJ4AQhE809jV6G/gbVuHlVasWO6+Emfi8JOfXOKef1i3bp3YE9Oxg4FDEEggAYQigUFPx2UTBk0PHz4sixYtlOE3DfOGlbTn8MAD94tfHKyMnke3WSAAgWgTQCiiHb9ArPc35v5GXivXW1nnzJ4t/ftf501Id+7cSZ54YpJs3LjBG1bSvFbW6rPfgRhJJRCAQMEIIBQFQ1+4E1sDXl2qvYalS5fI7bePlMsu+6mbb9BvPPTq1VNmzJguJSUl9BIKFzrODIGCEEAoCoK98Cc1kdC5hOLiTTJt2jS5+upe7vvSehtr+/aXuyemFy58TfTLcZbf0sJ7gAUQgEC+CCAU+SKdh/NoI151qdqw6+/S0lJ57bXXZMSI26R1q5au19Cgwfmi35aeNeu5at/OWrWequfhNwQgEF8CCEWMYmuNuaXqmm4fPXrU3aH04IMPyJW/6OKEQXsNnTp1dHcprVy50r2Z1cppygIBCEDACCAURiImqQnDO++8LY8++ohc1f3X3nCS9h5uvfUWeeWVl2Xv3r3eXINfIGw7JjhwAwIQCIAAQhEAxFxWYQ13TemXX34py5YtE+0xdOv2354wtGjRXG688QaZM2e2bNmyJZdmUjcEIBBjAghFyINbVSD0BXp/+9vf5MUXX3R3JnXs2MEbSmrduqV70d7cuXPdp0N1oloXqyPkrmIeBCAQUgIIRZ4Do422LlUbb/9+M0n3/aO01L1Yb9Kkx6Vv32u9B910jqFLl04yZsxoKSp6RXbs2MFQkoEjhQAEAiWAUASKs+bKTBz8qZaw3/v27ZXly5fL5MlPyuDBg6RNm9Zeb6FlyxYycOAAd2zVqlXu7atajgUCEIBArgkgFLkm7KvfBEHfk6Sf+NRJ5YceetD1FC5u0dwThaZNm8g111ztjv35z0tl585vewtW3p/6qmcTAhCAQE4IIBQZYLWG2orqb10stW395sL69Z9KUVGRTJz4qFx//RD5eYcr3Id7dOhI10sv/akMGTJYfv/EE/L6669L1Vdx27kstbrdCfkHAhCAQB4IIBQZQLZG+8SJE65hf++9d13v4Mknfy+33HKz9OhxlehQkYmBpvpm1Wuv7SPjx4+TF16YJ/qtaBUSq6tqmoFZFIEABCCQEwIIRYZYtWF/7913K4lBq5YXy1VX/dqJhb4079VXF8iGDRvcW1erCkFtvzM0i2IQgAAEAieAUGSIVBt6fYGeTixv+/vfpaysLMOaKAYBCEAg3AQQigzjU12PIMOqKAYBCEAg1AQQigzCU51I6D4WCEAAAnEkgFDEMar4BAEIQCBAAghFgDCpCgIQgEAcCSAUcYwqPkEAAhAIkABCESBMqoIABCAQRwIIRRyjik8QgAAEAiSAUAQIk6ogAAEIxJEAQhHHqOITBCAAgQAJIBQBwqQqCEAAAnEkgFDEMar4BAEIQCBAAghFgDCpCgIQgEAcCSAUcYwqPkEAAhAIkABCESBMqoIABCAQRwIIRRyjik8QgAAEAiSAUAQIk6ogAAEIxJEAQhHHqOITBCAAgQAJIBQBwqQqCEAAAnEkgFDEMar4BAEIQCBAAghFgDCpCgIQgEAcCSAUcYwqPkEAAhAIkABCESBMqoIABCAQRwIIRRyjik8QgAAEAiSAUAQIk6ogAAEIxJEAQhHHqOITBCAAgQAJIBQBwqQqCEAAAnEkgFDEMar4BAEIQCBAAghFgDCpCgIQgEAcCSAUcYwqPkEAArElcPr06bz7hlDkHTknhAAEIJAZARUJW7UG/3ZmNaZWCqFIjRO5IAABCBScgAnDgQMH5NSpUwhFwSOCARCAAARCSODkyZPSuXMnGdC/nxw7dsxZaALiN9ftk7M9EMtjqT9vbdv0KGojxHEIQAACISFgjXxRUZHUq1dXevToLocOHfJ6FnrcFt3esX2nvPrqQiktLXV5tCdy7/j7ZeOGjZYtpRShSAkTmSAAAQiEg4CJxYoVK6R+/Xpy5ZVdZP++fU4I/BZ+9dVX8tJLL0mrlm2kZ4/eor979+4jvXtdLW+9tdKftdbtGoXCDLJUa/NvV1e7Hg9ysfosDbJu6oKAn4Bd21xrfipn/+bjzMViX9nzcP9Smz/88ENp1KihtG9/uZSUlHjts4vV6W9jt/wvK+Rff/BvMnbseNf7yMSrGoVi08aNouvajz6SdevWybZtfz+zbpNt2767fvHFF3Lw4MHA17Kyskx8owwE0iJgjcWRI0cCv4Zz8XeR7zr37NlT7d99dW1B1PZt2rRJPvroI9fWaXsXhVXt1XXunDly/vk/lrZt28jnn31WSSz0mj5x4oT88If/LpMnT0nr78GfuUahaN68qehap855rDDgGuAa4BoI+TXQs8dVZ4agzk5iV5yqkAsuuFAGDvyfb0XkzMiQXwhq265RKLQro6v2JP7617869TqX6ur+t99+W5YuXRrwukRefvlP8sILL8iLL4ZrbdKksVx//ZDQ2RU2TlGyR6+z+fPnB3wNB/03UZj61qxendf/ac+bN88J0+LFi3J6Xm27tEcRtV6QjfDMmDFdfvzj/5SuXX/lTVpb71jT559/QUbdMVp+9KP/cEJx9OjRM2JSmzycPV6jUFg2O6n+9m/b8Zr2+/Nksm3n0zQsi9nUpnUrefrpp87JJCz2YkfqBCy2qZdIRs5CcNEGXEcztEHM5WJti6W5PFeQdau9CxYUSd26deTqq3t/5+6nPV/skT179srkyU/J9pId8k///C+yZvW78tjEJ3IjFOqcQbQ0SIejWlebNmeFIqo+YHdlAlzflXkU6pfGQecJvhWKbTk3I0pxV1t1nTlzhuMzcOAAKS8v9/7DqsfWrH5Pvv/9H0j/fgPdHIXu69z5F9K6dVvZvfsLrz1PFWxKPYpUK0taPhOKpPmNvxDIB4F8CkU+/AniHCYSkyY97kTid78dXukJbTuu6fr1G+TUqYpKAuI/no49CEU6tKrkRSiqAOEnBAIkgFB8F6Y29BUVFXLddX3lrrvGyDfffHNOIdDBer8w6LZUeVfUd89Q/R6EonouKe1FKFLCRCYIZEQAoagemzb4estrPheEIgvaCEUW8CgKgVoIIBTVA7JeQvVHc7MXociCq/+upyyqoSgEIFANgbN3PeV+Mrua07PLRwCh8MFIZ1NV3XoUhVD4dGwlLwSiSOBsjyK3t8dGkU2+bUYoMiBuwqB3HqxatSqDGigCAQjURED/xnbs2C733jteDhz4R01ZOZYHAghFhpD1QmaBAARyQ8D+M5ab2qk1XQIIRbrEyA8BCEAgYQQQioQFHHchAAEIpEsAoUiX2Jn81jW2NMNqKAaBxBKwvx1NdbE0VSBWPtX85MucAEKRObvvPvWYRV0UhUASCZg4pNvoW34rn0R2+fQZociQtn405rmZM+WXv/wv91GnDKuhGAQSScAa+s8+K5YHH7jf/R2l2uhb2WXLlkmTJo3S7okkEniWTiMUGQLcvr1E3njjDfdiLv36HwsEIJAeAW3wVShGjbpD2l7SJqUG30Ri37597jkmhCI95pnmRigyJKcX7K5duxCKDPlRDALW6D/77LS0hEJfhDdixG3y2MRH6VHk6TJCKLIArd8I1/fl06PIAiJFE0tAhUIX/UJbbT0KExVNVVjef/99mTF9uhOKxALMo+MIRRawEYos4FEUAmcImFCcC4hfJDZu2CBTpkx2w1RaToeeWHJPAKHIgjFCkQU8ikLgDIFUhUK/9Xz77SPl5MmTCEWerx6EIgvgCEUW8CgKgTMEUhWKCRMekpKSEo+b9Sj8PQ7vIBuBEkAossCJUGQBj6IQOEMgFaHQb0LrfOC5Vq1DBYMlNwQQiiy4+oWCizQLkBRNHAH/34tOSutkti7+/brtXw2S5bEehZWz/ZaPNDgCCEUGLO3itdtjP/jgg0oXeAZVUgQCiSNgf0d6F1OrVhdXEgU7ZmlVOLrfhEK3WXJLAKHIkO+WLVvk/vvvc13hm2/+nXz66acZ1kQxCCSTgDbwq1evlu7du0ndunVk1qznpLS01BOMxYsXSf369aS4uLhaQCYU1R5kZ6AEEIoscZ7rfzxZVktxCCSKQHW9Ar3L6X9XrHB3OVUHo7oy1eVjX/YEEIrsGVIDBCAAgVgTQChiHV6cgwAEIJA9AYQie4bUAAEIQCDWBP4frotIccJcuSoAAAAASUVORK5CYII="></p>
<p>curve drawn over correct domain <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the sketch shows that <span class="mjpage"><math alttext="f" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> </math></span> is concave up <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math alttext="{f'}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> </mrow> </math></span> is increasing.</p>
<p>this means the tangent drawn using Euler’s method will give an underestimate of the real value, so <span class="mjpage"><math alttext="f\left( {1.4} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mn>1.4</mn> </mrow> <mo>)</mo> </mrow> </math></span> > estimate in part (a) <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> The <em><strong>R1</strong></em> is dependent on the <em><strong>A1</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.iv.</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">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iv.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2xy\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^2} - {x^2}">
<mn>2</mn>
<mi>x</mi>
<mi>y</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation and show that a general solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + {y^2} = cx">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> is a positive constant.</p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that there are two horizontal tangents to the general solution curve and state their equations, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{y^2} - {x^2}}}{{2xy}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mi>y</mi>
</mrow>
</mfrac>
</math></span></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = vx">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mi>x</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = v + x\frac{{{\text{d}}v}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</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="v + x\frac{{{\text{d}}v}}{{{\text{d}}x}} = \frac{{{v^2}{x^2} - {x^2}}}{{2v{x^2}}}">
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mi>v</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</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="v + x\frac{{{\text{d}}v}}{{{\text{d}}x}} = \frac{{{v^2} - 1}}{{2v}}\,\,\,\,\left( { = \frac{v}{2} - \frac{1}{{2v}}} \right)">
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>2</mn>
<mi>v</mi>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mi>v</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>v</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Or equivalent attempt at simplification.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\frac{{{\text{d}}v}}{{{\text{d}}x}} = \frac{{ - {v^2} - 1}}{{2v}}\,\,\,\,\,\left( { = - \frac{v}{2} - \frac{1}{{2v}}} \right)">
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>2</mn>
<mi>v</mi>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>v</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>v</mi>
</mrow>
</mfrac>
</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="\frac{{2v}}{{1 + {v^2}}}\frac{{{\text{d}}v}}{{{\text{d}}x}} = - \frac{1}{x}">
<mfrac>
<mrow>
<mn>2</mn>
<mi>v</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{2v}}{{1 + {v^2}}}} {\text{d}}v = \int { - \frac{1}{x}} {\text{d}}x">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mi>v</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {1 + {v^2}} \right) = - {\text{ln}}\,x + {\text{ln}}\,c">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for LHS and <em><strong>A1</strong></em> for RHS and a constant.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {1 + {{\left( {\frac{y}{x}} \right)}^2}} \right) = - {\text{ln}}\,x + {\text{ln}}\,c">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> M1</strong></em> for substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \frac{y}{x}">
<mi>v</mi>
<mo>=</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span>. May be seen at a later stage.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + {\left( {\frac{y}{x}} \right)^2} = \frac{c}{x}">
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mi>c</mi>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for any correct equivalent equation without logarithms.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + {y^2} = cx">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>x</mi>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[11 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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{y^2} - {x^2}}}{{2xy}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mi>y</mi>
</mrow>
</mfrac>
</math></span></p>
<p>(for horizontal tangents) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</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="\left( { \Rightarrow {y^2} = {x^2}} \right) \Rightarrow y = \pm x">
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mo>±</mo>
<mi>x</mi>
</math></span></p>
<p><strong>EITHER</strong></p>
<p>using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + {y^2} = cx \Rightarrow 2{x^2} = cx">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>x</mi>
<mo stretchy="false">⇒</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>x</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} - cx = 0 \Rightarrow x = \frac{c}{2}">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>c</mi>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{y^2} = \pm cy">
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>±</mo>
<mi>c</mi>
<mi>y</mi>
</math></span>.</p>
<p><strong>OR</strong></p>
<p>using implicit differentiation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + {y^2} = cx">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + 2y\frac{{{\text{d}}y}}{{{\text{d}}x}} = c">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>c</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Accept differentiation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \sqrt {cx - {x^2}} ">
<mi>y</mi>
<mo>=</mo>
<msqrt>
<mi>c</mi>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0 \Rightarrow x = \frac{c}{2}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p>tangents at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{c}{2},\,\,y = - \frac{c}{2}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>hence there are two tangents <em><strong>AG</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="{x^2} + {y^2} = cx">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - \frac{c}{2}} \right)^2} + {y^2} = \frac{{{c^2}}}{4}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>M1</strong></em><em><strong>A1</strong></em></p>
<p>this is a circle radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{c}{2}}">
<mrow>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</mrow>
</math></span> centre <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{c}{2},\,\,0} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>hence there are two tangents <em><strong>AG</strong></em></p>
<p>tangents at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{c}{2},\,\,y = - \frac{c}{2}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Illustrate graphically the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} < \sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo><</mo>
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down a lower bound for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</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>Find an upper bound for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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="\int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} = \mathop {{\text{lim}}}\limits_{R \to \infty } \int\limits_4^R {\frac{1}{{{x^3}}}{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>R</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi>R</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> The above <em><strong>A1</strong> </em>for using a limit can be awarded at any stage.<br>Condone the use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to \infty } ">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
</math></span>.</p>
<p>Do not award this mark to candidates who use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\infty ">
<mi mathvariant="normal">∞</mi>
</math></span> as the upper limit throughout.</p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{R \to \infty } \left[ { - \frac{1}{2}{x^{ - 2}}} \right]_4^R\left( { = \left[ { - \frac{1}{2}{x^{ - 2}}} \right]_4^\infty } \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>R</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>4</mn>
<mi>R</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</msubsup>
</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=" = \mathop {{\text{lim}}}\limits_{R \to \infty } \left( { - \frac{1}{2}\left( {{R^{ - 2}} - {4^{ - 2}}} \right)} \right)">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>R</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mn>4</mn>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{32}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> <img src="data:image/png;base64,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"> <em><strong>A1A1A1A1</strong></em></p>
<p><em><strong>A1</strong> </em>for the curve<br><em><strong>A1</strong> </em>for rectangles starting at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 4">
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
</math></span><br><em><strong>A1</strong> </em>for at least three upper rectangles<br><em><strong>A1</strong> </em>for at least three lower rectangles</p>
<p><strong>Note:</strong> Award<em><strong> A0A1</strong></em> for two upper rectangles and two lower rectangles.</p>
<p>sum of areas of the lower rectangles < the area under the curve < the sum of the areas of the upper rectangles so</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} < \sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo><</mo>
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>a lower bound is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{32}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Allow <strong>FT</strong> from part (a).</p>
<p><em><strong>[1 mark]</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \frac{1}{{32}}">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>32</mn>
</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{1}{{64}} + \sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} = \frac{1}{{32}} + \frac{1}{{64}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>64</mn>
</mrow>
</mfrac>
<mo>+</mo>
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>64</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} < \frac{3}{{64}}">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mn>64</mn>
</mrow>
</mfrac>
</math></span>, an upper bound <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Allow <em><strong>FT</strong> </em>from part (a).</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>changing the lower limit in the inequality in part (b) gives</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} < \int\limits_3^\infty {\frac{1}{{{x^3}}}{\text{d}}x} \left( { < \sum\limits_{n = 3}^\infty {\frac{1}{{{n^3}}}} } \right)">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<munderover>
<mo>∫</mo>
<mn>3</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo><</mo>
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</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="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} < \mathop {{\text{lim}}}\limits_{R \to \infty } \left[ { - \frac{1}{2}{x^{ - 2}}} \right]_3^R">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>R</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<mo></mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>3</mn>
<mi>R</mi>
</msubsup>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} < \frac{1}{{18}}">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>18</mn>
</mrow>
</mfrac>
</math></span>, an upper bound <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Condone candidates who do not use a limit.</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.</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>This question asks you to explore cubic polynomials of the form</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> <strong>and corresponding cubic equations with one real root and two complex roots of the form </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>z</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo><mo>=</mo><mn>0</mn></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p> </p>
</div>
<div class="specification">
<p>In parts (a), (b) and (c), let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn></math>.</p>
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>z</mi><mo>+</mo><mn>17</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></mfenced></math> for <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>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i</mtext></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>On the Cartesian plane, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mo>-</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> represent the real and imaginary parts of the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<p><br>The following diagram shows a particular curve of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced></math> and the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mn>80</mn></mrow></mfenced></math>. The curve and the tangent both intersect the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math> are also shown.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≠</mo><mi>r</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mo>(</mo><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> are as defined in part (d)(ii). The curve has a point of inflexion at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="specification">
<p>Consider the special case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mtext>i</mtext></math> are roots of the equation, write down the third root.</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>Verify that the mean of the two complex roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</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>Show that the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and the tangent to the curve at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, clearly showing where the tangent crosses the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></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>Hence, or otherwise, prove that the tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow></mfenced></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce from part (d)(i) that the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this diagram to determine the roots of the corresponding equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced></math>.</p>
<p>You are <strong>not</strong> required to demonstrate a change in concavity.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence describe numerically the horizontal position of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> relative to the horizontal positions of the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>, state in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, the coordinates of points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.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>4</mn><mo>-</mo><mtext>i</mtext></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>mean<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>4</mn><mo>+</mo><mtext>i</mtext><mo>+</mo><mn>4</mn><mo>-</mo><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math> <em><strong>AG</strong></em></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><strong>METHOD 1</strong></p>
<p>attempts product rule differentiation <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for attempting to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn><mi>x</mi><mo>-</mo><mn>17</mn></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mn>4</mn></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> is correct, award <em><strong>A1</strong></em> for solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn></math> and obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>1</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>=</mo><mn>4</mn><mo>+</mo><mi>c</mi><mo>⇒</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>states the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and verifies that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> lies on the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is the tangent to the curve at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>(M0)A0A1A1</strong></em> to a candidate who does not attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to solve a correct cubic equation <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>4</mn></math></p>
<p><strong><br>OR</strong></p>
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mn>1</mn></math> and forms <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn><mo>=</mo><mn>1</mn><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to solve a correct quadratic equation <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>4</mn></math></p>
<p><strong><br>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> is a double root <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is the tangent to the curve at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Candidates using this method are not required to verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math>.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img src="data:image/png;base64,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"></p>
<p>a positive cubic with an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math>, and a local maximum and local minimum in the first quadrant both positioned to the left of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> As the local minimum and point A are very close to each other, condone graphs that seem to show these points coinciding.<br>For the point of tangency, accept labels such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>,</mo><mo> </mo><mfenced><mrow><mn>4</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math> or the point labelled from both axes. Coordinates are not required.</p>
<p> </p>
<p>a correct sketch of the tangent passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and crossing the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the same point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math> as the curve <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if both graphs cross the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at distinctly different points.</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><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(M1)A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi></mrow></mfenced><mi>x</mi><mo>-</mo><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mi>r</mi></math></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>a</mi><mi>x</mi><mo>-</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>a</mi><mi>r</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>AG</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><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>attempts to substitute their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR </strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mo>-</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mi>a</mi></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>A1</strong></em><br><strong><br>THEN</strong></p>
<p>so the tangent intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>attempts to substitute their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mi>x</mi><mo>+</mo><mi>c</mi></math> and attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>-</mo><mi>r</mi><mo>=</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>the line through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> parallel to the tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has equation<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>≠</mo><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>since there is a double root <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow></mfenced></math>, this parallel line through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> is the required tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.ii.</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>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>⇒</mo><mi>b</mi><mo>=</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em><br><br><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>±</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i=</mtext></mrow></mfenced><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>R1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>hence the complex roots can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>4</mn></math> (seen anywhere) <em><strong>A1</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p>attempts to find the gradient of the tangent in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and equates to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math> <em><strong>(M1)</strong></em><br><br><br><strong>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>80</mn></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo>=</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>80</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>16</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>80</mn><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>16</mn><mo>⇒</mo><mi>a</mi><mo>=</mo><mn>3</mn></math></p>
<p>roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> (seen anywhere) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>±</mo><mn>4</mn><mtext>i</mtext></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>±</mo><mn>4</mn><mtext>i</mtext></math>. Do not accept coordinates.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Accept “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn></math>”.<br>Do not award <em><strong>A1FT</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mo>−</mo><mn>4</mn><mo>)</mo></math>. </p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>a</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>r</mi><mo>-</mo><mn>4</mn><mi>a</mi></mrow></mfenced></math></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> and correctly solves for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p>for example, obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>r</mi><mo>+</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if the answer does not lead to the <em><strong>A</strong><strong>G</strong></em>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math> of the horizontal distance (way) from point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> to point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept equivalent numerical statements or a clearly labelled diagram displaying the numerical relationship.<br>Award <em><strong>A0</strong></em> for non-numerical statements such as “<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is between <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, closer to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>”.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><img 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Hlz0aBBA/gCAXi9Hu29U+LhViMiJVW6dNoXf/E5SpUqjTvuvENPdJJwZw8+MTHgiSjGGgL5/vvv46GHHsJpp52mc9GY0p49+ATFgCciJRX8pk2bsGzZcvzznw21cvf7fQz3BMaAJyIlvfYZb89A1apV8YdSpWL3UiJjwBOR2r9/P15/fRKuvPJKqzVDCY8BT5Sk7AOnci3tmVdfeUVD/pZbbtFqnhIfA54oSUlvXca4y8HV7du2Y+LEiWjWvDnKlC2DUChkPYkSGgOeKInJAh4yW+SMGW9j9+7duOeee0zcezjfjEsw4ImSmFTx2VnZeubq3Xc3Qrly5eDxeTlyxiUY8ERJTBbxWLN2Db7//nvcdtutOjRSZpIkd2DAEyUpObDq8/swaeJE3HHHHahx6aVauXs8Xj2DlRIfA54oSUm1vm/vXsyePRutWj2IlEDA3CutmSg8nHvGFbgViZKUDI+cOnUaKlSogIsqXSRHXM19su6qV6t7SnwMeKIkEo1EdQhkKBzGgf0HMGXKm7i9bl1dZFuOq0pVb19T4uNWJEomOvWvT9dXXbRoEZYtW4brr7sOPnMfuQ8DniiJyAgZac0czjqMZ599Fk2bNkWFCufqKk7kPgx4p4lNASI7oX0qudwpt60RDmZPNHdHouyRUv7JK0peO+NeG4etW39CixYtZDC8CQImvBt5THDYKfL7ohwbW1j++85/8c47M7QPGg5bCy7I3CDSO92y5SdMmz4NgZQUrcZ4xiHlh4x737NnN/7+95vM5UY8NWCguS+kKzcx4hOIJ2/7PQPeYWT0wsGDB3H48GHti2rBHttC2dnZ6N69O0Y9PwqpqdZCyDzjkPJDXl8Txo9H37598fbbM1CpUqXYdAVRnfudEkQeA54tGoeRwC5RogTKli2LUqVLoXSZMihjbsuldOnSSEtL1c/ZMl8IUX7JpGJDhgxB69ZtUOlPlWKjZeTAK19PbsSAdxi7KpfWi1yOVOhSxketlk1srpAjjxEdm3xA11eO+UXOTpXFtIsXL4FWD7bScJfqnYtquxe3KlESkOM1clh+2rRpaNmypQn54tYD5GoMeCIXk095MuZdKvUN69dj8+YtuP322/WTID8Buh8DnsjFpEUj4S7GjB6DSy65BGXLlYPXhHtex1dQ4mLAE7mdyfHPPv1M2zPWuHe9izNGJgEGPJGLSRtGht0+9VR/U71X1YsEuxxYZYvG/RjwRC4ma65+Mv8TrF+/Hl26PqonNOnoGQl3BrzrMeCJ3ESGQ8rQyNhF1lsdOWIErr76alSp8ld5MLc1wwre/RjwRK5izVlkV+cfzZ6NrVt/RsdOnRAIpOhjXMwjeXBLE7mKNe2AVOpZWVkY+/JYtGr1ACpWrMiOTBJiwBO5iLUik3Vi09o1a7X33qhRI31Mg5+SCgOeyEWk/SInMcmKTWPGjME111yDUqVLxw6sxp5ESYMBT+Qi9slLy5ctw/vvv4/6DRroSBpdY5UJn3QY8EQuc/jwIfTr10+r9+rVq8En0wCb3Ge8Jx8GPFGCk4VghFW9ezB0yBBs3LgJj3V7zBo5Y+6VkLere0oeDHiihOdBWNZajUTx4w8/YMqUqahbty4qnFNBz1i1WzOcEjj5cIsTJTgJcRkhE46E8cILz+s6Au07tNeqXXvvlLQY8EQJTk5e8vm8WLVqFd555x30798Ppf5QKnc9X0pe3PpECU5a61Kpj3vtNdSrVx833Ph3vZ9TERADnijBSZDv2LEDs2fPRvP7m8Mf8MfmgOdB1WTHgCdKcFK9y8iZyy+/XKcksKYDNrs2R80kPQY8UYJbuWIF3nrrbdSrVy92j5XtXh+HRiY7BjxRApLYlvHve/bsRa9evVC79hW4/vrrtV1jT0ugM0eyD5/UGPBECUaL8khEh0MOyxyKTZs24YkePeDzB6wnEMUw4IkSjBblpkrfuGEj3nhjMlq2bInzzj0PXp9MNBa2nkRkMOCJEoyuyGTK+BkzZqBMmTJo1LiRjpqRKYJ9HPdOR+GrgSjBSIfm0MFDeOut6Xjk34+gdGw6YGnZWOU9kYUBT5RgpA0zatQonHXWWTrnDNHxMOCJEsySJUt0MY/WbdqYyt1U7UTHwYAnSgA6mVg4gv3796N3r9647LJLUb16DRPwbMnQ8THgiRJCFLLe6ttvvYWVK1eibdt2SEkJIGJCn+h4GPBECUDGvi/+YjEGDBiIJ554HJfVvFzvt+acITo2BjyRA8kUAzocMnZ77949eOyxR1G7dm20fOABBPx+HTlD9Hv4CiFyJI9W7fawx48//hgHDhxEjx49fjH9AKcioN/DgCdypGjuyUsyLPLVV15F06b36dBIrtJEecWAJ3Igact4dNIwLz768COd7/3+Fi10OgJW7ZRXDHgih5Ie/DcrV+DJJ/ugdeuHUKxYce272715ohNhwBM51IH9+9GzZy9kZGSgQYOGRxbRZgVPecSAJ3IgacNMmjQJy5YtxRNPPIGU1BQd867tGT36SnRiDHgiB9q4cSMyMzPRtGkz/LVKFb1PTnQSslITUV4w4IkcQtovskqTjJoZOWIEatSogfYdOphHrFWaZLZIjn2n/OCrhcgh7JEzCxcsxIIFCzBo0CCULFkSXvbc6SQx4IkcIBqJmurchwP79mPIkCG47fbbUap0GXN/WA+uEp0MBjxRETl6orCo+QqHQpgwYTy++WYlmjRurINlpN8ulT3RyWDAExURCW8JeTu/ZZbIZ555Bg880Arnnnee9txl1AxPbKKTxYAnKiIyDYGW6aZ637ZtO9q2bYs777wLrR58kAdTqUDwVURURCKmdJcFO2TUzAvPj9IDqk8/8zSKFy8WewbRqWHAExURGdcuwyJ37tiJd9+dib59+yLgD8AT+yI6VQx4oiLiN2EeDAZ1rpnLLrsMVapW1WGS0rJhvlNBYMATFZFQMAeZQ4dizpw56PRIJ/jlDFUZOeP3c+QMFQgGvJOZnVx3dL2Wq9hOL1es8BKO9NrlbFVZPFtaM2+99TbGjn0FQzMzccEFF+j877KhuWmpoDDgHcgKdXPDYzaPx6M3f4ND5xKODHu0RLFxwwYMHDgQDRs2QJ06dXTUzNFDIjk0kgoCA95hrCrd7Nyyg8eCXm7q/s6dPqHJPO4S3FlZWTpDZPHixdGuXTukpaVxlSaKCwa8I0XNx/gQwrrTW/W73Z2hxCXhHswJ4tOFn2LRokXo268vTj/9dL2P494pHjymYsxbdETDsRsUT7I1Pv/fIixcuFC/l/lJIhH52XuQk5ODL774HBMnTUJKSqo8W0ODnEt2L7nIdpLrnTt34t577kGzZs3QqFEj+Pz+3OdxW1KeefI2PxED3mHko7ru7KaikzMdZU1Oqe5kzHROdjbat2+vB+XSUtMk8xkKDicHVu1tJEMiH27TRodHjhw1klU7nbw8BjxfYQ5jV3pROY3d0EUeYhkeich7sRyIM+FvqnqGewLQ+snapq++8go+/vgTPPzwwxr8RPHGgHcYK+Ct2zLqQj/CS0UvYR7LczlYp2OleWDO+WQkFKJY8vXXePbZ59C0aVNU+lMl+AMB63GiOGLAO1Ag4Ndgl1C3Mv3Ir1IJen1eeMybgHXWIzmZvAmvX78e7dq1xx133IEuXbvodAS57+JEccSEIIqzIYMHo3LlP+Hpp5+25prxemLtNqL4YsATxdFnn32GBQsWonfv3hrscpEzWXmAlQoDX2VEBUhaaNZ0BGFs+3kbunTprMMhK1Q4NzYqyqPHVuQ8B6J4Y8ATFSA73LOzs/H449010Js2vU8PoMiBVT2uIrdj49+J4okBT1SAJMAl3Hv26IFvv12NUc8/jzPOLA9ZVFuqe6LCxIAnKkChUBj9+vbFzJnvYdiwTFxyySXmXhnyZIU/UWFiwBOdArsqDwVD2pr53/8WYfLkyejWrRuqVL1EQ10OqPKgKhUFvuqIToEEuExBEJUvE/aZQzPRvHlzNG7SmC0ZKnIMeKJTIAdVvR4vAoEUjHttHLZv347WMt+MnqnKgKeixYAnyid7KKTVWrfmmRkz+gUMHjxYpwAuW7asPIltGSpyfAUS5ZMcSFUemS0yhPHjx+Oppwagc+fOuPLKKzXY7d47UVHiK5Aon+zgDpug/+HHH/HMM0/jnnvuwX1N7+MEcOQoDHiifNK5ZMIRhCPWkMhata5A165dEQoGcxfwIHICBjzRCVg9d2uUjFyHQyFdV7Xvk0/i4MGDGPifgUgvlq4HVuWEJiKnYMATnYgJdl14xfCa6j07Kxvdu3fD1KlT8dygQShdujQCsXCX6p7IKRjwRCcg8+7LQVN7jpnBgwfhww8/xJAhQ3DO2efoEnxS2cs8/azgyUkY8ER5YJ205MGwzEyMHfsK+vTpg6uvucaaAth6hj6HFTw5CQOe6BhknLscSJUpfuW25PvcOXMwevRodOrUCQ3/eTfS0tKsJ5tMt4dGEjkJA57oeCSvTXDL0McvPv8cffs+iYEDB6J1m9Z6oNWq6Rnq5FwMeKJj0DNVDQn36dOno0mTxnjggVb4R716umBHICUFERPyYEuGHIwBT2RYPXYJdOmlS+Fu7RpyMLVnz56oV6++rsxkLXQuoS4HVX3yG/V5RE7EgCcyJODlzFQT7+a21XNfuXIlunTurOHes1dPrdqlordb7dJzZ9+dnIwBT2To3DF+WStV1k31YuWK5WjXth3atWuHJ554Qse6S9XOQKdEwoAniomacJe1Uv/7zgw0bdpU55dp0bKlnqUqI2ok3O1WDlEiYMBT0tK2jAl1oUMhzfXkN95Aly5dcf31N2jIS+DLY3ISk1b5sd48USLgq5WSkoS7VOUyIkarcvPftGnTtB1zyy234Mm+TyIlNVUfY1uGEhUDnpKSVuOmKpcDq2Lu3Dno3q077r77bvTr1w8lSpQw91rBz4CnRMWAp6Rij2/X61hP/e2330b//k/p9AM9evbUnrtW9cJkO+eXoUTFgKekkZOTYzLdi1AwZAIcOHTgIIYPH66jZTp16ogm/2qCtFRr+gGp2nMvPJmJEhQDnpKGHjCVg6omr7OyDmPAgAEYNWoU+vXvh9vr1rXaNcxychEGPCURSe+oLszxwvPPY8qUKRruMhxSRsfIakw6pztTnlyCAU+uZffOdQhkNIpINII9e/eiV4+eeO+99zBx4kTUr19f2zDmF/lP3wPYkiG3YMCTa0lQ52Rna4BLwK9buxZN72uKOXNmY/SYMah6SVXreR5vbs1u992J3IABT64llXtKapqezLR48WI0a9oU+/fvw9ixY3H22edYC2TrMMgjo2uI3IQBT64llbi0ZVZ/+y0ebNUKZ5xxBl5/4w2cd/75CPitmSBlLLyEO6t2ciMGPLmKVOu6EpMJbbm9ccMGdOzYETVr1sJLL72Ms84+G36fqdy91tQDcpGzWRnw5EYMeHIdmfI3FArhrelvoXGjRrj66mswdOgQlC1XLnclJh0IT+RyDHhyFTmYmp2djccefRRdunRGy5Yt0f3x7la/3RTpcs1anZIFA54SmgS6DoHUBbKt4ZCDnnsOH3zwAUaMGIlmze/XMe4pKSlateuCHV6OlCHnktewsA/82y1HuQ7r7bx/+mTAU0KzdwYpy0PhEEaNHIk335yC0aPH4Kabb0YgYCr2WJjbPXcip5NAl6k13pv5nk5+J7e3/vwzOv/73+a+mbFnnRgDnlzh0MGD6Ne3HwYPHoL+/fujZq1apnJnmFPisYsWOfj/95v+jrVr1+CHdevw6itjcdppJfWxvGLAU2Iz+8Kc2bNRv34DLFnyNcaPH4+bb7lF2zKs1ikR2QEvr18J+Tp16mi7sVWrB9GrV2/cettt+nheMOApodg9SPnIGswJYvr0afi//3tId4KJkybhyquu1LaMVO8MeEpEditRzrCWVk2FCudi9epvUaJEcRP+RxZ9zwsGPCUUn0+m+w3qDrBgwXz07t0bjzzSCV26dkFKSmrsWUSJK3f9AVPJR8xl79492LLlJ2zfvh055rVvH3bKCwY8JRQZSSBnn77x+uto164dmjdvjgdatUJqWhr8cnYqUYLz+nzYs3sPli5biqVfL0GVKlVRrFgxfPzxJ1gwfz6ys7NizzwxBjw5jvQgc0fHGLnDxcz1li1b0KZ1a4wYMUJXX2rfoYPO8y5lzdG/hyhR/PZ1G8WhQwcxa9Ys7cH/8aI/YsDAATqP0uU1ayLNFDN55TF/eN72iqi1diUVPmsLRZF1+DDat++A4SOGI1XaES5uMetY36i1rJ7Viwxjx/YdaNasKUqWLIkhQ4ciIyNDH8tPT5LICWRxGanUZec+qempPXn7tMoKnhxHVl3Sk5FyR8JEsXbNGl2YIyPjLB3jfuaZZ5r7rceIEo2Eu3XANL7VCQOeHEeD3XzJh0s5yPTxvI/RqFFj/b5f//447Q+nWU+U6oflOyUkU5iY12441n6MlzwHvNRJdq0kR3m50nwhMy8GCTPpQ0vQyZebmH+SvqbsfrtUNzJh2OTX39CDqZUqVcLrr09CRvny8Hq88PsDerDVZT8GShK6D5vX+qz3ZmLXzl36vb1v248ffX2y8tyDj4SD1g372aZwkiqL4iz2I5bNJD34Tp06YejQoUhNSzf3mY94btkG0oc0/0Z5eckLPysrC0MGD9EDTa1bP4R69RsgNSVgfh5S3RMlNi3RzIv9y8VfmMLldZx++hlo1LgRzjvvvNyT9LSINi/2Y35KzWMPPs8BHw7l6LX0jsLBEHwBazUcijPZthJ85pKTnYP27dthaGYmUgIpuuGlneEGchBVXtjySWXn9h3mjawj1q9fj8lvvqkvfnmx6eOKEU+JLRQKImD2YXndyzGnBQsW4Omnn0azZs3Q8J93m2dY7ccjr/lfKeiAf2HUSL0+fPiQfpSw/mLuaIVBXgSysaVl8e677+L22+vmbvzjvgASjszhHkY4HMKcOXOxfPly1K9fD+eff74OFZPXmnVQyj7wSpS45DUs0asBbyp1GeorJzLJmdmtWrVC2/bt4TP7tjzvmK/3gg74jz54X6/lo7OcCi5hIzubU8jqPRMnTkDjxk30hBcn/b+dKtm+spkOH87CK6+MxQMPtNIzOoXXm7cN7UTyb7InBJPtt3LlNzq+vVSpP6BPnz6x8b7WjuAGsqPKfrN9+zb897//1e3oRvJGHDSf8mfOnGmKkdtNXgRij7iH7JNS6I4ePdpsxwfM/uiPPZJXUqFbw39lH5bXxY8//oBXX31Njznd2/Q++UHqa+aYRVxBB3zEfKRQ+m5i/ZZjvK8UGXkX7N69OwYMHGi+Mz8Y624XMAEX+3lLD75jx07IHJap85vrO3uChp/+X8sL2Ly4Jdw//OADPPXUU7jttluxatUqjH3lFQ12eXG7ZlvqvgOsW7cOmUOHYsiQIfq928i2lXZiZuZQdO7SxUX74hHyb5TMaftwG22ZyptYfv6dMjpM9l8ZRbNyxQpMnzZdpySQN/2//PUvsd1afpGAP8afXNDj4GXEglzkNSoH9jRcHHSxFk+Wk7H0J3PM5yTmRa6sn7c1Lty8OMJHFgI48rwEuxgS3rt370L3bt3Qtu3D6Nz533jkkUe0NylVjdWaMY71+xPwIr1W658jlby5/avH3XKRN2b5konhnJYTBXYxJHhzcoLWYjLiWM87zkV+RvJ6mDBuPIaZN4gb/34jnhs0GJUr/8kKd3kDkMrdSvqTlvc+hvxF8pea/zn7zCsNeod8Sa5LHys3DF30JT9pZX7+9vEPud8nragE+ZIXtL1dZJSM+OmnrejSuTNmzHgb//nPf3B73boaCtIClI/5crafsP+MhP8y201GRsgnFnnzOuoRV30JqwixBgcc/ZhbvmQftAstmdXU3ifz+iXb3+cP4KKL/ojhI0bobKjyhuGXTwLmRygFq7362KnIe8DL3yqXGOudWW445yLhlzuW9BiPJ+zlF8zLw2x0/flb3ybERXcIs21ygtZoLJkd795778HixYt1ml8ZBukzVbv826zjO9YbmAT+sf68RLzIz0BWnZJrrUiO8Rw3XOTf5zdv0tK2cN2+GLvIv0v+jRLKgZST+HfKlfm9V9SujfT0dOv72D5tX+ubhr2fn6S8B7zDSXjIQQv5aK8/bHIc2S7yot23b7+p3P+t0/tOnvwm/vrXKvrpJByxKlt71Ix1lp97tqW8RqVCk3+rTpDmUvZ2luvcNpvLSPBKm03exOSTZr6D2LysI+b3xTuvXBPw8oKqWrWKqfhC5rtTe9ejgmd/upr90Ww0bNhAJwqbOGkiKlasqKOe7DCQir1mzZoImUrfa3YaDUQ3Mf/OkiVPQ7Vq1WJ3uJEHwWAQlSv/Oa7hVZSk1SaZfumllyIirSjzfb6Y3yyZJa+HeMrzKBrOJll0rC2UeLNJ2kNpZUfIzs5G/379MG7cOLRu3RpdunSBz8VVLFFcFfQoGqL8koNJcrAoOytbRwpMnz5d15SUoXMMd6L4Y8BT3MgHjxxTub/44ouYOHESnn/+ebRo2UKD360f3YmchAFPBUZ67ELCW4aQyQkczZo2xeTJk/HCC8/jir/9TXuzHj2f4pf9Je3RHxX60ouXP4fvA0QnjwFPBciazlh67xMnTsS//tUEp512GqZMnYKatWoh4LeGzsnB01+THr0Eetj83n379uHQwQP6/TGeSkR5xICnAmSd2DJt6lT06dMbjRo1wnODBqFcuXJ6/+8JpKToSIS5c+fivnvvw5w5c2KPENHJYsBTgZG2y8IFC9G/f3906NABjz72GIqXKAFdnuwEw8ikUD+wfz9kIY9NmzZqK4eITg0DnvLl6EpcbkqvXU5IktPvJ7/xhi7O0axZczzUuo01IZo8zzwuZ6X+HnlzKFGyBGrXvgJlypSBTOQkB2Ptvj4R5R8DnvLHBLFMkqQ5b9JbTrfevm0b2rVti0GDBqFr10fRpk0b7Z1LaMtFT+g4AXmeX6Zcld8Y+17//Nj3RJR/DHjKF50A6ajQPXDgADp17Ii1a9dh2vTpaNqsGdLS0/IU6kQUX9wLKV8k3LV9Yir3nzZvRscOHbBnz16MnzAB5cqdbh4/Mi0BERUtBjz9LjuopS2jrRn5xoT4D2vXolWrB/H1119j8ODBJtzLwl5l6siEYfkjc4hL68ceHnmikTdE9PsY8PS7pNUiB1JlJIzHXCTkV3+7Gvfcc6+OVx83fjwuqlRJR72cygyJcpBW3kys1o41X7Z8WmDGE508BjydmKmm5cxSCfeZ776Le++9FzfeeAPefPNNVK5cWattGTEjIX2yrDV0ZYWcbMiK8wcPWou7y59NRCeHAU+/YY9Zl2u7gpacfWv6W+je/XEMGDAAffo8iTPLn5l7MFXC+JTm/jZ/0eHDh7Dos0WoXr06tm37GVt/+oltGqJTwOmCE4C1hQpvumCrco4trWcCPBQMYdLECejduw+eeOJxtGj5gDWaxnvUylKn6Oi/02oF2dW7DLNkGU/0C5wumE6eCW4T7NI2CebkYMjgwejXrx9atGiB++5rqu8ruiBwAZIevxzBtdcttRZ4lzeQ2BOIKN8Y8PQbEqoSssFQEAOeGoB3330Xo0Y9j8e6PaYrLMloFyuNCy59pb9vsw6uWh8s2aIhOnkMeNIQlYseSI313fft24tHOnXCZ599inHjx+H6G67XcJdMtyprc4n9/oIgvXy75SN/h8w4Kbd5whTRyePeQ0cCW5LVVOb79u5Bp46dMG/ePD2ges4551iVtBTtBRrrRBRPDHjSA5qy2IYc6Ny1cxceeqg1Fi/+AqNHj0a16tU11GX190hETnRiy4QoUTDgSUepyOiVvXt262yQq1evxksvvYzLa9aS/o22TuRxry6gzQqeKFEw4JOUVu0mtPVivt+1cyfatm2nKzBNfnMyLrv8cp16wDoBybxQ5DaHKxIlFAZ8sjKVuLTVJcC3bd2qUw+UKvUHjHr+BVSsWFFHteiQRbZkiBIWAz5JRaPWiUqrvvkG//rXPdpff/yJJ+DXhTmsSl1+5SgWosTFvTdJ2EMhj769eeNGtG/fTkN86NBMnH32OTpUUnry0pKR50gLh4gSEwM+icjJSxrYHg/27t2LTp06oVKlP+H1N15H5T//WQ+o2jNC2sMmWcETJS7uvUlCDqpKz0X66jKXe5PGTVC+fAYyh2WidOkykvnm1WB+ibJiJ3ILBnyS8GjLBVhnwl0mLJOee/fHu2sbRh7QSl2re74kiNyCe7NLybh1u4cuGS6jYmT91I4dOmLr1p8watQoZGRkIBAIaM9dWBOM6U0icgEGvEvZsz1aPfQosrKy0KxpUxPuW/Hiiy/ijxddZLVt5DzV2HOJyF24Z7uUjpSJHVSVpfV69uiBtLRUTJg4AX+tUkVHt+tIGZl+QEp8InIdBryL6FmpJtRtUplnZ2Xj2Wee0ekHXnzpJVSs+EcN9FhThtU7kYtx73YRa2hjbMSMkZ2VhaFDh2LChIno0bMnihcvnjv9gP1cwflliNyJAe8mJqjDobBey0HV1157DaNGjUS3bt1Q8/KaWt3brRkicj/u6S4SCYV0UY5wOISpU6dh5MiRGD58OFq0bAl7MQ3ttx/VxiEi92LAJzJtrRxpr0hsB4NBrFi+AgMHDsBzzz2LO+68M/aI2dgyBYH5PazgiZID9/QEptV47pmnHvj8fmzcsAEdO3ZA3bp1cc2115rnyPsAe+xEyYgBn8h0eKNVxcuJTTu2bUOvXj1NFR9Cm4cfNhW7D6FQ0HoaESUdBnwC08o8Vp1L333IkKFYunSpuR6CM04/Q++XycPkgKv1RkBEyYQBn8CkRSMVeigYxKuvvopZs2Zh+PARuLzm5XpQVXru8iYQSAnY7wNElEQY8AnMPklJTmKSETNjxozBlVddaZ3sxEQnSnoM+AQmFbzM696hQwfceeddqFatGkzNrpW79OSJKLkx4BOOtRpTKBxCOBRC7169sGvXTjRr3sxqy+j8MrHpf4koqTEFEoyEu9fjxeHDh9GzR098+ulnGDlqFCpUqGAddDU4zp2IBJMgwcj87TnBHAwfNgzTp0/DCy+8gJo1a+aGOxGRjQGfYMLhMFasWIGXX34ZXbp0RfUa1RHRge4MeCL6JY/5yJ+302CiMpaa4k02h4S4HCyVZfYgq+h5ZLm9KLKzs9G6dWs9gFqyZEk8N+g5pKalxH4nESUNjy924/exgncg6wQmcyNWmIfkRCW5DoWwY8cOLDcVfLfu3bVdQ0R0PKzgHUYW7Vj1zSosXbbMVOph+HwBHdfu8/mwffs2nR3yqf5P4c677rLmdudbNFHyyWMFz4B3GAn4jRs2YtOmTTr9gGwdj9mYa9Z8j75P9kXGWRmYM2cOUlLT5Nk65p2IkgwDPrHpiUrSqol6kJMTxP33N8f6H9ebgD8L48aNQ1pamj5stqD1G4goebAHn5jk/dY6C/VIZf7RRx9h/ifz0efJJ1GuXDnrOXl8Xyai5MWAdxgdPeOV3rq1abKysnTpPRk9c+Xf/qYTi8lZqgx5IjoRBrzDyKqpclBV+u8yauaFF57Xezt06oCI+ZKDqtKa8frML2zPENHvYMA7jAyRDIdMle7zYfq0aRgyZDC6dO2M9PS0I00bE+zaftcmPBHRsTHgHUZG0Uh7ZuP6Dejfvz/q16+PS6pV05OfjmrLExGdEAOeiMilGPAOI22Xw4cOo2/fJ1G8eHH8+9+d4TMVvSy9J9MAExHlFQPeAWRYpBxQDct1MIiXXnoJX331FUaMGImMs88yoW9tJp7URET5wYB3AJlrxieLY4fCWL5iJUaNGonevXujStWqbLsT0UljwDuALOAh4e7z+/D8qFG49trrcP0NN2q7xq7eiYjyi+nhALq8ninVZfHsL79cjG7duyEtNdWEuzWqhojoZDDgHSAcCePQwUM6S+TNN9+C8847TxfxkBNVOdadiE4WA94BJMRHjhqJVatW4f8e+j+dgkBGzshBVU5HQEQniwHvAHNmz8aY0WPQp08fnGuqdxkO6THhHg7LdMDcRER0cpgeRUTmm4mYAN+9azd69eqFq666EjfceEPuwVap6v3mWuahISI6GQz4IiPTikUx7+N52LNnD57o0QMpcmCVFTsRFRCmSRHQRbVNhb5v3z68OGYMBgwYiIoXVjQVfdhU9lxYhYgKBgO+kNjzt8vFns/95ZdeRtmyZXF73dsRiUY09DlqhogKCgO+kBwJbivEly1dhrFjX8bdjRrpPDOyqLZ9ISIqCAz4QiIHVa0KPoLDhw5h0KBBWr1fe+21sWcQERUsBnwhsdZZNfW714uZM2fi008XomvXR3XGSCKieGDAFxJpy8i0A999+y2ee24QmjVrhltvu1UreyKieGDAFxKvT37UHvTv/xTKlSuLLl27mtC3DrYSEcUDA76QSKW++Isv8PHH89ChQwc9sCpDIuWkJiKieGDAFxYPMGXKm7j11ltxzbXX6lBJGfdu9+aJiAoaAz5OpN9uX8TypUvx6aefoVv37joUUgI+kJLCuWaIKG6YLnFijXu3xrwfOHAA/fv3x913342MjAw9k5WIKN4Y8HFiBby5lmGR787EkiVLdNSM3x/QqYCJiOKNSRNHclLTvr17MXLkSNxyy6248MILrQdi4U9EFE8M+DjymK8xo0dj27ZtaNGiBXw+v7mXwyKJqHAw4ONEDq6u37AeY8e+gk6dOqHqJVX1fmnd2O0bIqJ4YsDHiYT4hPHjUb16dTS//365R3o2em2PrCEiiicGfAGR7JZx7eGQuZjrgwcPYu7cuejYsWNsWKRHz2aV4p1DI4moMDBpCoxJeI8J8NhC2e+//z7S0tJx8V8u5slMRFQkGPAFRELdOrEpimAwiGefeQYNGzZAIBCIzUNDRFS4mDwFxD54Go2E8Z+B/9GWzU0336z3hUOh2LOIiAoPA76A2EG+8ptvMHHiBHTu0hnly2eY6t3HhbSJqEgweQqIPe3vsMxhqFy5MureXlcPqOqASOshIqJCxYAvMB4sW74M8+d/gjZtHkZqWppW9ZFwhD14IioSTJ5TYK2xKhfg8KGD6NmjB2rX/hvq1Kkjj+pj1tBInthERIWPAV8AZBjkvI8/xg8//IhevXshLT1dQ53j3YmoKDGBToG23eUXrwdvTZ+OO+6oiwrnVNCqnWerElFRY8CfApkt0uPzYcumzVi5cqUuxef1+xAKhVi9E1GRYwqdAntopKyzeuGFFXFmRoZW9F723InIARjwp0AOom7ZvBmZmcPQuHFja0ik+ZXxTkROwIA/RUOHZiItLQ3XXX+dVvQ+v3Vikz0unoioqDDg8yOW2TJbpBxEXb16NaZOnYpmzZohPb2YNRzSBDuHRRKREzDg8yFqvvTEJVOhS4hPGD8B5cqVxV3/uEsD/+hgZ8gTUVFjwOeDhrb5T+Z8//677zF9+nQ8+uhjOK1kSZ3vnYjISRjw+SB9dY/Hqwt4fPDhB6hWrRruvOtOBFJSOe6diByHAZ9PUsRLq+bDDz5EkyZNNOxDoaDOGklE5CQM+HyQ/rscYF23dh127dqlI2fsfjxXbSIip2HA54MMgZQm/IQJ41GjRg2kp6VZD0R5UJWInIcBnw9WDx5o1KgR2rZrl9uWKej2jPw9uZeIefcw10LeRPR9hEPsiSgPGPD5IAErLZnKf/4zLrroj7HAtUK3oCp4fROJnSilt2UuefNHxzLeut98RSJh6w4iouNgwDvQtp9/xrffrsa6devw/XffYc33a7B2zRqsWbMWWVlZOhUClwEkohPxmIowbx/4o6wYC8uC+Qswd+7cX3wykOucnBx8+eWXmDZtGvwpKYiEwrHjAkSUVDx52+8Z8E5jtoa1QezNYgW8bKbsrMPo2LEjhmYOQ2pqSoG1hYgoweQx4Pk532m0apfLkf6+3h27jkSsA73mEesOIqLjYMATEbkUA56IyKUY8ERELsWAJyJyKQY8EZFLMeCJiFyKAU9E5FIMeCIil2LAExG5VN6nKiAiooTCCp6IyKUY8ERELsWAJyJyKQY8EZFLMeCJiFyKAU9E5ErA/wMPMafex7m+kAAAAABJRU5ErkJggg=="></p>
<p>a positive cubic with no stationary points and a non-stationary point of inflexion at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Graphs may appear approximately linear. Award this <em><strong>A1</strong> </em>if a change of concavity either side of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> is apparent.<br>Coordinates are not required and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept need not be indicated.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) (i) was generally well done with a significant majority of candidates using the conjugate root theorem to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mtext>i</mtext></math> as the third root. A number of candidates, however, wasted considerable time attempting an algebraic method to determine the third root. Part (a) (ii) was reasonably well done. A few candidates however attempted to calculate the product of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mtext>i</mtext></math>.</p>
<p>Part (b) was reasonably well done by a significant number of candidates. Most were able to find a correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> and a good number of those candidates were able to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mn>4</mn><mo>)</mo><mo>=</mo><mn>1</mn></math>. Candidates that did not determine the equation of the tangent had to state that the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is also 1 and verify that the point (4,3) lies on the line. A few candidates only met one of those requirements. Weaker candidates tended to only verify that the point (4,3) lies on the curve and the tangent line without attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<p>Part (c) was not answered as well as anticipated. A number of sketches were inaccurate and carelessly drawn with many showing both graphs crossing the <em>x-</em>axis at distinctly different points.</p>
<p>Part (d) (i) was reasonably well done by a good number of candidates. Most successful responses involved use of the product rule. A few candidates obtained full marks by firstly expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, then differentiating to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math>and finally simplifying to obtain the desired result. A number of candidates made elementary mistakes when differentiating. In general, the better candidates offered reasonable attempts at showing the general result in part (d) (ii). A good number gained partial credit by determining that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>(</mo><mi>a</mi><mo>-</mo><mi>r</mi><mo>)</mo></math>. Only the very best candidates obtained full marks by concluding that as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<p>In general, only the best candidates were able to use the result <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> to deduce that the complex roots of the equation can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo></msqrt></math>. Although given the complex roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i</mtext></math>, a significant number of candidates attempted, with mixed success, to use the quadratic formula to solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math>.</p>
<p>In part (f) (i), only a small number of candidates were able to determine all the roots of the equation. Disappointingly, a large number did not state <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> as a root. Some candidates determined that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>4</mn></math> but were unable to use the diagram to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>3</mn></math>. Of the candidates who determined all the roots in part (f) (i), very few gave the correct coordinates for C<sub>2</sub> . The most frequent error was to give the <em>y-</em>coordinate as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mn>4</mn><mtext>i</mtext></math>.</p>
<p>Of the candidates who attempted part (g) (i), most were able to find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> and a reasonable number of these were then able to convincingly show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi><mo>)</mo></math>. It was very rare to see a correct response to part (g) (ii). A few candidates stated that P is between R and A with some stating that P was closer to A. A small number restated <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi><mo>)</mo></math> in words.</p>
<p>Of the candidates who attempted part (h) (i), most were able to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced></math>. However, most graphs were poorly drawn with many showing a change in concavity at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> rather than at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>. In part (h) (ii), only a very small number of candidates determined that A and P coincide at (<em>r</em>,0).</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>
<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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>This question will investigate methods for finding definite integrals of powers of trigonometrical functions.</p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_n} = \int\limits_0^{\tfrac{\pi }{2}} {{\text{si}}{{\text{n}}^n}} x\,dx{\text{,}}\,\,n \in \mathbb{N}">
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
</math></span>.</p>
<p> </p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_n} = \int\limits_0^{\tfrac{\pi }{2}} {{\text{co}}{{\text{s}}^n}} x\,dx{\text{,}}\,\,n \in \mathbb{N}.">
<mrow>
<msub>
<mi>J</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>.</mo>
</math></span></p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_n} = \int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^n}} x\,dx{\text{,}}\,\,n \in \mathbb{N}">
<mrow>
<msub>
<mi>T</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_0}">
<mrow>
<msub>
<mi>I</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_1}">
<mrow>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_2}">
<mrow>
<msub>
<mi>I</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use integration by parts to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_n} = \frac{{n - 1}}{n}{I_{n - 2}}{\text{,}}\,\,n \geqslant 2">
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</mfrac>
<mrow>
<msub>
<mi>I</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</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>Explain where the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 2">
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</math></span> was used in your proof.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_3}">
<mrow>
<msub>
<mi>I</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_4}">
<mrow>
<msub>
<mi>I</mi>
<mn>4</mn>
</msub>
</mrow>
</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>Use the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2} - u">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>u</mi>
</math></span> to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_n} = {I_n}">
<mrow>
<msub>
<mi>J</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_{5}}">
<mrow>
<msub>
<mi>J</mi>
<mrow>
<mn>5</mn>
</mrow>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_{6}}">
<mrow>
<msub>
<mi>J</mi>
<mrow>
<mn>6</mn>
</mrow>
</msub>
</mrow>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{0}}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>0</mn>
</mrow>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{1}}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the fact that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^2}x = {\text{se}}{{\text{c}}^2}x - 1">
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_n} = \frac{1}{{n - 1}} - {T_{n - 2}}{\text{,}}\,\,n \geqslant 2">
<mrow>
<msub>
<mi>T</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain where the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 2">
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</math></span> was used in your proof.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{2}}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>2</mn>
</mrow>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{3}}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>3</mn>
</mrow>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_0} = \int\limits_0^{\tfrac{\pi }{2}} 1 \,dx = \left[ x \right]_0^{\tfrac{\pi }{2}} = \frac{\pi }{2}">
<mrow>
<msub>
<mi>I</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mi>x</mi>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</msubsup>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_1} = \int\limits_0^{\tfrac{\pi }{2}} {{\text{sin}}\,x} \,dx = \left[ { - {\text{cos}}\,x} \right]_0^{\tfrac{\pi }{2}} = 1">
<mrow>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</msubsup>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_2} = \int\limits_0^{\tfrac{\pi }{2}} {{\text{si}}{{\text{n}}^2}x} \,dx = \int\limits_0^{\tfrac{\pi }{2}} {\frac{{1 - {\text{cos}}\,2x}}{2}} \,dx = \left[ {\frac{x}{2} - \frac{{{\text{sin}}\,2x}}{4}} \right]_0^{\tfrac{\pi }{2}} = \frac{\pi }{4}">
<mrow>
<msub>
<mi>I</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</msubsup>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u = {\text{si}}{{\text{n}}^{n - 1}}x}">
<mrow>
<mi>u</mi>
<mo>=</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{v = - \cos x}">
<mrow>
<mi>v</mi>
<mo>=</mo>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{{du}}{{dx}} = \left( {n - 1} \right){\text{si}}{{\text{n}}^{n - 2}}x\,{\text{cos}}\,x}">
<mrow>
<mfrac>
<mrow>
<mi>d</mi>
<mi>u</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{{dv}}{{dx}} = {\text{sin}}\,x}">
<mrow>
<mfrac>
<mrow>
<mi>d</mi>
<mi>v</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_n} = \left[ { - {\text{si}}{{\text{n}}^{n - 1}}x\,{\text{cos}}\,x} \right]_0^{\tfrac{\pi }{2}} + \int\limits_0^{\tfrac{\pi }{2}} {\left( {n - 1} \right){\text{si}}{{\text{n}}^{n - 2}}x\,{\text{cos}}{\,^2}x} \,dx">
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</msubsup>
<mo>+</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<msup>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</math></span> <em><strong>M1A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0 + \int\limits_0^{\tfrac{\pi }{2}} {\left( {n - 1} \right){\text{si}}{{\text{n}}^{n - 2}}x\left( {1 - {\text{si}}{{\text{n}}^2}x} \right)} \,dx = \left( {n - 1} \right)\left( {{I_{n - 2}} - {I_n}} \right)">
<mo>=</mo>
<mn>0</mn>
<mo>+</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>I</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow n{I_n} = \left( {n - 1} \right){I_{n - 2}} \Rightarrow {I_n} = \frac{{\left( {n - 1} \right)}}{n}{I_{n - 2}}">
<mo stretchy="false">⇒</mo>
<mi>n</mi>
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msub>
<mi>I</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mi>n</mi>
</mfrac>
<mrow>
<msub>
<mi>I</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>need <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 2">
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</math></span> so that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^{n - 1}}\tfrac{\pi }{2} = 0">
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
<mo>=</mo>
<mn>0</mn>
</math></span> in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ { - {\text{si}}{{\text{n}}^{n - 1}}\,x\,{\text{cos}}\,x} \right]_0^{\tfrac{\pi }{2}}">
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</msubsup>
</math></span> <em><strong>R1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_3} = \frac{2}{3}{I_1} = \frac{2}{3}\,\,\,\,\,\,\,\,\,\,\,\,\,{I_4} = \frac{3}{4}{I_2} = \frac{{3\pi }}{{16}}">
<mrow>
<msub>
<mi>I</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<msub>
<mi>I</mi>
<mn>4</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mrow>
<msub>
<mi>I</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1A1</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2} - u \Rightarrow \frac{{dx}}{{du}} = - 1">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>u</mi>
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>u</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_n} = \int\limits_0^{\tfrac{\pi }{2}} {{\text{co}}{{\text{s}}^n}} x\,dx\,\, = \int\limits_{\tfrac{\pi }{2}}^0 { - {\text{co}}{{\text{s}}^n}} \left( {\frac{\pi }{2} - u} \right)\,du = - \int\limits_{\tfrac{\pi }{2}}^0 {{\text{si}}{{\text{n}}^n}} u\,du\, = \int\limits_0^{\tfrac{\pi }{2}} {{\text{si}}{{\text{n}}^n}} u\,du\, = {I_n}">
<mrow>
<msub>
<mi>J</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mn>0</mn>
</munderover>
<mrow>
<mo>−</mo>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>u</mi>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>u</mi>
<mo>=</mo>
<mo>−</mo>
<munderover>
<mo>∫</mo>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mn>0</mn>
</munderover>
<mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>u</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>u</mi>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>u</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>u</mi>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> <em><strong>M1A1A1AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></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="{J_5} = {I_5} = \frac{4}{5}{I_3} = \frac{4}{5} \times \frac{2}{3} = \frac{8}{{15}}\,\,\,\,\,\,\,\,\,\,{J_6} = {I_6} = \frac{5}{6}{I_4} = \frac{5}{6} \times \frac{{3\pi }}{{16}} = \frac{{5\pi }}{{32}}">
<mrow>
<msub>
<mi>J</mi>
<mn>5</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>I</mi>
<mn>5</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>5</mn>
</mfrac>
<mrow>
<msub>
<mi>I</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>5</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>8</mn>
<mrow>
<mn>15</mn>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<msub>
<mi>J</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>I</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>6</mn>
</mfrac>
<mrow>
<msub>
<mi>I</mi>
<mn>4</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{0\,}}{\text{ = }}\int\limits_0^{\tfrac{\pi }{4}} {1\,dx} = \left[ x \right]_0^{\tfrac{\pi }{4}} = \frac{\pi }{4}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>0</mn>
<mspace width="thinmathspace"></mspace>
</mrow>
</msub>
</mrow>
<mrow>
<mtext> = </mtext>
</mrow>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mrow>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mi>x</mi>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</msubsup>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{1\,}}{\text{ = }}\int\limits_0^{\tfrac{\pi }{4}} {{\text{tan}}\,dx} = \left[ { - \ln \left| {{\text{cos}}\,x} \right|} \right]_0^{\tfrac{\pi }{4}} = - {\text{ln}}\frac{1}{{\sqrt 2 }} = {\text{ln}}\sqrt 2 ">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
</mrow>
</msub>
</mrow>
<mrow>
<mtext> = </mtext>
</mrow>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mrow>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</msubsup>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_n} = \int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^n}} x\,dx\, = \int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^{n - 2}}} x\,{\text{ta}}{{\text{n}}^2}x\,dx\, = \,\int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^{n - 2}}} x\left( {{\text{se}}{{\text{c}}^2}x - 1} \right)\,dx">
<mrow>
<msub>
<mi>T</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^{n - 2}}} x\,{\text{se}}{{\text{c}}^2}x\,dx - \int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^{n - 2}}} x\,dx = \left[ {\frac{{{\text{ta}}{{\text{n}}^{n - 1}}x}}{{n - 1}}} \right]_0^{\tfrac{\pi }{4}} - {T_{n - 2}} = \frac{1}{{n - 1}} - {T_{n - 2}}\,\,\,">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>−</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</msubsup>
<mo>−</mo>
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span> <em><strong>A1A1AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>need <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 2">
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</math></span> so that the powers of tan in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^{n - 2}}} x\,{\text{se}}{{\text{c}}^2}x\,dx - \int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^{n - 2}}} x\,dx">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>−</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</math></span> are not negative <em><strong>R1</strong></em> </p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_2} = 1 - {T_0} = 1 - \frac{\pi }{4}">
<mrow>
<msub>
<mi>T</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msub>
<mi>T</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_3} = \frac{1}{2} - {T_1} = \frac{1}{2} - \ln \sqrt 2 ">
<mrow>
<msub>
<mi>T</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>ln</mi>
<mo></mo>
<msqrt>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>This question asks you to investigate regular <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>-sided polygons inscribed and circumscribed in a circle, and the perimeter of these as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> tends to infinity, to make an approximation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i}\left( n \right)">
<mrow>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</math></span> represent the perimeter of any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>-sided regular polygon inscribed in a circle of radius 1 unit.</p>
</div>
<div class="specification">
<p>Consider an equilateral triangle ABC of side length, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> units, circumscribed about a circle of radius 1 unit and centre O as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_c}\left( n \right)">
<mrow>
<msub>
<mi>P</mi>
<mi>c</mi>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</math></span> represent the perimeter of any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>-sided regular polygon circumscribed about a circle of radius 1 unit.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider an equilateral triangle ABC of side length, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> units, inscribed in a circle of radius 1 unit and centre O as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The equilateral triangle ABC can be divided into three smaller isosceles triangles, each subtending an angle of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> at O, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Using right-angled trigonometry or otherwise, show that the perimeter of the equilateral triangle ABC is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\sqrt 3 "> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </math></span> units.</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>Consider a square of side length, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> units, inscribed in a circle of radius 1 unit. By dividing the inscribed square into four isosceles triangles, find the exact perimeter of the inscribed square.</p>
<p> </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 perimeter of a regular hexagon, of side length, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> units, inscribed in a circle of radius 1 unit.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i}\left( n \right) = 2n\,{\text{sin}}\left( {\frac{\pi }{n}} \right)"> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an appropriate Maclaurin series expansion to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{n \to \infty } {P_i}\left( n \right)"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math></span> and interpret this result geometrically.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</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="{P_c}\left( n \right) = 2n\,{\text{tan}}\left( {\frac{\pi }{n}} \right)"> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_c}\left( n \right)"> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\,{\text{tan}}\left( {\frac{\pi }{n}} \right)}}{{\frac{1}{n}}}"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mfrac> </math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{n \to \infty } {P_c}\left( n \right)"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the results from part (d) and part (f) to determine an inequality for the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</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">[2]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The inequality found in part (h) can be used to determine lower and upper bound approximations for the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>.</p>
<p>Determine the least value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> such that the lower bound and upper bound approximations are both within 0.005 of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">i.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>consider right-angled triangle OCX where CX <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{x}{2}"> <mo>=</mo> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{\pi }{3} = \frac{{\frac{x}{2}}}{1}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mn>1</mn> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{x}{2} = \frac{{\sqrt 3 }}{2} \Rightarrow x = \sqrt 3 "> <mo stretchy="false">⇒</mo> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i} = 3 \times x = 3\sqrt 3 "> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mi>x</mi> <mo>=</mo> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p><em>eg </em> use of the cosine rule <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = {1^2} + {1^2} - 2\left( 1 \right)\left( 1 \right){\text{cos}}\frac{{2\pi }}{3}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mn>1</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>1</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>M1A1</strong></em> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt 3 "> <mi>x</mi> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i} = 3 \times x = 3\sqrt 3 "> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mi>x</mi> <mo>=</mo> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Accept use of sine rule.</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="{\text{sin}}\frac{\pi }{4} = \frac{1}{x}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = side of square <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt 2 "> <mi>x</mi> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i} = 4\sqrt 2 "> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo>=</mo> <mn>4</mn> <msqrt> <mn>2</mn> </msqrt> </math></span> <em><strong>A1</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>6 equilateral triangles ⇒<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 1 <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i} = 6"> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo>=</mo> <mn>6</mn> </math></span> <em><strong>A1</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>in right-angled triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\left( {\frac{\pi }{n}} \right) = \frac{{\frac{x}{2}}}{1}"> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mn>1</mn> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = 2\,{\text{sin}}\left( {\frac{\pi }{n}} \right)"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </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="{P_i} = n \times x"> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo>=</mo> <mi>n</mi> <mo>×</mo> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i} = n \times 2\,{\text{sin}}\left( {\frac{\pi }{n}} \right)"> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo>=</mo> <mi>n</mi> <mo>×</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </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="{P_i} = 2n\,{\text{sin}}\left( {\frac{\pi }{n}} \right)"> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></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>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{n \to \infty } 2n\,{\text{sin}}\left( {\frac{\pi }{n}} \right)"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x = x - \frac{{{x^3}}}{{3{\text{!}}}} + \frac{{{x^5}}}{{5{\text{!}}}} - \ldots "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mrow> <mn>3</mn> <mrow> <mtext>!</mtext> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>5</mn> </msup> </mrow> </mrow> <mrow> <mn>5</mn> <mrow> <mtext>!</mtext> </mrow> </mrow> </mfrac> <mo>−</mo> <mo>…</mo> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2n\,{\text{sin}}\left( {\frac{\pi }{n}} \right) = 2n\left( {\frac{\pi }{n} - \frac{{{\pi ^3}}}{{6{n^3}}} + \frac{{{\pi ^5}}}{{120{n^5}}} - \ldots } \right)"> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mi>π</mi> <mn>3</mn> </msup> </mrow> </mrow> <mrow> <mn>6</mn> <mrow> <msup> <mi>n</mi> <mn>3</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mi>π</mi> <mn>5</mn> </msup> </mrow> </mrow> <mrow> <mn>120</mn> <mrow> <msup> <mi>n</mi> <mn>5</mn> </msup> </mrow> </mrow> </mfrac> <mo>−</mo> <mo>…</mo> </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=" = 2\left( {\pi - \frac{{{\pi ^3}}}{{6{n^2}}} + \frac{{{\pi ^5}}}{{120{n^4}}} - \ldots } \right)"> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>π</mi> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mi>π</mi> <mn>3</mn> </msup> </mrow> </mrow> <mrow> <mn>6</mn> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mi>π</mi> <mn>5</mn> </msup> </mrow> </mrow> <mrow> <mn>120</mn> <mrow> <msup> <mi>n</mi> <mn>4</mn> </msup> </mrow> </mrow> </mfrac> <mo>−</mo> <mo>…</mo> </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 \mathop {{\text{lim}}}\limits_{n \to \infty } 2n\,{\text{sin}}\left( {\frac{\pi }{n}} \right) = 2\pi "> <mo stretchy="false">⇒</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> </math></span> <em><strong>A1</strong></em></p>
<p>as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \to \infty "> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></span> polygon becomes a circle of radius 1 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i} = 2\pi "> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> </math></span> <em><strong>R1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider an <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>-sided polygon of side length <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span></p>
<p>2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> right-angled triangles with angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{{2n}} = \frac{\pi }{n}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> at centre <em><strong>M1A1</strong></em></p>
<p>opposite side <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{2} = {\text{tan}}\left( {\frac{\pi }{n}} \right) \Rightarrow x = 2\,{\text{tan}}\left( {\frac{\pi }{n}} \right)"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> <mo>=</mo> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1A1</strong></em></p>
<p>Perimeter <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_c} = 2n\,{\text{tan}}\left( {\frac{\pi }{n}} \right)"> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{n \to \infty } 2n\,{\text{tan}}\left( {\frac{\pi }{n}} \right) = \mathop {{\text{lim}}}\limits_{n \to \infty } \left( {\frac{{2\,{\text{tan}}\left( {\frac{\pi }{n}} \right)}}{{\frac{1}{n}}}} \right)"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \mathop {{\text{lim}}}\limits_{n \to \infty } \left( {\frac{{2\,{\text{tan}}\left( {\frac{\pi }{n}} \right)}}{{\frac{1}{n}}}} \right) = \frac{0}{0}"> <mo>=</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>0</mn> <mn>0</mn> </mfrac> </math></span> <em><strong>R1</strong></em></p>
<p>attempt to use L’Hopital’s rule <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \mathop {{\text{lim}}}\limits_{n \to \infty } \left( {\frac{{ - \frac{{2\pi }}{{{n^2}}}{\text{se}}{{\text{c}}^2}\left( {\frac{\pi }{n}} \right)}}{{ - \frac{1}{{{n^2}}}}}} \right)"> <mo>=</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mrow> <mtext>se</mtext> </mrow> <mrow> <msup> <mrow> <mtext>c</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\pi "> <mo>=</mo> <mn>2</mn> <mi>π</mi> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i} < 2\pi < {P_c}"> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mo><</mo> <mn>2</mn> <mi>π</mi> <mo><</mo> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2n\,{\text{sin}}\left( {\frac{\pi }{n}} \right) < 2\pi < 2n\,{\text{tan}}\left( {\frac{\pi }{n}} \right)"> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo><</mo> <mn>2</mn> <mi>π</mi> <mo><</mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </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="n\,{\text{sin}}\left( {\frac{\pi }{n}} \right) < \pi < n\,{\text{tan}}\left( {\frac{\pi }{n}} \right)"> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo><</mo> <mi>π</mi> <mo><</mo> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find the lower bound and upper bound approximations within 0.005 of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 46 <em><strong>A2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">i.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">i.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y = {x^p} + 1">
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−<!-- − --></mo>
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R},\,x \ne 0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> is a positive integer, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p > 1">
<mi>p</mi>
<mo>></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation given that <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> when <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>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate(s) of the points on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> satisfy the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^{p - 1}} = \frac{1}{p}">
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>p</mi>
</mfrac>
</math></span>.</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>Deduce the set of values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> such that there are two points on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span>. Give a reason for your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{y}{x} = {x^{p - 1}} + \frac{1}{x}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p>integrating factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^{\int { - \frac{1}{x}{\text{d}}x} }}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ = }}{{\text{e}}^{ - {\text{ln}}\,x}}">
<mrow>
<mtext> = </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</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="\frac{1}{x}">
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{x}\frac{{{\text{d}}y}}{{{\text{d}}x}} - \frac{y}{{{x^2}}} = {x^{p - 2}} + \frac{1}{{{x^2}}}">
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mi>y</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</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{{\text{d}}}{{{\text{d}}x}}\left( {\frac{y}{x}} \right) = {x^{p - 2}} + \frac{1}{{{x^2}}}">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{y}{x} = \frac{1}{{p - 1}}{x^{p - 1}} - \frac{1}{x} + C">
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mo>+</mo>
<mi>C</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Condone the absence of <em>C</em>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{p - 1}}{x^p} + Cx - 1">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>+</mo>
<mi>C</mi>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p>substituting <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1 \Rightarrow C = - \frac{1}{{p - 1}}">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>C</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong> </em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to find their value of <em>C</em>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{p - 1}}\left( {{x^p} - x} \right) - 1">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[8 marks]</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>put <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = vx">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mi>x</mi>
</math></span> so that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = v + x\frac{{{\text{d}}v}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> <em><strong>M1(A1)</strong></em></p>
<p>substituting, <em><strong>M1 </strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\left( {v + x\frac{{{\text{d}}v}}{{{\text{d}}x}}} \right) - vx = {x^p} + 1">
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>v</mi>
<mi>x</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\frac{{{\text{d}}v}}{{{\text{d}}x}} = {x^{p - 1}} + \frac{1}{x}">
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}x}} = {x^{p - 2}} + \frac{1}{{{x^2}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \frac{1}{{p - 1}}{x^{p - 1}} - \frac{1}{x} + C">
<mi>v</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mo>+</mo>
<mi>C</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Condone the absence of <em>C</em>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{p - 1}}{x^p} + Cx - 1">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>+</mo>
<mi>C</mi>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p>substituting <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1 \Rightarrow C = - \frac{1}{{p - 1}}">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>C</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong> </em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to find their value of <em>C</em>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{p - 1}}\left( {{x^p} - x} \right) - 1">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[8 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>find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> and solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{1}{{p - 1}}\left( {p{x^{p - 1}} - 1} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</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{{{\text{d}}y}}{{{\text{d}}x}} = 0 \Rightarrow p{x^{p - 1}} - 1 = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<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="p{x^{p - 1}} = 1">
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><strong>Note:</strong> Award a maximum of <em><strong>M1A0</strong> </em>if a candidate’s answer to part (a) is incorrect.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^{p - 1}} = \frac{1}{p}">
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>p</mi>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> and their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> into the differential equation and solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0 \Rightarrow - \left( {\frac{{{x^p} - x}}{{p - 1}}} \right) + 1 = {x^p} + 1">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</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="{x^p} - x = {x^p} - p{x^p}">
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>−</mo>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>p</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="p{x^{p - 1}} = 1">
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><strong>Note:</strong> Award a maximum of <em><strong>M1A0</strong> </em>if a candidate’s answer to part (a) is incorrect.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^{p - 1}} = \frac{1}{p}">
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>p</mi>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>there are two solutions for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> is odd (and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p > 1">
<mi>p</mi>
<mo>></mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p - 1">
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> is even there are two solutions (to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^{p - 1}} = \frac{1}{p}">
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>p</mi>
</mfrac>
</math></span>)</p>
<p>and if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p - 1">
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> is odd there is only one solution (to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^{p - 1}} = \frac{1}{p}">
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>p</mi>
</mfrac>
</math></span>) <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Only award the <em><strong>R1</strong> </em>if both cases are considered.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore properties of a family of curves of the type</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> <strong>for various values of</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <strong>and</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>, <strong>where</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math>.</p>
</div>
<div class="specification">
<p>On the same set of axes, sketch the following curves for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>2</mn></math>, clearly indicating any points of intersection with the coordinate axes.</p>
</div>
<div class="specification">
<p>Now, consider curves of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mroot><mi>b</mi><mn>3</mn></mroot></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="specification">
<p>Next, consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></math> has two points of inflexion. Due to the symmetry of the curve these points have the same <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>)</mo></math> is defined to be a rational point on a curve if <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> are rational numbers.</p>
<p>The tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> at a rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> intersects the curve at another rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> be the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mroot><mn>2</mn><mn>3</mn></mroot></math>. The rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>)</mo></math> lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mo>-</mo><mn>1</mn></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>Write down the coordinates of the two points of inflexion on the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering each curve from part (a), identify two key features that would distinguish one curve from the other.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By varying the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>, suggest two key features common to these curves.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <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><mo>±</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</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 deduce that the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi><mo> </mo></math>has no local minimum or maximum points.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of this <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mfrac><mrow><mi>p</mi><msqrt><mn>3</mn></msqrt><mo>+</mo><mi>q</mi></mrow><mi>r</mi></mfrac></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the coordinates of the rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> where this tangent intersects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>, expressing each coordinate as a fraction.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>S</mtext><mo>(</mo><mo>-</mo><mn>1</mn><mo> </mo><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> also lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[QS]</mtext></math> intersects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at a further point. Determine the coordinates of this point.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXUAAAFrCAYAAAAuIRkjAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAGunSURBVHhe7Z0FoFTFF8a/13SDNAhIKA2KfxWQUEAalS5FujukkX50t1IKSDeKggiYGDTSKErDI1//5zt37xMREB4vNs5PL3f37t19u3fmfnPmzJkzXpEGRJfIcMcDRVEU5Ynw8nE8eDK8HXtFURTFDVBRVxRFcSNU1BWXI8pfGBmJiIgI2eRpRKQ5ZL3KY0/iWVQUV0VFXXE5IsIjEG42SjmF28vLW0Tcy9tLXg8PCzPHvGRTFE9DRV1xQSwLPDLKSret80iEhobC28fHOqSWuuKBqKgrLogX9vz4AyZPmoRm776Lc+fOieUeFHQNzd9rjp1f7+Qpf7tpFMWDUFFXXA56VQoVLoLKVapg586d+GnPT/D18YaPtzfu3LmDX375xYh8uLpfFI9ERV1xQbzgY0Q8W7ZsyJkzF27cuC6eliRJk6Jly5YoULAAfHx91fuieCQq6opL4m2scj8/P6RJkwaXLl0Wd8u5v87h+PHjKFHiRUQYS11VXfFEVNQVl4NOFUa9cJA0R46nceXKZYQEh+Cjjz5Ew0YN4e/vJ6JvR8Moiiehoq64HvSVGyOcop4gQULcvnMHy5YtRdOm75jnCTScUfFoVNQVF8Ryq/j4+CBVqlT4+aefkClTZqRKnUqOK4ono6KuuB7UdIc1ntoIefny5VGyZEl4e8dMQiRFcWVU1BXnxzHgyR1dLmHhYTLx6OzZs7h16xZatW4NX38/8bUriqejoq44P/SPi2XOnTe+++47mXD0xdYvULtOHRkUpeJHaLSLokDzqSsug5XzJRwvvPA83nvvPTQzm6+Pj8SkU/EZxkg/u6K4JDGUT11FXXF6WEXtaBY+vn79OhIlTmxUPhK+fr4IDwuHl8Na9/bRzqfioqioK4qiuBG68pGiKIpyLyrqiqIoboSKuqIo8QsdwI/qBI6+s9hjUFFXFCXekCE9iVU1jyMeYQlCnmdO4dKFyv3RgVJFUeIdypDH5+vRgVJFUdwBWugUdM5BCL5zx3H0Phjzk6/zPC5dqNwfFXXF5eAkI1p2XMIuJCQEHy9ejF07dyEsLExudm587Uk6oUrcIAItZRmO7du2o2LFijhy5IiUpbzGBcaZGsI8DwkJRtOmTTFo4EDzPNTxCcq9qKgrLgh9sFbaAB9vH0yaNAn9+vUTMafF5+1tJfuiBag4N5wBzAaYyxAONGJ98+YtpE+fXl6LcseYtpnt865du/HFF18iW7bsOnP4IaioKy6HJdiRCAkNRd++fc0N7otkyZKha5fOCA62uudUAm+98Z0eq+GNxJAhQ3Dy5AmMGDkCCRIm/DvjptV+4+qVK+jfvz8KFMiPmjVrWq8p90VFXXE9xICLxNrVa7B3717Uq18P3bt3k6yNixctlu66JeyKs0LXGMuJDrKNGzZg0cKF6NqtG8qWLSvFGxlpuc/E127OGzduHE6fPoUJEycieYrk8hnK/dHoF8XloGCf/eMsKleujLlz52LPT3tQqFAhsdbfevMtrFm7BtmzP20taefowSvOAwUnIixc8vTcvnUbZcqUgZ+fL9auXYsUKVP+7XYx0JceFBSEl196SdaenTNvrhF901Mzou92LhiNflE8FabfHTRoEOobC73488Vp1onVlzdvXjRr1gyDBg4yZ1E6om+vKLEHB7rpGgsNDcWmTZtMD+sP8aenSJVKBkbvhg04xT4o6Dq6dOki76Xmsw4o90evjOJyHD9+DAkSBKBt23Zy03OgTXywRty5YMZTT6XDoYOHROgV5yQiwpSbsdanTZuGJk2aoFz58ggzIu/j+09r1c/PDx99+BE6d+6M/Pnzw9c8t8IZtcF+EOp+UVyeGdNnoHCRwijxQglTox0HFaeFDTEb3MDRo7Fu3Xps/fIL+Pn6itslPCxMrHi6zqhM8z/6EOPHT8D27duRLHmyf7hm3A51vyiK4opQmE+fOo05c+aiTp3a8DECTiGn0MuCJwYK/82bNzB16jRUr15NImK0xX40VNQVRYlTKNg9e/aQwc7adeqKiyUyPEIWOmGoqreXEXnzePny5Th16hTeeKOyDKTyfOW/UVFXFCVO2bB+g0wk6tChI9KlSys+clrpjGphRAw9wlcuX8a4sWPxv//9D4WLFpHxElmLVvlP9CopTgFvZBne4f/y+J97xXVh+YU6pv2fPHkKvXr1RIUKFdC8eXOxzjkDmMsSck8o8CNHjpLZpYOHDLYsecdx5b9RUVecA4duMyqCRHJvjrnzuJinQB+6CI3ZT5s6VeYTDB8+HAkTJ5IQxX9gyvzw4cMy/6BDhw7In78AWwXrJW3cHwkVdcU5cIi3F+OXQ0JMzfQ2VpyXxC27dcSDByB+clOuf509i9WrV8nM0fQZ0iM8NOxfLhXTN8OKlSuQK1cu1G9QX6x3ijknGqn75dHQq6Q4BbzxKeCrVq6UPCBzZ8/GxYsXRdjVQnM9osrM7PmYrpfx48cja9aseOutt8SVQpcLj/N1ns7Zo4xVX7d2nUwsS5kypRy3I2Poc1f+GxV1xSm4eeumTDDZtWuX3MjTp09H2TJl8dlnnznOUFwK07uiYDOtLhvmWbNm4ZNPlqBXr95icdtWN7NssrzDw0KNcEdKTHqSJEnwbrP3pIfGThqlXM5XTX8kVNQVp2DHVzvQsmULDB82HP3698OGjRvh6+uDgQMGiAWnuBhGqa3cLF7YunUrhgweglq1aqFU6VLW6zYi1JHw9vXF0aO/GfGfibp168qMYXW7RQ8VdcUpKFK0CHLmyiU+VYpB8uTJ8eKL/0PoffyuivNju1+Crl1Dv7798PTT2dHn/T7iSrkXWvN0u4wcMcKUtQ+qVa8udcBKB6A8Lnq3KE5BhgwZjHWWQMLXKOJ+fv6m+x6GIkUK/0vUmTNENvpj5ca3RCEi0kq5K0cidOWj+IYJu9atWycpc3v37o00adNKufwDRxnt3bsPO3Z8jUaNGiFVqpQSFWOHOCqPh4q64nSw2810q7/88gs6dur8L3GONPc6/bQ8j0LOV2nhcTBNzjUbj6moxx9sXMNMwztjxnQ0btwYlatURaQ5di8sIQr48GFD5VnTd5rC19c07D6+1ovKY6MJvRSnIzg4GMOGDkPOnDnRoGGDf+XNpuB/+OGH4o6lv/3HH/cgQ8YMyJI5iyX0RtwbNW6EFClS/MvKV+KGGzduok/vXrLe6JKlS5EoUSI5fm9oIuXnw3kfyqpGY8eNlcgY+7g1UOpB1noMJfRSUVecBjtsbcqUKciUKRNq1KwhflV7zVFabqyu4RHhOHb0qLxGi/3TZctMA5BL/PJMDkU//DPPPGMsPl95XYkbmALZ1uCJEyZg2LBhWLV6NV588UU5RjG30iT/HaZ64cIFxwIYJbBg4UIp47vLTEX98VEzRnEKbt++I/7xb7/9FmnSpEG1alXNzc08IHf5xnl/m41inSdvXuR7Np8sjJE2bTrkyJkDefPkxTO5c+PZ556Fn7+fCnp8YER4185dCAwcI4m4ihcvLodt69zHx9vhL+eqVN6yAAbLs0fPnvI6c7/YFrpHCXoMoqKuOAX+RoSDrgfh+++/R526dYwgWzNLb968iT/P/ikWHAVDV7xxTugGowV+PShIXClp0qTGiJEjxd1yb5kxfJFly8Z6y+bNeP3115G/QAHpeSlPjt4hilPw7TffoFbNmti0aSNqv10b9evVQ8MGDVH5jTcQFh5mWelGBP4VPaE4BRTvO8HB6NC+A86fP4cpU6cidapUjlf/KdaS78UI/fJPP8XBg4ekEZBOmGm0n8QbrFioT11xCvb+uhfXr1+3ljOjeLNamhs/YYIAFChYMKr7fj905aP4QcZAjBCLO8WI+tIlS9CxY0fMmj0bVatWdZz1bzjmcevWLZR5tTRq166NHj16Guud5W5elHK3zvM4dKBUcScsEecwqQUFw3a58IGKuvPBsEWWC8uO7peyZcqgZMmSGDR4MPz9/R1n/RsOlnKm8PTp0ySv+jO5nzFHaaVbZe6xYyE6UKq4FQ5BZ1IvTiySQTLrf8WJsVf/nztnLq5dC0LnLl1kAtnD+PPsWXzyycdipXOAW0qZjfoT2JfK36ioK04BlzOTNLuO6AeZGWpuck4uepiVrsQfLCda1b/++qvkR3/33XeRKlUqiWR6GGPGjMGtW7fRvEUL88wh6MZE54CqRiw9OXq3KE4BQ9kknM3xmGFuHHy7d+KR4jzQRXb06FG0b9dOJoo1adpEyo3JuO6Gks1GmtsP3/+AZcuWSdqA5557znK5sNAdPTPlyVFRVxQlWtBN1r5de1y8eAmLFy+WGbyWB+Wf6iwNNePSzX7u3DkoWLAA3m32rpyrjXbMo6KuKEq0WLNmNX799RcJSUz3VDrH0fsb3HStnT5zBp999jn69u2HgAQJxDhXN3rMo6KuKMojQxcK3S7cmH+nUqVKeLt2bcsSp0rTN+79d7w5/5XzIyMwZ/YcWabupZdegrc515qYJKcpMYiKuqIoj4Qt1OFhYeIT57yCsWPH8hVrkNsotCXsFpIW2fGeLZu3YM6cOWjRsoWY8jzv74ZAiUlU1BVFeSQo6hThOXPmYsH8BejVqxdSpEzpiE6yxNuG5/EIhZ3iP2jQQOTP/xyqVKlinaDEGirqiqI8Epw0tG/fPgQGjpasiuVfe11yposFf09+FzlmtvDICIwfNx6nTp2SRkCJfVTUFUW5L/SFc6YoBZpLzoWEBMvSdIwnHzV6lCRh4zqy9I3fu0oRI2Ngjl2+dElCGCtXroJSpUubc30dZyixhYq6oij3hS4UCjbFnVb3+HHjsGvXLlkYPGeOnI6z7o+Pr68kX6Mv/a+//kSHjh3wdxIIJTZRUVcU5b6Ihe5Ih7t9+3bMnj0bHTp0QL169eTYw2C0y80bNzF9+nQMGfIB8ufPb3ndzWcqsYuKuqIo/4TCaza6WTjjkwtIDxs6FJUqvYG+/eh+sdID3AstejYE0hiER2DKlMnyfk404nvERaPGeqyjoq4oShQShkiMCEt8udnmzZ0rA50DBw2UOHWmAqDo340MjBoo3jTJz5w5LVEyDRo0lIyNlt/d2xEpo8QmeoUVRYnCEmea01Yc+eFDhzFq1Gi079ABqVOlFnEOM5a7eVHOt2FjIFY69+a/CeMnyCBrterV7tV/JZZRUVcUJQquC0srnW6Tw4cOokmTJnjuuWfRqGFDR9x5hCxHxwlI/4CeFb7PPFyxfDmWL/9UJihxAXEa70rcoaKuKEoUFPOwsFAEXQtCs2bvyfPpM2YgWfLkEr4o2TSNSosL5i5o1TOO/dy5vzBgwAD873//wzvvviOviUtGiTNU1BVF+RsKtp8fFi1ahAsXLsji0RkyZJDjD8Ny20Ri2NBhuHr1Krp37wEfb+Z2odVvnaPEDSrqiqL8TUQkrl65Ism6atWqhQoVKlhhjf+hzBTvs2fPYt26tahbtx4KFS4k65ZyoFUN9bhFRV1RlL/x9sKQwUPE1TJg4ECESbSLZXE/DFrqzKnOgdR27dqZ9/gZQXekEFDiFBV1RfFg7FS69Iffvn0bQ4cMwaefLsPgwYNlBSNfI9IU9PuJOt9jvTcc586dw5JPlmDsuHHInCWLEXdvEfh7fe9K7KOirigeDIWX1rTEo8+bJzNAOWu0bLly/2md82VrghLQo3t3ZDFizglK//U+JXZRUVcUD4eRK8eOHcXwYcNRqlQptGnbVmLM/wsRb/P/1zu+wvbtX6Fx48YOoXecoMQLKuqK4mHYfm5a53wcGhKKqVOnImXKlBg7dhwSJkwobpOo2aUPgO+9dfOmTE5KnTq1se7L6oxRJ0BLQFE8DIp5cHCw7BnZsmDBfGzftk1S5GbImEHOoTjfL7/L3fC9YwIDcfDgQYwcNRIpUqQU612FPX7Rq68oHgbFnLHoZMH8+ejXrx969OiBp3PmkGOPyuFDh8TCb9iwIcqXLy+CzkFTjXiJX1TUFcXDoCUdaiz1b3bvxvDhw1GkSBHUrFXrsRMoLly4EFmzZkPHTh2tFAKRETLwaqfrVeIHFXVFcXPs6BbmbaERTYv6ypWr6Nmzl/jPJ02aZPaJ4H3PknT3ws9gGl6uahQUdB0bNmxAt25dkZJuF/O5FHR+NuPalfhDRV1R3B7LBvfypthGIvhOsEwQOnPmDMaOHYunc+YUtwnzujwMirmfv780EhPGjzcNQQJUqVrVSgKmOA1aGori9kQaWaewR8qEIQo5BzdnzZ4lIYyMcuFkof/Cx1jgzM64c+dOSSPQpUsXsfRpwT+270aJNVTUFcXdifzbBbNh/TqZZDRj5gxUqFgR3sy86G3NGL13gJPuGhu+xNcvXriINm3a4JlnnkGNmjXFHcMGge9XnAMVdUVxc6xlKyKxy1jY3bp1Q/Xq1SU1Lq1uptG1uTcUkW4VLktHtwtPo7DPnDXTWPth6Nq1i6xoxE0F3blQUVcUD+D4sePo2LEjkiVLhh49e4pQ21kUH4RluUfCx89XHh/Yvw+zZ83C888/j3Lly1snKU6HirqiuDnHjh1D3bp1kCpVasmTnjp1KvgH+FOv/xMR/vBw3Lx1C126dMXTTz+NMWPGqnXuxKioK4qbQQObFjgjWm4ZMW7RvDnSpk2HpUuXIHfuPJICgKJMXX7Y7E9a58wBQ9/6ogULJD/M6NGB5rPSmldV1J0VFXVFcTMo1hT00NAwdGjfXmLKOTCaPEWKh7pb7oWCzwaAOdVXrFiJ2rXroPjzxcWnzk1xTlTUFcXNsKNYBvTvLxOEunfvJmlxKeiP4HGJgp9D18uWzZtx4sRxdOrUEcHBIZJiQN0vzouKuqK4GRTjwNGBmD//I9SrVx+13nxTBJ0zPv8jR9e/CLp+HYMHD0GtWm8ibbp0Er4ojYOj4VCcDxV1RXEDGC9Osb129SratmmDjz76CH379sPwEcPhZyxr24/Ogc+HIQm5zJ6fx0yOvXr2wrVr19CiRXP5DH6W7ZZRnBMVdUVxAyiyISGhEoe+Y8cOWcGoZetWIsCP6yqJNMLOPDBLlyzF6tWrMGTIYGTJls3xquLsqKgrihtw1VjoYwJHY9u27Zg6bRpeLvmK+MN9fHwfa3CUGRZpzdM6/+CDD1C8+POoVr26JOxSXAMVdUVxJRziKoOYRqy50VUyauQoybZIS52zRX2Mhc7ZnqGhIQ8NW/w3kQg3n7lixXJj+Ydg4KCBSJAggSbtciG0pBTFRZDBSS+zd1jefM6JRdWqVsOaNasxfPgINGnaxBoQdYgwhf1xoGV/6cJFzJo1Gz179kTBggXNUbpv1FR3FVTUFcVFoG9cdN0INq3pUydPommTpmJRr1q1Co0dgv447pZ7Yfz5kCFDkD59erRs1VIaBy5+YWV5VFwBFXVFcRGsMEKmzw3Hrz//LOGKFPGlS5ciZ65c4kP39qaVHv1FKrZ9uQ3Lly9HnTp1RND5WZTz/1qvVHEeVNQVxUnhoCUFnHta30zCFWa2tWvXokGDBrhx4wamTZ+GVKlTiQAz3JBx5Eyl+6iwoeDf4BYUFITBgwfJRKWq1arK6wycYc9AJxu5DirqiuKkUHDv3qirK5Z/irZt2iJDhgwi7nny5JH8LNGFjYUItukEzJo5E4cPH0H//v2RMEHCf+RTV1wHFXVFcVosC50cP34Cffv2xaBBg9Gvfz+sXLUaWWMgdpyCzr+xd+9ejBs3Hu3atUX58uWlAdGhUddERV1RnATbIpdZnWKZe4sVzhzmr5Uvh40bN+Kjjz5EixYtkCxZUvGnc6Pb5XGwGwrLB29JwIgRw1G4cCF07dYNAQkCROzV5eKaqKgripNgiSjF1FsiWs6fP493mjY1Fvr7eOGFElj26aco/sILUWLLXXSFlw0HP4Drk/7888+yjRg5EokTJ5bXOQFJRd01UVFXFCeB1jnzrdCSPnXqFN5+60388MMPGDBwoCz0nCvXM4gMN1Y2wxqfUHBlpqn5rLCwUPTv1980Gi8gb968jlcVV0ZFXVGcBIrs7l270LpVK1SpXAVFihTBl9u2oZV5niBhAnNGpESicL3RJ8GOpuHn9OvXD4cPHxK3CxsVxfVRUVeUOOJuf7m9iWVujl2/fh2tW7dCo0aNjJV+Eh9+9CHGjB2LrFmzSrIu+r5pndNAt/3g0UU+y9sLixctwrx589CuXTuZOfqkn6s4B1qKihJHWC4TLxn85Er+3Ljc3OzZs1HylVewc+cu8Z+vWLlSFneOrfS2ERHh+P30aYwfPwHFixdH6zZtowZPFddHRV1R4giKOScG3b59Gx9//Ilsr5V/DQMGDMRTTz2FTz75RCJbEiVKFCN+8wfBHsLIkaPw119/ifuF38nqCagcuANaiooSR3AyDwc+69SujV69esqWMmVKzPtwHlauWoVnn3sWXpyWb8Tc24c5V2LHx82Y9JUrV6Bjx454oUSJKLdLLLUhShzjZSpO9GtOZLjjgaI8GTJwZ2oiq6NMczcKQ43hc8uX/GDFmTF9BgoXKYwSL5SgdyNO4feTG8jsOdDp7esjjzml38fPFyF3grF3314sMVb4N998iwsXLqBKlcpo0KChvD9//vzwC/D/O195LHx/fkd+H342ffqNGzeW1LzTpk9H0qRJo0RdiWe8op+z525U1BWngBEZxpTlKJ4MHIqIS9X0gg+F8iHEt6jL9xX3Bf+4NUPz5MkTWDB/Ab788kscO3YU+fLlw7vNmqFihQpIljy5vI/4+nLiEH+7vDVWkO/IiBfzPRctXoRxY8dh2/Zt4uYJCAhwnKXEOzEk6tpEK06BpYdGEI21e+TIb9i4YYNoOi3fJ7E74gIKJr/h1StXsfXzz9GieXOUK1sWH344DylSJMeEiROxafNm1KpZC8mSJRMBt3sfTGvL3xybjRGvn/lLOHX6NIYNHYaaNWsgRcqUjr/v3NdWeXxU1BWngNYtxXH9urWoWqUyPv30U8k4CFrwMcS9ER73JqyiwNkid6/YyQzMu/b262fPnjWCvQnt2rRFiRIvoGHDhvjll5/RpEkTfPbZ51i1ejVq1KghLg7/AH/T67DDE+3NGqQkMSWw934Or2u42QYNHGj+lpf0GMxJjvQCcdy1UWIddb8oToHlU7fitqtXq4Zs2bJh1uzZpo6ZSkof+0N4VPcLP5/T4u3P49+0XCbmbVGRH38Lu/1aBL/XnTu4cP4Cjp84jv379uPAgQOyXbhwHhkyZMQzz+RCkaJFUdRsefPkRcJEicz76U2KH7tJGjD+DPMdGHUzcsRIzJs3F4MHD0Z90/CwsZTfx/8dv1OJZ9SnrrgTYaFhIi6M+qj99ttInjx5LIi6+ce8zqn2snpQuONvevtEPQ4ODpG8KxcvXMCvv/6K3bt34/Dhwzh48ICs1k/SpUsnVvnz5u9VqlgRyVMkj0qqRUuefnJOv/f18f3P7x4b0J3Dv2vf2szrUvmNN9ChQ0f07NVTLoSXjzWgS1TUnQQVdcWdEMvSiAvlhSF/9D3HiqhHmh6B+TuXjGj/cfYsDh06hOPHjuHSpUs4c+YMTp8+LYtPXLhw0TGT00siVLiqfqFChfDcc88hZ84c8A8IcESUWLM8rc+n79o8Mf/TQpfn8SCY9i0dbsSdjVWb1q1x8uRJrFm7VtYstRsbidD5j0FoJQ5RUVecDasqWYN/lpvDjgixqhhf5ybraBrBYSZAOddxnG+npf6mDCgmRb9+/UVgz58/h6tXr+LmzVvGkr6DO3eC5fNoUd+8eVMs6jRpUiNz5ixiITNZ1fXrQbh27ZpYzjdu3JSZm/ysoKBr5vMu8NvIZ4i/2ZyTPHky5MiRQ9w+uXLlQsaMGZEv37PImy9vVOZCV0HcSo5GZZ+5NrVqvSkTmwoULCCirpa5k6Kirjgbd1cl+qgtoTbHzGZPpgkNDYMv47eDg3H58mWcPHHSWMincfKU2U6ckPC/gwcPmXNpZVLw7cFMK1qEepQiRQp5Tks6QYIEsgwbHwcE+Isrhe+hsPv7c3k3Hg9A0qRJjHCnEPFOnDgJkiRJgqdzPI1nnnkG2bJmlc+k71x84OYrW+4LfnXL7x5fvvHoYF/zm6Yhq1mjJtKlS4sFCxfKcW6xlX5AeUJU1BVnwx6EpOVLUef+0sWL+PPPP3Hq9CkcOnhQrGpGjJw7d86IbgASJkyIDBnSi6gmTZoMTz+dHQsWLECmTJnQvUcPBJhzOOsyYcIEci4tTcvCNwJs/gYbi3lz56JgwUIoXKSIESzrxpAFJOgrjwgXd4ORM/lOdL9IA2G+p+17pmjbg6bc5Hdwz//M3zFH5TxXgd/5zu076NqtKzZv2ox169ZJj4PlwWvnbX6b4oSoqCtxCmuJQwskFNAhDCKujq4+uX3rtrG6T2L7tm1YuXKVRIhQTOjCyGeEpVixYij+/PMokD8/0qZLB38uliyWI4XVqk/16taTgVIuqmxbyHdbyhRg+zn/6oxp0yTyhD5163uYzfH9KM70Hcv5ju/PY+Tuqh91zOG6sCYUccq+HHZ6ZEzCIL/JbGPGjMWkSZMQGDgab9euLb+Pr3Fjg6c4ISrqSlxhCx21MsJYuhRIscb5orF+6ce+eOEiJk+eJLHZDPNjaGLevPlQu/bbKFy4CLJnz25EPK0Iii2gd0NRkqpoXnurVi1juafEnLlz5Bj/3v3eYxOfM0qdhTDTcIlgm57K999+h7p166JkyZKYNWe26e34O85SnBoVdSXOoKCaakK3BEXTmkEZid/PnJHu/fr167Fnzx7xab/yyit49dUyeKHEC8idO7c12ca833LJWOGD9FPzM2RykQNpOIzgMx1t1SpVkCpVakl0xfebSmo1Kg9ARV2KR+LROVZRuXIVXL16Bes3rEe6dE+JS+phjaLiJKioK3EFa4hVTSLFH77ty23YtGkjdu7cKcLNnNyFCxc2Yv4q0qRNJwJN0be6/NZ76fs2Si7ibPve7xYaij6348ePY+2atWK5ly1bBs8++ywSJU78UFFSUbeuMa/ZxAkTMWPGdMydNw8lSpjrYfivno7iJKioK7EFxTXKZy0C7I3fjhzBqFGjRMw50ebNN99Ex46dkCVrFjkvPvFUUbf96CwrPj596jQqVqyIcePGoUKFCo6MkeYE3uKq6c5PDIn6g/u0ikcTGhoq1h8n5UyfNhVvvPEGdu3aJWlbvzSW+vARI5AxUybH2UpcI+4saXitAVAOUHfo0B558uQxgv661TOSXpLVa1I8BxV15V/Ylt/cOXPw+uuvY/DgIahevRq2fLYFg4cMQeYsmR1descblDiHA850c3HGKCV7/vz5+O6779C5c2ej85bYi4X+BB1xxTVRUVf+xf79+1G9WnW8/35fZMuWFXPnzhXLnLHjjNnmYKf4aFUv4hU2vIw7/+6bbzHClE+ZMmVR4sUSjrJh4XBcQ/3pnoaKuofC7jtFIczhZmFIHHOfDB40WHKv5MjxNJYuXSqLIFd6o1LU9HKGzNlrWkoXX4kXRNBNGTACqVOnTihQID8mTZksOWmktaWOU8xVzz0OvSs9FUfP3NvXV3KozJkzB6VLlcKiRYswdtw4TJk6DaVeLW3E356mrzgTbIjpUx83dqxklpxtyi9VipSi4epx8WxU1D0Vc/fT2qMCjAkMxJDBg1GoUGFs2bIF5cuVF18tZ1VyrzgfTJGwb+9erFq1CqNHj0KqVKmiVolSb4tno6LuITA00bbg6HqhqlOwe/fqhenTZ8gA2ydLPkHWbFnFrWLnB9c1LJ0HCjbz1Ui0y+1b6NC+PV4tUwZlypYR9xgLmC4y9aF7NirqHgL1nOFtnHVIUfhm92689eabWL9+A6ZNm4ZOnTpK9j4JlXOIwoOm9CvxA2fhMk0DUwq/9957MhGsd+/ejlct6GdXPButAR6CiDMtPSMMWzZvwdtvvy2x6Js3b0bFShXhYyxzWoESKmfOU5wPDlCzjNi72rZtGyZPmYJcuZ5xvKooFirqHoIY3Oaf48ePoVevnihXriyWffopMmTMIL718NAwiWyRgVHVdKeEeXF27d6FFStWok2btpJjh70vhpgqio2KuhsjPlgj2NzT7XLq1Ck0bdJUFkqeOXOW5C/ngJufL9PfMlTR20r+5EK5w90Zlp2Un2NP8Z4yeQqef/55dOveDX5+vuIyUw+Zcjcq6m6OLewXLlxA40aNZD9+/HgkTJTQEm+x4K1zFeeCLjMuyM0Zouw8cdYoFxmZPGWyiLmi3A8VdTeGYs4ZhXt/3Yu333pLBH7ZsmXIkzePeWwEX6JgFGeFQs6Gl1FKSz75BB98MATdunVF1ixZZexDUe6HirqbISFvRswJu+snThxH3bp1kCBBQiw1gl6ocCF5jV12Wd7NoAOjzoeEoIrLxRtfbd+Ovu/3ldTGTZu+Y8Se4alaZsr9UVF3IxjNAnGpWGtuMvxtzuzZIt7ssqdPn0FEIipM0d6pU9ZpkHIzDTMtdJbV2T/+QIcOHfDUU+kwdOgwGftgeWmJKQ9CRd2N8PX1s6x0c9PT0vvx+x8wf/4CjBkzBnly53G4Y/iaTv13Rlg+EmduFJvlF3wnGN27d8fNm7cwefIUpEuXVs5jI62WuvIgVNTdCutGZ+jbkd+OoFWrlnjnnaZ47fXXpMtOweDUf4q+4nxIg2v2MrhttqnTpuLHH/dgtultWW4zq9wYxsjBU0W5H1oz3AiKAYXhelAQ3n3nXZk63s1YekzPSkGX7IqOx4rzYfeiWEZf79iB8ePGy9J0pUqXtkIXHS4ZuywV5X7o3e1G8GbnNP9xRgxOnDghi1skTZqUai/HFedG0gAYUT9lyq5Hj+6oVKkSXi1TVuxz9bYoj4qKupvBdSpnzpyBt99+C2XLlRMxoOWnIXCuQVDQdUmudvHiRXTq3ElcLZFG1TVbpvKoqKi7EbTGly//FGnSpEGv3r0drnOj6kbZOfCmODuRGDsmELt27caAgQORO3duKVO6ZOiaUZRHQUXdxWF3nbMOefP/9eefWLRoMUaOHIWMGTOKdW5nWtSp/84Jyy04OBjhphwXL1qEBQsWmPIbgfr160vZcVyEm/a0lEdFRd0NkNhl89/IkSPFunvttddE7BXXgLnrD+zfjyFDhmDQ4MGoW7ceOHFMwxaV6KCi7vJYFvi+/fvwySdLUKVKFfj6+6mouwjsRZ07dw6tW7WWlacaNmwo5RcWFioNtaI8LirqLgwtucgIawHpyZMmy+QUxqTTh67pWJ0TlpVY4DTCzcbsmd27dcOff/6J999/33KZeXuLy4UzghXlcVFRd3GoDd99+52sLdqgQUOkT58ezImusejOhx2yyDQAly5dlK1J4yb46aefEBgYiAIFC1jjH45BUS1DJTporXFprBwvI0aMQOrUqdC1W1c5xjFRmXWoOBmR8DJCffvObWmAuX333XeYOHESqlatqhEuSoygou7CsBt/4MAB/PzzT+KL9fHx5UH4+PpaXXzFqaDlHRYSikULF2L//n2ycXCUC0czOknLTIkJVNRdDcd9z648n6xYvgKhoWGoVq2asfRMgfpYRaohcM6BuFvMFhZmjX38/sfvYpm3atVatvoN6ovYs7zU3aLEBFqLFEVR3AgVdReD9jkzLbK7fuvWLSxdugRly5ZFzmeeibLiFeeDvagrl6+gVctWyJHjaUkFwE1RYhoVdRdDBN101SPDI/DDDz8YYb+NNm3bIDw0NMr1ojgPnOJPX/nRo0dRr149XLlyGaNGjYafv59sihLTqAq4GMyLHh4WjrDwMHz5xZd4xljoL7xQQqaZS650JX4xAk7fOcc8KeZsgC+cv4A6tWvj5s0bWLRoEXI9k0sGs7kpSkyjou5iyOQUY5EzvzYt9YqVKoK5tTnVXEPi4h8ubkFR50IlDC9lXh6mb7h69SrGjhuHPHnzShmKBW82RYlpVNRdEFrqXAjj8OHDqFu3blQoHGOglfiFE78k4oV9qohwrF+3DmvXrsGMGTNQtEhRK+uiKS/O+NVZv0psoCrgYjCFLrvtO3fuRObMmWQGqRUyp/m2nQFfXyPWXt6IMA3v+PHj0alTJ3Tt2g3lX3tdEq8xbJE9KsmTrhPElFhARd3FkOh0IwarVq1GwYKFRCQ46YjuGCXuodXNjb0neWwaXe6HDv0AgYFj8N5776FFyxY8M0rQubc3RYlptFa5GHSbh4SE4IsvvsCzzz4r/lmjJOpPjyco4CwUCjQfMgJp06ZNmD17DmrWrImevXrJOSrgSlyhNc0F+fXXvbhx4wbyGVGnhcjl6sRKpKoocYqIOQc9zbWPiAjHX3/9hQEDBuCNN97AmLFj4B/gL+4yjUxS4goVdReD4vHD99+J7zZfvnxGVCwLneKu1nrcYzemHAC9fv06mjZpglSpUiFwTKBEJLFczAlRDa+ixDYq6i4GhZuZ/XLmzIkUKZJHRbz4+qlPPa7goLQVi24JupeXt6TPrVK5shH2G5g4aSKSJEkiZcXwU+7tTVFiGxV1F4O5uH/66WdZJYeWoBL3cFCaom5UWp5v/fwz1K5dG4kTJ5bJRXny5InysStKXKOi7mKcOHkC58+fE+FQ4ofQ0FDTM6JrJQwH9h9AmzZtkSVLFsydOxdZsmaVcxgFo5a5Eh+oqLsAVlff4U//4QdjBfqgxIsvWtaiEifIpCGHy4VWOJ9T3Hv27CkLlMydNxcZM2emmsv5dIeppivxgYq6k2P5bI06OPa7d+1G0qRJ8eyz+WTwTYl9rMbTS1IA2H70/Xv3oUb16iLcS5ctkzEOWu4sEzuXvVrqSnygou7kiKjLYKiVU+S3335Dvnx5kTBRIhF6JfaxG1U71nz7tu146603ce1aEGbOnIXMmbOIFe/n7y97RYlPVNSdHLH1KN5GWIKDg3HhwgWxCjlgKmaiEif4+DHWPAJ7ftyDFi2aI1OmTJg//yOkeyqdaWzDxSVGQdf8O0p8ozXQybFFgsvXUdSvXw+SQTnJG6KGeqxhjWNwIMM84c48vn7jOjp06ICsWbNi0eKPkTNXLrHeGQ3DRUto0avLRYlvVNSdHGtwzhSUEY1r164hKCjIdPczi2Woqh57GHmWfYRpPCPNf+f++gs1a9SQGPUFCxciVcqU4j/XwWrF2VBRd3po/VmW46VLl8USzJI1m2VFKrGHueYMSwwODsG6detQpXIVaUgXLFyADBkzwj9BABVfLHVFcSa0Rjo7Rly4qhHFnNYi3TBZs2RxvKjEFFHuFj52jFeER4Sj7/t90Py991Cy5CtYsXKFrDRFG17seHOObdErysNg3WK9Yg4geXxXfWMO/pg00lTUnRyKuURUmDK/fOUyAgICkDRpEtP116KLWZjj3HFjUafNw3Fjx2Hp0qVo0aIl+vfvb657MikPWufcm//Fl64o/wXnNcg9ayrN7Vu3cfPmTYSGhMhchwsXzsvzmEKVwQVg3DNF5OqVq7IoRsKEicQ1oMQcFGjL3RIs8egffTgPkyZNQvv2HdB/4AAkT5FCbXIl2jBT582bt7B40SKULl0a8+bOk2NzZs9G8WLF8OmyTx1nPjkq6k6OiLeYhMDFixeRPXt2eEmSKMcJSoxAK51WN+cB1K9bDx98MBSjRo1Cl65d4W0uNqOQOGCqKNGBbpdEiRKirqlb06ZPw9atn2P9uvUoUKAgli9fgdKvlnac+eSoqDs50r03gkNx//PPP61wRvrgHH52JfpItkVzbbk4NH2cn3/2ucwS/f33M1i6bCnerv22CLr0lMz5tttFUR4XGg224VC4UGFZiDxlypQoWqwoXijxArJlzeY488lRUXd2bDev0ZKQkGBTEVJI5eDgaZQPWIkWDEkMvn1bfJ1fbd+Otm3boFix4li5ahUKFiwo0S4hoSGOsxUl+jA9MwdEZd6JuZczZMiI0LBQK3qKPcEYHCNTUXdyorr8puBv376DVKlTS2uvFuOTExIcLNd1/LjxaNy4MUqVKoV58+Yhdeo0koWRVzhBgoRixSvKk2D18mCMiDs4f/48bty4jh++/x7+AQFSv8JCQx1nPjkq6i4E1yZNljSpVBBp4ZUn4uaNm3ivWTOMGjUS5cqVw/gJE4yIJzA9oAgRfDan9IVqA6o8OVYYo4+vDzZt3Ijatetg9+7dCA0JxZ4ffpReY0wRY8rAL8zwHHYxwriEVzShR4HvZzx2mLmhPB0KCv3pXBaNos4MjYK5Tup+eTjix+S1Y3ywqZ+8ftzfvnMHG9ZvQLVq1XDlyhUsXLQIs2bPlmtLVwwXH/H39zcNp5c8dzVR52/k/SPpgs01sJ7/vXk6Ui/MrcPrI3MSHgP72lqf8Xe8OcX5Yezftx9NGjfB/I/mo2LFiij+fHEcOXIE06dNQ/oM6eETg0ZajH2SPYgUHh4mFyu68P5hgiReKB8vtUYJrwnzc/O6UmwoVPTNqQX5cHjjsvWjdWR5sbxw8MBBVK1cBe+91wyvvf4aVq1ejTKvlolKl+sOsF5w4z0pwm7qi/bs7kIqhkOUHfnvHxW+x7rvzOa4zhxwlzr2EPLmy4ux48aKm++p9OmRO3du7Nq1G63btJYw5ZgkRi11/tDTp07j2292WwejASsgV8o3D6z70NMxF8HSJl6X6wgQ94CjUpprpTwYWtq8QsxzTut8QP/+qFKlMk6cOI4xY8agT58+xkLykRvb3RpIGka8J9lYXbl6RXy2/I1qqRuVkgbOSxYKf9xy5/k0XMm2L77A8ePHefQ/P4cx6enSprPcLOb+5T2cJGkSSQYX0w3uE39aVCUxX/KzLZtRt25d/PbbUetYdDAXJ3B0IK+8Q7w8HEdd4ZWge8vfP8A6YIjJEXNXhveTZXXZ3WHrOF1WvIF/+fln1K1TB/Pnz8drr72OL778ErXNc19fP7nZ/Pz8H/vmdmZ4LXgNGPbKWYuBo0cbMfGVY4zC8HTEk2AuxuRJk6wb6zEw8i0izHGXCxcuovbbb2PD+vVS73h/2pp1b+PJsqA1z4XIKeysb96mLOjKiele4hOVML8Qb6SdO3eic+fO+Oyzz7F6zWq88+67YkVGZ5OaJ7Cbw4tx//M8ZbMG6lj4tLpMq24uCisOK4XkkbjPezxp4w3KMRhLlK3rQkuKFtSsWbNQo1o1vPXWW8iaNQs2b96MGTNnIFPGTOZc3p7WzWanYbjf57viJkJl6oiPLExuWef8jey5MHLqfu/xpI1iah5Zwsv76T7nPGgjrDn8780338SGjRuxa9cudGjfAd988400ovdzP9vWOOsnNztdM0U+pnmiT7x48QLeadLEdGsHiIUeaLq09A/xZpGwu2hsfLN1f/IC3v8cz9qsVpwVgb47U7WkpefVYeW8/3s8Z7Otpj9+/wM//vgjFi5YIAm4SpUsJdP8c+TIga++2oEJEycid57cDv+ntX7o/T7PPTZvsQo5Sc2az2DqCp/zgRGm+7/HczbeOxRoEVjz7H7nPHBz6JO9pU2XDsNHDMc77zTF0A8+QNOm7+CvP5l4L/7cXF6moPkbo0Wdt9/Ejh1fo1ixokiSJKlEFzBygDdZdD+WF/rQoUPIl+9ZuWj3a/U8C3MRTMWjlU5LgFkCkydPJteFg6bxWXmcAy8cPnxIcs0zTS7rna8RMF6f5557Tmbt8dqFhoaY+uQt1yzMMenDfa+dVWd4C/K3njhxErlzPyPXhsdosSvAsWPHkTNnDsezx4fX0zK2wo3u+YrB8PXXX+Oll16WjJ587bHwihk3zBOJ+qqVy9G1S1fUqFED7zV/D77m5qEVxO4vrcjowAszefJkWWEmpgcQXBPrRiT16tVD7969JF8E/XAUpceuOG5HpKwTajf+33yzG0eO/GbE2w+FChWSepQrVy5xE7Kqs14yTFGunYif+8GBX4bIWW4pIDBwDLp27SJ1hnXJ4+uM+fl0R02dOhVt2rR9zOthuW2oc3YjSSOWnzB37jwsWrRIcgbVqVvHOv1xcAZRDw8LwW9HjqBPn/flh06YMF7WbuSgDH2/0cFcJvTp3RsfDB0m3R1vN73xHhWrcCxBeuXllzF6dCBeLvlKVHxtdK+zO0EjgDk0SpQoIReM/vQZM2Zg7dq1YsFzkehOnbsgW7ZsYqXyJmZ9pRHijrDOiNCYBoypJd7v0wejAwPlN8sgXbTveDfB3DKcA/PB4MEYMHCgefro9xAbTBqbNCLssa4zZ06jR/ceMlDKFM2FixRhAYjx8Fg4g6iHmS4tQ6doAWzd+gXmf/SR6Xr8DzVq1sRTTz3lOOvx4A3X9/2+GDp8mAwEejqWFWH5817+3/8wZuxY/M/sZXDPIU6ejRdmzZyJIkUKo3jx5+V60adMUbt06RJWrVqFHV99hZ9++glvvvkWatepjTx58sgNyXPcEZnDwAfmzg41Pd++77+PESNHOK6LMRA8vM7IdTDXgD7wvv36yTV5ZFi/zEZjir7zlStXYNu27WjW7F2UfvVVSS9hCbqVBO6xcAZRj4yw4jUJfygTvU+cMFFcKP3693O88njw66xdswZVqlSxKqf5XE+GbgN7cKtUyZIYOnQoXilZSixOsbzMNfJkeA0o6oULF5Fsd4SDYLSSRMDMxnq0ZdNmvP9+H/z++x+oX78eevbshXRPpZPXWF8ZjWBHGrl6lZPfzbphhCXEsRxfrTdrOX6f1eh5MrSyeW02rl+P115/XcYBHxXbiKJVPn3adFy4cAH9BwwwRyLh5+vHPnWUZj22djmDqCPyb0ualcfX3/woc0NxcIYr9EQXTrn1k88yXUgPr4CEvlHOKK1UoaKEjpYrVxY+pgKxIinAjOkzTJe3MEq8UMLUaMdBA0XM2yHWbBzPnzuHSRMn4sMPPzI9yXSyf+65Z3kbyg0odc6cS1y93tE9R0OALRQnXzHjJDWGjZan1xsJTaRRxPphLsrjlLUll1YlCw0JNnXGClagcfXEBmgMiXqM1Vz/AH+Eh3KA1EsiDKILKyMFTKwNxzFPRiqR2aQCmopDXyDDHPX6/Ddys8q1i5CUE3QJDh4yRJaoY7RWvXp18dWOHdLr4f0ogm4euPp1ZZWRnooRrwhjEHDWrDkqgu6uLqfHgY0dy9qU+mMJOpGejuOxnzFceR9a19d5iDFR54+jGHMggQmUogsvsiVk5nHU5fNcpBKxEpprkiBBAG5cD5JjAq+T8kB4nex6yaokN6Dpdr/4v/9h4+ZNePfdZmj27rto0bw59u3bR9mTeicnuzRWhAbrjeQkMT+H14KGgfX7PBvWA+oMr4/tTnlUeMtZl9AYVeZ60mCQ/DFOdF1jTNRFfMzGm4Zxwo8DfZqSldHsjx09iqlTpmDmzJn4/fffpdtM9wNf80Rk6rHZ8/ffuXMHly9flpuUXWpp/NwE3lysA1zdieMy06ZOwx9/nJU1Q5/kd9oNIPd2ng3GFCdNkgQdO3XE4sUf49ixY6herRqWfPKx1Df+vai9uXFd8TpLhIa5b3hdf/3lV5w6ddr08sI8NvOp6Eio+f0OHWEu87FjxmKZ6bXduH4DwSEh8tp/iTyrk7V5iV+eBoO4XkwD4SzEmKhHG3OxGVPLKBqmG+DU2yVLlmDgwIGoUKECdu/cZa6g41wPhILy3TffoG6dujJjUkTdHGNlsgXLHeDv5AAWk24xNelrr72Gtm1aS3pS/t7YgCJf4sUS+HT5cpQsWQoDBgzA2rVrzJ+z4o55g4tP/jGtufiG15IGwNVrV7Fm9RrUqlULp0+dtMTHU+8lYzBwSj6r0pjAMWjatClWrVqJLl26SpK3y5cuQdIHuIGdFO+izmvIjYOjH86bh1WrV2HTpk1YvmK5WGnt27eL8ifTkvM0ODKfO09eVK9e3dyQ3rhy5aqInx2n7i6wO/zJxx/LDNBcz+RCtuzZpFHv2aMHQkxvJTag+LFh5N+cPWcOKleuYhqSNrIS0u3bt8UkYy4P1j1Xgt/X39SbRAkToWzZssiQIYO4RCV3kIv9lphCXC2mvA/s349Lly5i+/avsHnzFnTr3h0HDx6S8EYOnrpDoxf/om4uNLuJdLV06twZWbNkRYKECfF88efFaj979k+JN7aC/R1v8iB4fVKlToWsWbOKAF28eNHsTbG52bW4fesWAgNHo2jRYvL7aEUz7vfnn39xpDeNeXhtCWcHBgT4Y9CgQWLVjh49Ch07dJTvxHNczVLn9+UvS5AoIZIlT4bUqVOLS5TH3al39zhIxIuB6XZ79OyF5CmSIyBBANq0bo0iRYpgy5YtjjEH1zccY03UeTPwQsr+IRuhH545GLjYL9PJ8qbmAE/+AvmRKlUqJE+eXCxWdxo4lcgEc5NZfrx/XhO5KXnMcS5nAFqDpRGy0j3jM0TY3Yg9P+6RBit79uzyOyk+bOAZScVlv3hdZAWjGOyh2D521iqGzyZNmgTjxo/HtGnTsG/fXjRt0gRXTc+I4YAsE8lKavbODq8d7yn5XVKJ+L0t94OnwmggXpNXSr4ieiLzEsw14XFqT5EiRa264Ab3VeyKepQsWYblfTdaDmaTGVhm7yXCZsV9btq4Ce3bt5d1I+Vm4hvcBF4b9lDYJbZ9uPZmXzsrvvifRXT48BGpjO42K/DUqZMyKJwsWTK5BizvRIkTyaDm72fOyDky9mKuVWwgoW2m/vG61jTWOlOqcrGW114rL4u+8GaPCHevwWlPhZIjufbNnm42Gg3du3eT+udMA57RJRZFnRfPioih64S3wv02/mNZoQ4ryJxPIfv1119x7dpV1Ktfz5xkve5O8LdySjEHiK38znddEwP9f7Qs7rUMmSyNA2CskFEnuwHMsEiLWAaAze9mBWKZ0zqXsudzc15sxQSznvLvMPaYF5dx7EzXy0l0tWvXxoYN68VNo7g+LGvpyZj9iuUr8PLLL6NI0aKmrB3Go4sTa6Iu+ZzNTXLnTjBmzpiJhg0aRG0N6teP2piHfd6cOdJK0lonbD2nTJ4seU7YVaJ+UejZOLgLs2bOQuNGDdGkcWM0bNjwH9ekYYOGGBsYaIVfUcmijAerkTx//jwH881L7qPqSZIkMf9aYa0UcP5W1gk2ekmTJpO84Oanm33s1AG6u1jHxPfKBtfU35w5c8qi1LlyPYNWLVvJpCV+L9bru11mzodVYayv5j515EmhYNvlxfDOK5cvY8eOrzBi5EjeWabuWUv+uToxlibgXuTmkAsUKTlh7hbke/8ks+Wxq229B/h48cd48X8vSlY98a/7WGlEZSkodpPdAA7YSKw5n/Bn39XtY+VjHonEiRNbx831YtKgtm3b4vLlS5Lis2KlSnJ53aESEi45V7VqNXz8iSn7F/8nv/vs77/jpZdewrTp01HJ/F67YaeVdTcPShPwJLBs7NmGV65cQaBpZD9evNh007ujRYuW1qITppz4nZytTvL+4mWwb8Ea1WugS5cuKP1qaes1N6kzj4M1IdJKtBVujAWuWbto4UK8+dZbSJ0mTVR5c0Z8vJWns6UJuBdLpCxLJnGixGJt2Rv9pndvjDzgheTEiDmz56BgwQIyYEYo5MePHZcbyJ0GBxMlSoRkyVPIJJgUKVP843okN8e5KC0rGa8fhZ//pjGVj+/jYCmPyAtuwnP580v2yf379kuZ88Y7fOSIDJAXNV1jEXTTsNNFEyeYukbx499lyOOggYMwZcpUDB8+Qhatvn3rtgy23WugOA3mu9va7bTfMQ6xx+nMA9Pbi8SE8eMlmyx7iOIKNUbUoQMH3MIbEGsqKWJkNssq4N6yGqzNciNYm7f4ltnVnjplMlasWCFrSXK211hz8zBumJEITLbErri7wN8ulh5H4Y0Vcfc1sfMw24OhrIyM2WfjR4E5ceKEXLeYskqdAfPL0btPH+zc+bXUG9aJ5Z8uR/PmLZA+QwbpxVnXJW4Eys6CyVX47e9TtlxZ9OjRA+ONIIweNQrBxtpzRsFkHTL/yJ7XzRm/Y1zDe4jlGRQUhHamx8uFVBbMn48pk6cgcNRoNKhfTzJ4ivC7OD4DOXUz2jy4stwtUg/frKnwX3zxBebOmSPCzWnbx44dlT27vuzuJk2WVC443+MO8HfYv8feR22O1+0B5H1792H9+vVyLW7duinLBnJlFb7G89wBNlLsiZw7d06mcG/9fKtkUuRUfka98FrY1+defvzhRyP86ZE5U2bzQY6DTwj/DsuF3XUpH3PMz89f3DyHDx3G0qVLTFncRunSpa0elREM9iR4ojOUCSfzXbt6DXv3/ioTbThWkcFcoxQpUsjv8URohXN8j6tj0f159OhRs/2GkydPiHuma9eukpgw3sovhjwRseZTf1SshDiMcrByWts3hg0LgqPSdgPiaRWSxcMblL+b62zSr75v334Jw6JfN/7lI2awx0woPhwoZ9RJlKg+QMxtYsOn/i9Y/eQ7RIq1x2Uc2dDSFcMJSxQDOc2UV3zXUd7RDP3kt71z+47cV3zC7+ZnHtsBCZ6EhA+b+4Xje5z78A/ZM4/pkmHCPE5AircoJ2f3qT8yprKxq0sriDCkzd88jtocaXzZdXIGCyiuYdWT9RDNPiBBAhQuXBinT5/G6TOnrbvXTWAd4M+hz5zzEihE3CjoT2J3xBjyPaxJYZx+P2XqFIwYMcJswyUqJvhOsNP4Y+15D7xqdFvynqLPmILuPjXm8WBDSwNSjAVjEd+tMaxz9kIZdIe6OvEu6rzYFGvqNfeWlcOqZ22MS+ZrTj0oFZuY38zfz0k4pGy5ctLAffnFF/LcXbDK1pr1KLOHpU5YjXj8W77Wd5PvZBofaXB8fNGocWM0b94cPXv2xKBBA0U0RDzM+fGZWZTfQTbzfaPEynxvVqToLgjv6jDQgP+JocBrYR7bG40m65jZ3EBjnKKE7Rs46uY1j//erNdJfN/c8YF9g9qPn376aaRJk1qyF9rXxR1gmKL9O++uC87wG//9HaxBWx5/z4h61apVMH/+fBlApcVOreA74tsIsb/33fXHikrzPPjb7VBYe/awvXEn10aeuv71sUpbcRkY0khh//XXvWKxK/EArV3Ra/phE8rM0zfeeANjxgTiww8/lLEPyWEk0q4ocYuKuovBmYwvvfQyfvvtN9y4ft1xVIlLxAI3es2IF/qv2aWfNHkKOnfpgmHDhmLBggXymqdaxUr8oqLuYlAmypQpIyFZvx09KtY6o2PcYdKEq2D30O0Zp3bqXs42HTduHAYPGoRVq1aLX52hcrZvXWZMK0oso6LuYlBIChUqJCkEfvrpJ7EaGapl+02VuIXXn35YmUjm44Nab76JBg0aoHu3btj25Rc8QXy5nFug3hglLlAlcDE4gp8wYULJLHfo4CFHdJA10BzfA3OeiFx300tiY0vNZs+pS9euyJw5E9q1a4c9e36UXhRdNO4wCKc4PyrqLoYtC6+++ioOHjwoIk8x56K6KhpxT9Q1N+2pzKUw4s7cPVOnTpNB7ffea45z585LuKM2ukpcoKLuclix0sy4d/z4MZkKbrtglPhBrHRTJuICc7hb8ubNixUrVzos9ra4fvOGCD/LSjbzPpV4JTZQUXcxTGdf9lmzZEHGjBmxZs2au6xFlYn4xrLGjbD7+eLp7Nmx+OOPceXyFTRv1gzXb1jRSnIOB0114FSJBVTUXQwuFsFuPlfoKVGiBJYsWSKZBBXngGXDAVPJjmia4KRJkuL9vu9j586dkr739u07YqKL24wPFCWGUVF3Mdi157qmEWHhKFasOI4cOYxz58/T/FOJcAIknUW4tSADUx6wVDj+0alTZ8yePQuTJ02U1Zso/twsy95KOKUFqMQEKuouhuWTtaxBWuoMlfvow4/kNR0odQ6YQsAOMbWmp/uiU+fOGDZsGCZOnISFCxZIWXGjjjM6hn752FpUW/EsVNRdDBmU8+KKSBHIlDkTqlWrhpkzZ+LixYuOMxRngqlcOYjq7++HNm3bibBz9aRvdn8jG5thOcc8sK12RXkSVNRdDMZEixKIxQ7Uq1dfBH7lylXWCYpT4W2sdA6IWrNJI1G3Xl1Ztq9t2zayHT92TM6x4ty1p6U8OSrqLgZvfqq5ZbEDRYoWkVXvmdM7NDRMpqTTP0t/rRL/sIwk3JH+dSPs9Ln36dMHN2/ekq1Vq9a4HhSEUJ1xqsQQKuouDH2yzJfdqFFj/Pzzz/j8888sMaevVnvyToct7Lnz5Mbw4XTDDJMJZN26dZXj2hArMYGKugtDEeAgXKPGjaRLP2rkSIQaS53Woaq688HyshbP9pIcMdwCxwTKGqKcoBQSbKx1RXlCVNRdGA6Y2mu70to7cOAgPt/ymVjwXL9UcS4YCWNHL9nRL7Vr18HUaVOxevVqTJ8+TSJhmF6ZjbIOnCrRQUXdhWFAnC0O/3vpJZQvXx4rVqzgeBzNQuskxamgsHOzy40izlTKQ4cOxbRp03Hm9GlzVqSMj6ioK9FBRd2loZ/FEge6YWitf/vttzh1+hTNeOsUxanh2pgU+SZNmiJv3jxo1qwZrl69Km4alquiPC4q6i4M7Tje91yIgSJQsGBBlCxZEl06d8aNGzetkxSnx/a1jw4MxF9//YXGjRrhPGcJK0o0UFF3YSjo1oxFxjhbK/BwwI3W+qSJExwhjuGWH1fdMU4J0z5wTMTO7Pjxxx/j0KHDaNWyJW7duiXuGUbG2KsnKcp/oaLuZuTNlw+VK1fBlClT8PNPP4nfXXyz2pV3CfIYYR8xcgS++eYbSSdAK96UoLpilEdGRd3N8PP3x8CBA5EsWTL06tULIY4MjioKroG/Kb+qVavJpCQOnAYFBWnZKY+Firq7Ybrq6Z5KZ4R9EA4cOICVK1eagyoKroA1+cjKE9O1W1ekSZManTt1VteL8lioqLsZHHCjj7ZO3Tro2asnAkcH4uSJE1b6gPBwyb3OgVXF+bBCHa1eFXtaDHPcseMrdOzQEcF3gmXJQrscNdhReRAq6m5M27ZtUaxYUTRubEVThIaEwsfPD5z8ojg/z7/wAkabRnnz5s3o06e3dLiixFxj2JUHoKLuxjDrX/v2HXD69Blj7XXArdu3JQpGfbSuAQe4mUqgSpUq+PTTTyUyhlEyVHYtQ+VBqKi7MbTlChQsgEGDBplu/A58MGSwhMlRECTMUTZLPBTng40yy2rEiOEoWLAQhg0dij/OnJGsj1pmyoNQUXdjZDk1c/M3aNQQEyZOxPr169G6VWtcuXLFcYYRB41fd15ojJstQYKEmDN3DnLkyIG+ffshNDRURV15ICrqbgytPG6+psteu3ZtLF++HN9//x26d+tuDbYxsZSx+hRnxTS64aaMTOOcLm06zJ49Bz/88APGjhmj7hflgaioewLGqGO4XJ58+TBjxkyJqBjQv78VScFwORUIJ4XTjiJlkXEfXx8JVW3VqhVmzZqF3bt3O85RlH+iou7G2JY6wxwZLkeLvVTpUujXrz/mzZuHtm3aRg2esktP6z3CWIbatXcOrBQQPlJ+hGVZt149pEyZEt26dsNff/4lDbPGsSt3o6LuIcjEFiMK1OsGDRugU6fOWLduLRo1bIiLFy+JgNBq13BH54WNLickjRkzFr///jsGDx4s5SoRMYriQEXdQ/A2N774Z70sC7Btu7ZYvmKFrJNZt05tfP/99yIOaqM7LywfivjLr7yMkaNGYuPGDVi2dKljJqqiWKioewrGRJduPE1183+iRIlQokQJbNi0Ec899xzq1K6NLl26Ssij7X6hWPCxtVmWohJ/sDH2Nq0y16XlwHenTp0wZMgHuGR6WpwlTDeMCryiou4h2NESXkYYJOKFrpiISPj5+mLosGFo0qQJlhqrr3y58jh44IBD2K2VeWxx50Ic6r+NX1h+sjfl17ZdO2TOnBm9e/c2ZRRu9bQcjbDiuaioeyiRRqwZUUEBSJIkCQYMHIgZM6aLe+att97C5599ZgQ8VKxDijuhz515ZZT4h6LORrl161bihlm0aJEpn/CoxlvxXHwGMk9rtFGLwFWJ5M1vW3QO1wpzeTdo2BChRryHfvAB1q5Zi5DgYGTIkAFJkyYVvzzf42zC8eMPPyJ9hvTInCmz3f64PWyMWQ7P5M6NP/44i9mzZuO5555Fjpw55biKuwviFTM2tlrqHgp9s+Kj5ebjLXm82X1PnDixTE7atXs3ihcvjpEjR+GVl1/GjOkzcOvmTRF/Coq4ZO7y39rHYgy2N3dvyj9gubG8WG4TJk6QpQxbtmyFM2d+l7JgOdllxU3xHFTUlX/BCS8pU6bCB8Za/+zzz1Cp0hvm8RBUqlgR8z+aj2tXr4mARxix4HJ5ERGW1Rij1qHD7688GiNHjoCvrw/mzp2DCLrWHJEyHtN1UaJQUVf+hW3p0YLPli07JkyaiIULFyEkJBT9+vXFa6+Vx8wZM3D71m2xGKm9FJIYFWFal2phPjK58+TBsOHDsXjRIpw7d04EnVEyETrvwONQUVf+BS1uPz9fsfG4cZZp6VdfxY6vd2DmzJl45pncGDp0GF555WUMNRY810INo6jHkAbz7/1jIpQam/8Jr1mNGjWQx4h73759ERwSIgOnTDGg/nXPwstYZdG/FSM1btnToAXoZf47fPgwVq9ehV9++QV79uyBv38AKlSsgBIvvCBWY66cuRCQIIFlxZv30B1AWN1EZCIpNGaTx5Y/nse//fZbfDhvnqy1OtH0ECxXgmlg7hYmr39an/T3Fy5S2PztEh7bALBnxWv0/Xffo27dumjcuDH69e/niF6yfPCKk+MVMzODVdSVx4ahdKw0lh5brpqvv/5aIjC++WY37twJlgG8KlUqm62qiHzGjBkkHNLLy0oHTHy8fR1+X0v4GXXDwddXS5dG/gIFMH36dBF3ir6K+sPhNeUYh5e5tKNHj8b48eMwYMBAtGzdyrxolZXi5KioK/EBu/nevj4INd17b2/GudMK9xV3CYX5+o3rRti/kTj3zz/fiitXLsPPzx/JkydHqVIl8b+XXkbx4sWQKWMmczwgKu6d72VVpHjXr19PQihnzJwlrzFSR0X94XCwmhOQqN5hIaFirZ84cVxSQeR0hDkqTo6KuhJfiKtEXCeWhUiiRIPPJd42EteDruOXX37G/v37JQ/4rl27cO3aNWPZRyBVylRInyEjcubIIaLztNlzdmTOXLnQrm1bpEyZwljqM6Ji46PEmg9pjt71t2fOmCmi/sILL8j3YE+Cs2bpumHP4O6c8dJwmD2tWscB+RvyWfzY++SXl/dE/T6z/fuUeMf+OebXy7979+5F1apV5JosXLRIek6EvSrbFaY4GSrqirPDqsXKRZGlGgYHB+PcX38ZC/IkLl68iLN//ImLly5JxsE//zyLS5cu4/y5c7hjzkuRIgWeSpcOyc0+aZIkMus1YaJE4hsOCPATC5+WKZ/v3bsP6dKlRdas2eArA7zMRhkpMff+/n7SkwgICDDPE0lESIqUKc356eRvpEmdxmo4DN5sCOhaMr0PHmPbwQbMTn0rv8dsriCKTOewZfMWtGnTBoMHD0K9+vXlOL+7+tedFBV1xdmRaetGKFnFaOnaVrT8F0ERNcLJYzzZUQ1v3LiBd955R0SJghQUFCTL7/E4k42FhYbi0uUL0kBYuWhCTSNxQgQ8YcKEEnbJBoTW6s2bNxAaGmaOhZgewlVpSCho/FP8fMZ1U+TSpk2L9OkzIEuWLMidO7d5nF4eFyxYAMlNA2B/N/mujt/izPDriivG7KfPmI5ZM2dhy5YtSJM2jXmVk86c+/t7LCrqirNDK5fC0r59e+n2U2gsC9tYu0bQQ4zgUmBaNG+BosWLi2ByoJS5Z1IZMZ0+c6aIMCXIFlTi5U1hpTibzzfHZ4pPvQiKFSsqnysNCf+2eY3n8Uvwvfw+F85fMI3CZZw79xeOHj2KM6dP4+rVazh79g/TW/hTegvXr1/nW8VNxAyWzz9fHPnyPSvhghT9VKlTyfdwVvhb+d15bdmgVaxQAaVLl8bAQYPkGqj7xUlRUVecHVYtblevXOUzI5R/W4iWW8NHrPkkSZMgwD9ARNi8ATVr1kSaNGkwffo0I9BWKoOHWccxNVB669Zt3LlzB+eN4J8+cwbfffstvvvuexw5clh6CWyY6LLhyv70VVMon302n7h0+D1lENn8Jn5Vfl/+Rv5+Ppbsig5ff1wzb+48DBw4AHPnzZPvLO4Xc9dzXIG9FcVJUFFXXAGrdlnifrcvN6raGcHj4h08xx6wfJuWeqpUmD5jhpxy9/vuR0yIum3Zc+9FS9Z8D/s7UtDPGJE/fuy4LCayefNmnDp10rzihaefzo5XXnlFtteNRWz5rC1h5+tRvQY+sw7GOdeDgsz3KymDz2vWrpXIIrtRtVMKKE6Airri7Fg1i/9QzMz+XlET4bQOsxpS9LivUb06UqdOjTnGwqQgcqAyti11+7vSXUM3kf2cDYodTcNj9izXS5cu4eCBg9i/f59EmnDSFH3/TKxVosSLYhEXKVpErHiKJi11+S3/0UDFBhRwThBr2KAhXnyxBKaZ6xXg7y/XND56DsoDiCFRj/sapngQDteD0Q0RZUsVrU2sV+s10XsjdnRvfP/dd2Lp0g3y5RdWnPvDBD3mMF+C/zLyxXwXbrYFa/99+qgtS9wbadOkQelXS6N1mzaYNHky9vz0k7Hgt+CNN97A9u3bUaNGdbxoxL13r17Yv2+/+dy/Gy26abi3N4ousfcxDf9GsaJFJenXxo0bsWHDehHzsHBdKckdUUtdcQok06PDVcEaaYm9ET2zoxg+zMJ1pslH/O6M0Nm2fRuWfPIJdu7cJQOvb7/9NurVr4f8z+VHQMIEUS4ZX07cYqSK4b9+Z3QR4TafzQljnTp2wsmTJ7F6zWp5zY5fV5wAtdQVt4Jq6LAvRNwYteIQudgQutjCfGXzD1C2TFlMmzZdkqC1adNaVieqUrkKatWqiZ07vpbfyI2CLg1X1JtjHv4drnRFV9D7fftKCOjGDRskBQN7R4p7oZa64jTIJCWja7QsLSGnu8JK9MXtQTiXpf63W4UNE10c/C30t9P3vmnTJqxftx7Zs2eT8zm4yuyKXF2K58XGoCW/S7i5pgz35MDopEmTMHPmDMxfsABFixZ1qUbTrdGBUkWxcCZRfxi0ihlxwlmzzEdPPvpovmnEwlGuXDnJh546dRp5zkaMjZvkRHe4amJKfJmqoVzZsjJZa+WqVRINI2GZphGKpc6C8iio+0VRXAuxwo0NxciePn3el23LZ59J8q0NGzZI1M+2bV/KubS1ZKJWeIRsMdlaJTEiPnLUKBw7dhwTJ0ww38vXsuZNr0JxfVTUFSWOsBKMOQZDaRKbLVu2bBgwcCC+3rkTyZIlQ/169dG+XXucOnlKhJZa7iMThKLfob4XNg+vvPwKKlSogEWLFmPPjz/KcYq74vqoqCtKHCGx6kaoRdh9rAW/OWbACBSK+ydLlqBVq5bYtGkj3nijEtatXSfn0v3ysDGFx4Wf5+Pni2Ejhovr5f3335dcODHZcCjxh89Ag+NxNNBKoMQ/P/7wI9JnSI/MmTJbZqiTQmGO2vjcbLTa7WMU99KvlkHNGjXFOp8wYQJ+/fUXGURlkjFiD4HZE6KiA/8WM+okSpxY0hzMnTNHZvAWLlSILY+cIyGX5uN5rhJHOK79k6KWuqI4CSL0RkSzZsuKXr16YdmyZfjll1/x1ptvonOnzjLAafm+rUXBnwSZZGX+VqlSpdCvf39MnjQJt27fluPiJnI0NorroaKuKM6CEVHqKEWVFnyuXLmwZs0avGlEfeXKFaj9dm0cO3YMnKj1RPHlxti3Eo9ZScjq1qlregkBGD9unDQa4sM34q64JirqiuIk0DKmv9vXz0/EldY40/yOHTce48x27tw5tGrZShYUYSQNz42OuNszd6XxMH8jYaKEaNy4kYRXMp9NWJg1Icrh6VFcDBV1RXEi7Fh08Xs7XCB0idR6sxY+3/o5cuTIgUoVK2HqlCmycAhhzvTHhe4VCjo/n06W+vUbyGIhrVu3xpXLl01jwXQNaq27IirqiuKkcDCUwi4zbQ0pU6TElKlT0Lz5exg5ciSqVqkii348ie9b3mu2pMmSYs6cOfjrrz/Rt29f8bfbDYziWmipKYoTQ/eKNWhpblWjv5yQ1K5de8ycORPnjaC3a9cuWpa6Dd08kkzN/JcjZw6ZCLVu3Tr8uOdHhEuYo+JqqKgripNC3zd95w5jOio3u5+/HypUrIgFCxdIPhn62a9cviLvsUMeHxVa6j4+liuGi3P37NVLYuYZecPGhH770NBQEX7FNVBRVxQXhGJbuHBhTJ48CV988QXq16+HU6dOifjytejib4S9X79+WL1qlSzUTSwXjTxUXAAVdUVxUWiVlylbFitXrRRrukrlyvhpz4+Wqyaa+JqeQNlyZWV5vtatWiMoiItwM1pGLXVXQUVdUVwQDmJyozumWNFiWLpsGV566SU0bNgQhw8fkmn/jGCxXCePbrlLnLr5TPrqd+/ehTGBo+WYJvtyHVTUFcVFsfztXjJZKGXKlAgcM0ZCHt+sVcsI+2Ej6CFyzuNgR7wULFQINWvWxMKFi3D40CFdIcmFUFFXFDcgwljTiRMnxhgj7BT6Rg0b4cjhIwhj9MxjuGP+Hmj1QpeuXWVwdsiQDxwJvxRXQEVdUdwAzg7l6kY5c+XC4o8/RoYM6WWW6JZNm2X26KPDVAV063gjZ86cGDFiOHbu3ImNGzY6XlecHRV1RXEDaFH7+fqKm+S5557D8hUrkDt3HrRt2xY7vv5azmHM++3bt+Xxg/CWMEprpikjaapWq4Z69ephwoTxuHP7jnwGj6vl7ryoqCuKm2H70WfPnYOiRYugX9/3cfXKFXGtcPLSo0KBZ+Kv3n164/fff8fGjRsdYh8u650qzomKuqK4Gcy8mCBBAiRJnBgzZ83CnTt3JKcLo2F8HAOhjwIbgUjznuTJk6N69eoYNmyYpP/lpCSNW3deVNQVxc3gTNQ7d4LFVZIyZSoMGDAA27Ztx4D+/cTSfmSMqPsYy56fw4Rff/zxByZNnESvuwzMKs6JirqiuBkU7gQJAsTP7uvrIykFBg8ehCVLlqJDhw64fv26CPV/+cWtnDOWv75Q4UKy1B6Tfm3d+rkIvljyZmMcvGq886CirihuDmPPGzVqjEmTJmHlypWYOWOmDHZKTPpjiHH3Hj1RsGBBdOncBWf/OCvHGFnDz9E0vc6DirqieAAJEiZAxUqV0LNnT8ydOxfHjh0VC/txfOOMrOEi1UFB1zBz5gxxwVj+deZ8V1PdWVBRVxQ3xxZcRrO0at0aefLkQcsWLXHp4iWJbefr3DOlwMOJxPMvPI+qVauZhmEeThw/br03LMzoukqJs6AloShujh3iSDcJ/eOjRo/CjRvXZXLS6ZOnxL/+KBOU6KvnZw0eMhi5cuUUNw7fx6gaNhiKc6CirigeBvPDLFq8WBbZaNq0KYKCgkTw/yuGXQZFzZYsaTJMnjIFS5Z8ggMHDoigaxZH50FFXVE8EM42HThwAE6cOIHOnTojNOQRszkaUeeCGvny5ZOskKNGjZIGQePWnQcVdUXxMGhxc6tctSratGmDzZs3Y+sXW+VYWBjTAETcN9yRrhcKuG3Vv/XWW9i6dSu2mPebtzpCG9Vij29U1BXFw7B0l7NFw9G+QwdUqlQRgaMDxbfOBaet+HS6VB5uuVd6ozKyZMmK0aNHIzg42PHBarLHNyrqiuJh0NKm9Pr6+SFBQADGjh0rfvEO7Tvg1p3bCA8Nk3O4HPXDYIhjYGAgjh8/gQH9+0sEDa31R3LjKLGGirqieBicJcpEXYQLbCRNlgwjRo7C2rVr0O/996OE+VEiWl5+5WV06NAeixcvwoEDB42xHiFWvhJ/qKgriodhiy73FG/uCxcpjObNW+Djjz+RWad0pPxXRAtT9NJF06RpU2TIkAELF8x3+NRV1OMTFXVF8XAozF5Gi7t174bChQujf//+OH/u3CNkELDEm1kce/bqJal5b928iZCQYDmuxA8q6ori4TATI8MUma73o/kfIWvWLOj7fl9xpTwMiaIx+9DQMEnNmzt3bvTq1dt6UYk3VNQVxYORQVMvLmHnJbNNU6dOjQULFmDXrp2YPm2asbpDJCrmfoOf9LkztJEZIX19fGVN09WrV+PTZcvkfIo99xrmGLeoqCuKEgVFPt1TT+Hdd5th/PgJOLB/v8MH/3A/OXO4Fy1SBM899yzGjRvvmKXqSPSlmh6nqKgrihKFPTjasFFDJEqUCJ06dcK1q1fl2MOgRZ4wcWKJez9//hwmT5oMbyb5Mh9HwVfiDhV1RVGioJecLpenjLXOxF1HjhzBRK529B+6LJE04RGoWqUqSpUqjYULF+Dsn2dF0HlciTtU1BVFicLykSeQlABVqlZFv379MW/eXGzYsEFS83JxDWZlvJ+PnYSGheKDoR+If75Tx464ffu2lXNdiTP0aiuKcl+4SHWzZs1Qt249DB40SBawDg8Pk/zp9x/8jBQxz5wpE0aPDsTu3d/gq+1f/WcUjRKzqKgrinJfaI1zxmnfvn3l8aSJE60omQB/s/+ndNAXT4uc5zH9QNmyZWTpuyVLljgGWpW4QkVdUZT7QjFmNIy/v59Ew8yYMRPffPMNwkND/zX4yTh3Y5KL24ZWPN04w0cMN9b6bhw/flzcNsTeK7GHirqiKPfFdrH4+fujcZPGeP7559G2TVvs3bvvvoOfbACIWOZmo6X+yiuvoFfPnrgeFCTW/KOssKQ8GSrqiqI8AEucOfmIg6ezZs9CokQJ0ahRQ1y9esVxzv1hc8DJR506dxbf+rx584yVHi4fqcQuKuqKotwXTh6KcKxfSuubOV6GDRuO27fvoE/vh6cD4Hv8/P2QL19eY+EXx6xZs3Dx4sUHDLAqMYmKuqIoD4Q+crpfJL+60eOSpUqicePGWL16DXbt3ClWvLhV7hFr+tRplHPff8AAmWE6YsRIyzWjxCoq6oqiPBLiKjf7rt26yRqlc+fOQ3hYOEJCQx4q1oUKFULNmjUkX/u5c+cdR5XYQkVdUZRHx4g3V0uaO3cu9uzZg08++Vhi0x/mVqHgM3tj2rTpMHPGDMdRJbZQUVcU5ZGgbtMdw9j1rNmyokWLFkase+GLrV/IwKidN+beRasp6umeSocPPhiCjz/+GFevXJHP4nkPmpmqRB8VdUVRHgmudEQo0hT3ho0aIVu27Bg4cACuXjZCbV6jSNtL5dnQimcI5MuvvIJMjtmmnJnq4+OrceuxgIq6oijRguGNFHROLho0aKCxvEMtN8w9rhgOpvr6+UpDUK9+PSxZ8glOnjyF0NAQsfqVmEVFXVGUaEGr/LXXXsOQDz7AypWrsHXrViPojhfvghEwHGL18fZBjRo1kTRpUvTs0QMhIaESMqnELCrqilMgGQBNF51W3+ZNm/HLL7+I1Rd8J9iy/hSng26YMFNmXHiaYY6DBg6UrIzEDnO0Z54y5p2pBNKmTYPhI0Zg165dslA13S8sXZ4ne/WxPzEq6opTwK75V199hcpvVEarVi3xRqVKaNumjbymU8udEy/zH61wlk+v3r3E8mZUDBtoCrok+TJCfm+jXK5sOUk5wDztt2/fQkRYGINqEGmsdjvVgBJ99AoqTsGpU6cQGBgoIXJbPvsMVatWlQkuw4YN1XzczopRYoox1ydNnCgx3nnnHYlw2b9vv1jlErtOYZfo9r+hH71Pn964du0qNm3caITfx5xmWezaKXty9G5RnIJDBw9i5qyZyJ+/AHLlyoXxEyYgR44c2LBh479C5BRnweFiiYyQrI0cBM2YMRO6du2KG9dvSGNsWen/VGqKfdFixVCtWnWsWbNGXC48xnPZDihPhoq6EmtIF9zcz7Z/VfaOTXyp8rq1lS//GtKnzxAVDUHrr5i58VOlSmmtdXkX/EwKAT+DkRWEe3vKOl/jZyqxC10l3Og643VPnTo1pk2bhr/++gutW7c2wh6E8AhTLvcpCz9fP3Tv0R0///wzvti6VURdLHvliVFRV2KNqJvU7HjTSy/csUW9ZB7wNSZ/utufSuvv0KGDRhza/CvsjTHOXKSBH8GYaA7CiSiY9/N98gccwq/EDbTU2ZAWL14cI0eOxJYtW2TPhtZR1FGwMQ4zZZgte3aJhhnQfwBu3rwlr9mNtBJ9VNSVWINiTfGW7rUtvI6Nosu9REeYPc+1B0Rp13HQNHHixKhStcq/rG5vOZ/v49+w3sMcJHwnrUaKvjlJjitxA8uBV5yrIlWsVBHNmr2LWbNm48iR36RndTdMwcskYaEhIahevTqOHTsq6QPYUHt7a9z6k+Jlbph7LvljEKmtqvJgWLF4sw8cMECEOUyE95/QyuYamPkLFpBzaa3fvHETzZu/h1GjRyNjxoxynljmFHHzORcuXED3bt1l8go/88yZ00iXLp1jwWRf+YzAMYF4yhzTQda4gxY44845SHrlyhVUfuMNSeY1ZcpUOcZysf3ntMjZAPN5ndq1cfjwYWzd+gXSpE1rPilSzrXP8Ri8Yua3qqgrsQZvWG68YRndYGx0xysU578Hx+hLT5EiuTy/cvmK5N5+u/bbyG665zTHfX195DVpJcyeYv7HH3/Ijc/jixYuQt68eVG0WFE5xsUZMmfOZKV/5fuUOIeCfPjQYdSrVw8DBw5E5SqVHQLNFAN/N7Qs//379qFmzZooV64cJk6aZMqNCcLYuzMNgSeVn4q64uzwxpbKZSw43tB3C6xV6ThIau15LhuAGdNnyGSWxIkS8VURZgp9ylQp5Sa33C209MLgwxjpcPOemTNQtGhRiX1mdaaw8xzV8/gjnBFLpgAmTZyE8ePHY/nyT1GkaDHzil0+fxMSEoKFCxagT58+mD1nDipVqhRVjvee69aoqCvODkWaUMwp2tZ0cQura82ETqYOmRoYakR61oyZCLoehGLm5qegh4WG4tdf96JqtaookL8AdTrqvbzZw42g0+qbNnUqihd/HsWKF5PjIvzmb3qUIDgZdiN99cpVvPZaedNzyoKly5bJItZ31wO7N8eZqBUrVERAgD82btwkuWJYfh7V04ohUfcxXaOBjsfRIPrtgeL+8Ia0t3sF1n4ur3t7GUGfIRbdLz//Ym7qjdi8eZPZtoi/vF27dghIkEDOJ/bNbu/37PkJ6TOkRxYjHNbneZgYOCFy/Y08JDQ9rjSpU8tMUza2JUuVkgZXGl7zupSVsSsDAgJMeXphmRH+EiVelGyOdhl7DPeE7kYXtdQVp+DihQv00iDCWOz0qdK9Yvvhn0r/1ENFmi6bwkUKo8QLJUyNdhxU4h27x8Sya9qkKbZv34YdX38tg98c+DYnmLOMwJt/2WOjFFV4/XWZmDRq1Gh+goeJesxY6h50xRRnJnWaNGLRpXvqKfGfp02XTiJa0mfMIDe74lpIOCsF2RQdHw8fMVxmCI8cMUJEXF6gxe54xsab4j98+AisWb0GRw4flufK46OirjgNtjvFHlSVkDeH/1xxLehSkzEV0eVIpEqZEtOmTcf69Ruwc8cOEXS+znKm24UTzBjpUuLFEnj11dLo3bu3ed0h/2wYtGF/ZPRuUZwCscr4P/f2c4MKuutiN9LhRrwDEiYUS71CxQro378/7ty5I6J+t1hzdrC4YCpWxA8//IAdO76KmlTGxl2F/dHQO0ZRlFiF0S4hd4JlAlL9+g1klunc2bOthvseneYxLryRJk0ajBwxEqFhYRJJIz545ZHQK6UoSqxCi5y5fSjMJUqUQNN3mmLMmDHY9uU2o0BWj8yG5zI9BFP4/vrrL1i8aKFY8Bw0Vx4NFXXF5WA3nLNGuRe/7V1Yx6zj9qo7Svxiu2FomDNOvW/fvsie/Wn07NkD14OCxBJnemXu7fMrVKyEqlWroV+/frh4/rw5qqL+qKioKy4HrTZOTmFCqPutXM8p5hR2ib5QnI4ECRNi9OjR+P33P7Bi+QqJjrkbWTDDlGGHjh2QKFFibNy0SY7b4yzKw9Far7gcHDTjEmjM9Gdbd3/D4DhTsR2WoeJ8MIVAwUIF0bRpUyxevEgiZZjywe51Mb7d188PuXPnFv/6yhUr4WMaaB0ofTRU1BWXg6vqHDp8WAT9X1n8jJAfPXoMly5fkrUyFeeD7hUKdMdOHXH8+AmsXbNWUkJICKsRdPawwkK4zinQ5/0+spDG3n371FJ/RFTUFZfjxs0baNmiJQ4eOGAljoKVJ4YTWU4cPy4LVwcFBf0jx4jiPHDAlAKePn16U1atxG9+5vQZEXoep1XOgVU/P19kyJBB1qvt17cvbt++I4tbM8yReX+U+6Oirrgc6c2N3r9/P8kJc/PGDbHu/AMCcMM8bv7ee2jZspWk7bUnryjOhZ12meXWvEULI+TeaNas2X3XouV5XND6xx/3YNbMGaYjZoTfNNb8DOX+qKgrLgdv9FfLlBWf65QpU8TyYzTMh/M+RJYsWVGjRnVjuYeZbr52150RmVBm2lta5cmSJUXHjh1x9OhRLF2yNMqvfjf0v3OxjQULFuL6jZsyQE5hV+6PJvRSXA57cPTcuXOoUb06ypUrj2eeyYWxY8dh3bp1yPZ0domooDX3L5+7Eu9QuEXYDZQfpt2tVLESgoKuYfOWLUiVKpWUmyVNtOjDsfPrnbLgRpcuXdClaxd5ze3KVvOpK56K5VYxos1BNWOllylTRvyz8xcskHzc5o6nISgWveLcSONr9hwbadGiBbZu3YodX+9ApowZjTXux1KW8iQ7v/4atWvXQY8e3dHBWPcq6vdH3S+KoihuxJNZ6oqiKIpToZa6oiiKG6GiriiK4kaoqCuKorgRKuqKoihuhIq6oiiKG6GiriiK4kaoqCuKorgRKuqKoihuhIq6oiiKG6GiriiK4kaoqCuKorgRKuqKoihuhIq6oiiK2wD8H7HMPl/H1R5YAAAAAElFTkSuQmCC"></p>
<p>approximately symmetric about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p>including cusp/sharp point 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> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Final <em><strong>A1</strong> </em>can be awarded if intersections are in approximate correct place with respect to the axes shown. Award <em><strong>A1A1A1A0</strong></em> if graphs ‘merge’ or ‘cross’ or are discontinuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis but are otherwise correct. Award <em><strong>A1A0A0A0</strong></em> if only one correct branch of both curves are seen.</p>
<p><strong>Note:</strong> If they sketch graphs on separate axes, award a maximum of 2 marks for the ‘best’ response seen. This is likely to be <em><strong>A1A1A0A0</strong></em>.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>approximately symmetric about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn></math> with approximately correct gradient at axes intercepts <em><strong>A1</strong></em><br>some indication of position of intersections at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn></math> <em><strong>A1</strong></em> </p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Final <em><strong>A1</strong> </em>can be awarded if intersections are in approximate correct place with respect to the axes shown. Award <em><strong>A1A1A1A0</strong> </em>if graphs ‘merge’ or ‘cross’ or are discontinuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis but are otherwise correct. Award <em><strong>A1A0A0A0</strong></em> if only one correct branch of both curves are seen.</p>
<p><strong>Note:</strong> If they sketch graphs on separate axes, award a maximum of 2 marks for the ‘best’ response seen. This is likely to be <em><strong>A1A1A0A0</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><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Any <strong>two</strong> from:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> has a cusp/sharp point, (the other does not)</p>
<p>graphs have different domains</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn></math> has points of inflexion, (the other does not)</p>
<p>graphs have different <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts (one goes through the origin, and the other does not)</p>
<p>graphs have different <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis intercepts <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Follow through from their sketch in part (a)(i). In accordance with marking rules, mark their first two responses and ignore any subsequent.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Any <strong>two</strong> from:</p>
<p>as , <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><mi>y</mi><mo>→</mo><mo>±</mo><mo>∞</mo></math></p>
<p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi></math> is approximated by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> (or similar)</p>
<p>they have <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> intercepts at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mroot><mi>b</mi><mn>3</mn></mroot></math></p>
<p>they have <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> intercepts at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mo>±</mo></mfenced><msqrt><mi>b</mi></msqrt></math></p>
<p>they all have the same range</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis) is a line of symmetry</p>
<p>they all have the same line of symmetry <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p>they have one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept</p>
<p>they have two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis intercepts</p>
<p>they have two points of inflexion</p>
<p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts, curve is vertical/infinite gradient</p>
<p>there is no cusp/sharp point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The last example is the only valid answer for things “not” present. Do not credit an answer of “they are all symmetrical” without some reference to the line of symmetry.</p>
<p><strong>Note:</strong> Do not allow same/ similar shape or equivalent.</p>
<p><strong>Note:</strong> In accordance with marking rules, mark their first two responses and ignore any subsequent.</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><strong>METHOD 1</strong></p>
<p>attempt to differentiate implicitly <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></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>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>y</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mo>±</mo></mfenced><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></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>x</mi></mrow></mfrac><mo>=</mo><mo>±</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to use chain rule <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mo>±</mo></mfenced><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></math> <em><strong>M1</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>x</mi></mrow></mfrac><mo>=</mo><mfenced><mo>±</mo></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mo>±</mo></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math>, <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></math></p>
<p> </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>x</mi></mrow></mfrac><mo>=</mo><mo>±</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </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><strong>EITHER</strong></p>
<p>local minima/maxima occur when<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><mn>0</mn></math><br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> has no (real) solutions (or equivalent) <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>≥</mo><mn>0</mn><mo>⇒</mo></mrow></mfenced><mo> </mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>></mo><mn>0</mn></math>, so <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><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so, no local minima/maxima exist <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt to use quotient rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> <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>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfenced><mo>±</mo></mfenced><mfrac><mrow><mn>12</mn><mi>x</mi><msqrt><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></msqrt><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></mrow><mrow><mn>4</mn><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></mrow></mfrac></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><msqrt><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></msqrt></math> and correct denominator, <em><strong>A1</strong> </em>for correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math>.</p>
<p><strong>Note:</strong> Future <em><strong>A</strong></em> marks may be awarded if the denominator is missing or incorrect.</p>
<p><br>stating or using <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> (may be seen anywhere) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><msqrt><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></msqrt><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p>attempt to use product rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> <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>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo>+</mo><mn>3</mn><mi>x</mi><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for correct first term, <em><strong>A1 </strong></em>for correct second term.</p>
<p><br>setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts implicit differentiation on <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mfenced><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>y</mi><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>6</mn><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p>recognizes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</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>x</mi></mrow></mfrac><mo>=</mo><mo>±</mo><msqrt><mn>3</mn><mi>x</mi></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mo>±</mo></mfenced><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac><mo>=</mo><mfenced><mo>±</mo></mfenced><msqrt><mn>3</mn><mi>x</mi></msqrt></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><msup><mi>x</mi><mn>4</mn></msup><mo>=</mo><mn>9</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempt to use quadratic formula or equivalent <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>-</mo><mn>6</mn><mo>±</mo><msqrt><mn>48</mn></msqrt></mrow><mn>6</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>></mo><mn>0</mn><mo>⇒</mo></mrow></mfenced><mi>x</mi><mo>=</mo><msqrt><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>3</mn></mrow><mn>3</mn></mfrac></msqrt><mo> </mo><mfenced><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>3</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept any integer multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>6</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>).</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find tangent line through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mn>1</mn><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve simultaneously with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong></em> mark can be awarded for an unsupported correct answer in an incorrect format (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>.</mo><mn>25</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>8</mn><mo>.</mo><mn>875</mn><mo>)</mo></math>).</p>
<p><br>obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>17</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>71</mn><mn>8</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[QS]</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>79</mn><mn>42</mn></mfrac><mfenced><mrow><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>88095</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>solve simultaneously with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>28798</mn><mo>…</mo><mfenced><mrow><mo>=</mo><mfrac><mn>127</mn><mn>441</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>4226</mn><mo>…</mo><mfenced><mrow><mo>=</mo><mfrac><mn>13175</mn><mn>9261</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>228</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>42</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to find vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[QS]</mtext></math> <em><strong>(M1)</strong></em></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><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mfrac><mn>21</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>79</mn><mn>8</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>+</mo><mfrac><mn>21</mn><mn>4</mn></mfrac><mi>λ</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>79</mn><mn>8</mn></mfrac><mi>λ</mi></math></p>
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>79</mn><mn>8</mn></mfrac><mi>λ</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn><mo>+</mo><mfrac><mn>21</mn><mn>4</mn></mfrac><mi>λ</mi></mrow></mfenced><mn>3</mn></msup><mo>+</mo><mn>2</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>2453</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>28798</mn><mo>…</mo><mfenced><mrow><mo>=</mo><mfrac><mn>127</mn><mn>441</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>4226</mn><mo>…</mo><mfenced><mrow><mo>=</mo><mfrac><mn>13175</mn><mn>9261</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>228</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>42</mn></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This was a relatively straightforward start, though it was disappointing to see so many candidates sketch their graphs on two separate axes, despite the question stating they should be sketched on the same axes.</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>Of those candidates producing clear sketches, the vast majority were able to recognise the points of inflexion and write down their coordinates. A small number embarked on a mostly fruitless algebraic approach rather than use their graphs as intended. The distinguishing features between curves tended to focus on points of intersection with the axes, which was accepted. Only a small number offered ideas such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>→</mo><mo>∞</mo></math> on both curves. A number of (incorrect) suggestions were seen, stating that both curves tended towards a linear asymptote.</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;">
<p>A majority of candidates' suggestions related to the number of intersection points with the coordinate axes, while the idea of the <em>x</em>-axis acting as a line of symmetry was also often seen.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The required differentiation was straightforward for the majority of candidates.</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>
<div class="question" style="padding-left: 20px;">
<p>The majority employed the quotient rule here, often doing so successfully to find a correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>y</mi></mrow><mrow><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math>. Despite realising that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>y</mi></mrow><mrow><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math>, the resulting algebra to find the required solution proved a step too far for most. A number of slips were seen in candidates' working, though better candidates were able to answer the question confidently.</p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Mistakes proved to be increasingly common by this stage of the paper. Various equations of lines were suggested, with the incorrect <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn></math> appearing more than once. Only the better candidates were able to tackle the final part of the question with any success; it was pleasing to see a number of clear algebraic (only) approaches, though this was not necessary to obtain full marks.</p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Significant work on this question part was rarely seen, and it may have been the case that many candidates chose to spend their remaining time on the second question, especially if they felt they were making little progress with part f. Having said that, correct final answers were seen from better candidates, though these were few and far between.</p>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>ln</mi><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo><</mo><mi>x</mi><mo><</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>The seventh derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mn>7</mn></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>1440</mn><mi>x</mi><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>-</mo><mn>21</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>7</mn></mrow></mfenced></mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>7</mn></msup></mfrac></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></math> to write down the first three non-zero terms of the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></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>Hence find the first three non-zero terms of the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></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>Use your answer to part (a)(i) to write down an estimate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></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 the Lagrange form of the error term to find an upper bound for the absolute value of the error in calculating <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>)</mo></math>, using the first three non-zero terms of the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to the Lagrange form of the error term, explain whether your answer to part (b) is an overestimate or an underestimate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>)</mo></math>.</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>substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>-</mo><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>6</mn></msup><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</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"><mfrac><mtext>d</mtext><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> if this is seen in part (a)(i).</p>
<p><br>attempt to differentiate their answer in part (a) <em><strong>(M1)</strong></em></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>6</mn><msup><mi>x</mi><mn>5</mn></msup></mrow><mn>3</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for equating their derivatives.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>5</mn></msup></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[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"><mi>f</mi><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mo>≈</mo><mn>0</mn><mo>.</mo><mn>149</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept an answer that rounds correct to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>2 s.f.</mtext></math> or better.</p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find the maximum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><msup><mi>f</mi><mfenced><mn>7</mn></mfenced></msup><mfenced><mi>c</mi></mfenced></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="[" close="]"><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>maximum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><msup><mi>f</mi><mfenced><mn>7</mn></mfenced></msup><mfenced><mi>c</mi></mfenced></mrow></mfenced></math> occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>199</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><msup><mi>f</mi><mfenced><mn>7</mn></mfenced></msup><mfenced><mi>c</mi></mfenced></mrow></mfenced><mo><</mo><mn>1232</mn><mo>.</mo><mn>97</mn><mo>…</mo></math> (for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo> </mo><mo>∈</mo><mo> </mo><mo>]</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>[</mo></math>) <em><strong>(A1)</strong></em></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> <em><strong>(M1)</strong></em></p>
<p>substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>6</mn></math> <strong>and</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math> <strong>and</strong> their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <strong>and</strong> their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mn>7</mn></mfenced></msup><mfenced><mi>c</mi></mfenced></math> into Lagrange error term <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math> <strong>and </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math> <strong>and</strong> their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <strong>and</strong> their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mi>c</mi></mfenced></math> into Lagrange error term.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><msub><mi>R</mi><mn>6</mn></msub><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></mrow></mfenced><mo><</mo><mfrac><mrow><mn>1232</mn><mo>.</mo><mn>97</mn><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>7</mn></msup></mrow><mrow><mn>7</mn><mo>!</mo></mrow></mfrac></math></p>
<p>upper bound <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>000401</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept an answer that rounds correct to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>1 s.f</mtext></math> or better.</p>
<p><em><strong><br>[6 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"><msup><mi>f</mi><mfenced><mn>7</mn></mfenced></msup><mfenced><mi>c</mi></mfenced><mo><</mo><mn>0</mn></math> (for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo> </mo><mo>∈</mo><mo> </mo><mo>]</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>[</mo></math>) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>6</mn></msub><mfenced><mi>c</mi></mfenced><mo><</mo><mn>0</mn></math> or “the error term is negative”.</p>
<p><br>the answer in (b) is an overestimate <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>A1</strong></em> is dependent on the <em><strong>R1</strong></em>.</p>
<p><em><strong><br>[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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a gradient function given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>y</mi></math>.</p>
<p style="text-align: left;">The curve passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the same set of axes, sketch and label isoclines for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, and clearly indicate the value of each <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept.</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>Hence or otherwise, explain why the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> is a local minimum.</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 the solution of the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>y</mi></math>, which passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> does not intersect the isocline <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</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>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> on the same set of axes as part (a)(i).</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>attempt to find equation of isoclines by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><img style="display: block;margin-left:auto;margin-right:auto;" 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ulRbYX6HvlYbwQ5WQMACJadWLp4kQp9zFthQGm37sHkUjQhH/BB1olBAwcIq5PGAZowbZ6JqQ/N6P69O5rH7cC+PUGdMgNf+P6S388Beiy4wL5YDHwz8IABT+d9dKBdAjo+1BnzGsLf/+MAByDhNZ2DB/YJaYwq71cEGmuy9ZJ4LiBWef+eLql97Mg3SptncC0VVZeBT0oedGiTHH4GPMZMBjwZeMCAp/OAB98LJsx5c+eopuEm77YlW0XYRwl0BQudX0PG6u4KGmhMChbBcglRNPx1gK6m+rH069NLl0kA0Jwp0AUZkBTUl7VtBr4w4MnQKBmEijGPtVMUDxjwdCyP4EPx/pu1q1epttNQV6M+FMDCC3fvZ3mqf9KWnyDoAOAhAEAzVt+8npoHpEslRIx5ngd9AIf9uppqIWP1ju1fPKV1cU3rZj6d7MBrwNOxg+Up5o9gZitbXG1vwNNx/YnwVi2itVlWLF8mb40boxkAvEmN69nGj8/RRjnMcpjpbly/Jt1eeVmull9W0Ml2f/o1nqsBCM0uY/WokSPky+3bNCVPVF3SadmxmdqyMq8xSMcJklJoSwOejuMXNB14ZsvmTZpg0x8DIi7JZnbgASi4x4PXvbu35bVXX5ErRBsSkp0wXxrggsbEnzxuG9atVfByefgsVD7x+DaNp+MGS+LGN42oqIDfgKdjx9LOHV/KyOFDNepMNRhvPvPbLOPHm8QADCZzMr/m3JnTqrX41DVJxiv0COOeNGG8fLh0SWrCqpoCMcElBLIkzy7KsgY8HTtYipJJsgxwe982/jHgaWuLXPjCpbtxOdn27dmtjnuixhyIsPZOU5D40+VDS3+G99VoEIGmr2mW27duSM/u3eTs6ZMpfxFgFGUe45n8lRblmxtl+tTJsnzZUpeDL1gDyGtC6XWx4wheMOCJaCATukWllXSmQDDgyX0sOTBwfpRjRw/rMuKPHj1Imbei+o2IMkxoqhmxbW6Uirt35M0Rw+TEsaN6HEUjfB3QISCBLea6ZUsWy7szpqu/yC+1EC5v+wn73oAnYYMZEBkQtcMDBjzPN5aIWGMBtwH9+gg+GV2tM0gIGqWhIPgBHbQiQIj0N8OGDJadO7ZrBFpSYFA6QQoe1uVB2wHMyLnHRNak9Kx8Gm8Y8KQ1SDtCxRjH2imKBwx4cucRgAXQ+eH3vye3bl5X0xaahv9HtX1K02lqkAeVFRrqvO3zLW5BuCD/WhSN9OuAzLq1q2XG9GmasRp/TksQbZde1o4T9r0BT8IGM2Cyr712eMCAJ9lYAmwAFoT21fIyGTign9y4flVNXD6KDVMX5dA2wsKdc1zjz3lfBj8PAQCbN32q1zjPP3xvpn3o+GeoVlNXo8lDJ014S2qrH6doca+vcyY6di4mDxjwxGyodoSNMZq1n+cBA574vIDw9lkDrlwu01DnkyeOKUjEAQra3C9roAAEIDQ2yIhhQ2TTZxt19VDfL3G2HsTIvwYAbf7sU81+/fDB/acALg4tKxODDwx4YjSSgU7kF6MNNhdSS04xUrtcu3LZ2izLuFENo6lR7t25rZkETp88ock70XQAHp/5Oxtf4csBJCgPYEyfOkU+XLZE291rL9nuD19zz3RaD+v7DBzQX2prqhR0ALSk9MK0bT+DjDXgydAoWQaMMZG1V3s8YBpPfN5A0KNNDBk0UI58c8iZsjCrNQeh0hFzdQAmX5YghIUfzJOFH8xX0GkIska310/tnQdcWCaBVDiVFXc1KwG+I13ywlLgdOyHlAFP/MHSHsPa+dJsQ77a6Xv/NazA09IsixcukPKyS85EYxMLUwILsKG9aDfm5/Tv21v2792tPhMHIu6ab8/wuKJtOQYE/Hmvpbw3+12ZM2umTg6NG4wADZ7j+5Dt6VMnhNVjAURo+3lBXPPl/LNt+5xj3oDnORvQtKOUICi1wajCEkH6pEXu37srB/btlYP798nY0W/KJ+vXysH9e+UMExcb6swh/aRFw5BpC6LOmF+z48svXPqbGM5/eMuHOHvNsq62WpYvWyJTJ0/SlUOT8B8TQpn3wz2Y6S6XXdRlEgjj9oCWhJ6VTShHDXgSNpgBTckCTbpw8V/CAM/dO7fk882f6X/IoDdk2ZJFGhXF8tf1dbUa1pt+f6kdo0Ewv2b0myNk2+ebUwuyoaVEaRRcBxAIZ9Ztc6Ns2fyZJg+tra5KzJOOnqN54fxZ6d7tVbl+7YqjnXCJhFLrxw55XwMeA54OYaQSBGSishBgXnBqlFZzk66OWXbpgppympn7EaRvKfV2bmyol2FDBsnGT9Y78xpAwqTPwF8T1T4KOAFIoVG+8foAqautkfr62pS5M4pG+Do520ga+uMffl9T6/AB4ZOQanqeEuTpcPt06r4BjwFPpzJYEQ9eL5z81zpbhNeiBfOl/DI+Hsdbej6mOamY+gKg4H1oJ9rg040bZN2aVSlTFue89pLpvXVSaKoN23wy27dtlQnjx6om6cE/0/3hczzLH+tzW5vl9s0b0q9Pb12jR5N9RtTH32/bDpCZBjwd0IhFLFxtkMXnD+97sHBq12Zoegh0NMKySxdl2hSXdgae4lw23tIM0kHONcDJL+TmM1br/BqCDdA6AbgQsLRHF+2zNShHuDugc/zoYZ1P5D8S2rvXzscfB7HayoCngxvUQCirQInFlAXahgY8T48lAANQwJz10o9/qPnXABzO84/iBYDLmy8BGII3erzWTe5X3NVgAke/ORUkkI0eWg7+IQIJqh4/VDPd1wf3K31nCo2uTzb6du3pvo9sDwOehA1WoEIxkhHsvSIFYVQbGvA8PZYAmetXr+jyBmQnILiAc6zkGdWWXFeNqaFezZfHjx3RSZ13b9/SQA1ta8BLM1I3RK6Ho8DT3KjBDYMGDpA9u3amgj54Tpz6WJmn+/e52sOApwMb04R3SQ/gUgcer8Ug5PmjmXR75SUpL7vo1sJpcZkB/PWw4FLtJeXPcf4YzHRoPARqEHV2pbwsMIs5+tCBhn9uOj1H05UBXNB0evfsLpjrOPb1CJcL07D9TpSNBjyd2LgGRCUFRKUOPEStIdBJsslaOCzAdurkcV2500/GzCbM8bMQUKDmtSCBKBNxf/zDH8itG9eUl7x24kEnGz3Vrhrrdf4Q2Q3ISLBn906hnpnAKhstu9bBctKAp4Mb1MCmpMAmLJAMeOpUoKNZjBg2VPChYFYDAIj2C7dV+r5qIJrVgOwFTou5eeOars2DmQ6g0Si3tIwD6XT8McDiQQzQmfjWeJ3US10ANl/Ottn7pdPax4DnBTV8xEDstA6353aa0Cl14AE8mMzJ8gZ7dn2VMmcRSVZfW5O13Wk7eF41EWmVB5X35Qff+66UXTwvzYFZjHQ5RLvF0VZ8GbSb8WNHy5pVK3WODqCDRmbj6wXLPQOeF9wBBgRFIwRKDXhUSwmtY1P9+JGMGfWmLingBb83jYVzrKWEPj4d/DSq4bg5PWgkt27e0KgzAgpUW9L5NQ6Y/L2evj/2pje2XOPP/sx3puuEXn/s6sxzbdz7tnshWwMeY8AXwnhFCLilCDweGMjBhjnr0082pJz22fgKUFAtJwgQ8Oa1B/cr5PX+feXQwf3OvKY+I1LqRI9TBZwgBxsmvrlzZsuC+fPUp+MBMFud7Fp0G3dYGxnwdGFjF6Gw7TBGLIK2KT3gQatoEXwokydNlI8/Wq7mNQCAiLRsvKFtlZrL45J1PnpQqYk6iToD0FRL0TQ20eHX1IN0N5jiAB3qAhDiW2qoq81al2z1tGudJB8NeDqpYYtAkNqgS8YbpQY8CPXHDx/I/PfnysIF81Pr43gTXDb+8aYx5uHQbo8eVmrCz317dino4NeBDhNHKaPZpLOMKdWgWsmI0Cjr165R0AG8CDDgGvvZ6mPXkvH6c7eXAU8XN3iWwfPcnWm0X6hwKXbgccLdmchUmDc26Gqr78+dI6Sj4f3RUuBjvw3zNPf4Y18WoABcJk14S77Y+rmjkzY/x0ez+Xv9FjDhuRxrap7GBtmwbo2MGzM68A25esSdsOrp2rYLZKIBTxc0sgFCSuAU86AufuBxOdbQdBD6y5YulilvT5LamupY/Uv7yBNnVvNaEWvqjBk1UlYs/zAVBRdnzg98RB0AOMpjatu+7XMZNmSwmdYKQd4Y8BjwFDMYdOW7FTvw6MTOIGps5Yrl8s60qappACJxQpSdxtPqotianR+IxKGrV65QAAFE/D+sHWXqQ55J4lAAh3rt+mqHjBg2RBeZayQZaMp/ZOM7U/u98HMGPMaYL5wJC+ELLUYdix14eD8E/to1q2T82DGCtoLQV7NbbB+K8+kADCxZvXjhB26SaQAWTttpVZrZ+FK1HUCwtVm+PnRAsySweij+IObucD3b/XbtBcs9A54X3AExBJoNkvzsI/9VjeBltv7lSxfkavllmT51iuzeuUPX5GHNFwdIbf6NQulP3g//in9Ptp99+om8OWK4VD1+pMIdIPLXo96LgAE0I4Bh4QfzZeY707RtPEiEAwieoRnkf+N5PId72P/m0EEZNPB1zQvHee7jWpTGFFVXu97JY86Ap5Mb2IClaL88fcQVApD1Xd6d8Y6an370g+/LW+PGypxZM+XDpYs13LgQBZkPiUaIN9TXyu6dX8noN0cKYc/PAEMMPoce9636+COZNXOG1NZWK/B4v09UG3Evf8KmmZB66sRx6d+3jy47HnWvXc8zOWfAk2cdEmMA2yDKjz7DyY4A5AvbC0VMP3zNX7p4PnWO63ztF1q/4T/B9IXWc+jgAenTq4cwwRMAyUWjQKP59JP1MnL4sBQN6NB2cegxSZX25X/x/Dnp8dqrcvP6NfXzxLm/0Nq/qOtrwJMfQqyomaxIwVTTwDAPBQAK8omh/SxbskiuXrmspiBvGirEkF5vzjryzdfSt3dPqbh7W4U+78S1pDxL0lBAB+0JGh50VHOM4BEFJxKENjXKxQvnpHfPHprHDRq6WmnE/UnrauU7WS4a8HRyA9uASCygCmnQI/j8H+GIs33JooUKPJz376LLM+c5L1B/Dyi631gv586cUtAhaSfnAFjdxokaC96fdiC7AQEJbFkF1LdLe9unwCjIVk2i0Xv3WG7hNQUdwAuzm9IItXV7NO18Hsk6A5486ow8F0w2cLPzSqFHtSnoaFRYnQIEWhvmrBvXrirYJO1/NEI0vZogY/WxI4dT/i7AKxs9wApggQbtSt0eVFbIKy/9WE4eP2ZLGxS6rDDgyS5Msg0Ou2ZtF+aBQgceBD15zdDaLl+6qOasa1fLVaOLM08n3BbsM8+mpvqxjBszSj7duCFlXsM0GdYG0+/zx5jQ8DPh03lYeV+XW3DgZbnXfBsV7NaAx4RnwTJvnn31FTrwYLYCEO7euaXzYliADU2DtDRRGkomHoIeE0TJWF2vgOY0GK/FZLrHn9NnBtF0aDr9+vSWI98cUv9QIfrL/HvZNpC3BjwGPDYYOoYHCg14nIOfmf/O0c/29s3r0qtHdzl7+mTK3+PNXlF8Ei5H5BvzdObNfU+By2UZcOYz6LSn8YQBjkCCx48eyPChg+XzLZuC0OvMeeCi6mbXO4bHO6wdDXjyrEPy7Cu+wxitBN6r0ICHvgUAfLDA/Yq7utT00cNfa7hz0r5HO8I0RhDA3DmzZPmypVJHHjdpjUypQz1oP18nQIfwaZbQJnloGJCS1svK56GMM+DJw04pASFdjMKg0IAHkxd/AIMAgEEDB8g3hw5ISzBJM2kf8f5ErH24bIlOpkWDAjDY8o9DT6PmyOPW1KBRcN43FPf+OM+wMnkg8wx48qATDGhiCaV8FxiFBjyqZbS6tXBIsPnVl9sVhJIARbhPMKctXbxQprw9MQU0ulyCroeTHXg03Jw5Ua0tUl31SNMOffThstScH9N4ikxOGfAUWYcaiL0wEMt34PFaA9FiThNpkvraahk1criCDufU9xKY38Kgkmnfg4G/j+Shs9+doSHURK75iZ08F0BJp+Gv80xMa1wnIHtQ26EAACAASURBVIElq9GaPF2tk/H1M+2X3p4FdWzA8+yAKKgOtAGZNwMy34EHk5oX5jj/62qqdAG2NatWavaFpHwPWDjtqEm2bd2iWQkw2bnouOzjigmiCk6BGY79utoaWbZksZrpDGyyt1/Svsq78gY8Rd7BBkxdBkz5DjwIc4Cnoa5G59i8NW6MrF29MrWmTlLhRB43wOyLrVtk+NAhUltTFWQ2aItea4+m1qW5KZWCBxADdGa+M13nEhnwFLlcMuAp8g424DHgCXgA0NG1apobZfrUybJ08SIV8j4arT2QaO88OdcO7NujkWcsC4H2Q7ACGk9794TPU5Y5OYDMmtUr5e2JbymQaS63wPQWLm/7RSSrDHiKqDMNZGIJvI4SYOGvcvZV42lpliWLFgiTLzkX9mN01HPj0tH6tbqQafYVYFqaZfHCBTL//fdSAQDe/BZFl3K+LLSOHv5GXu/fT+7fu6u0tA2CnGmY8tLpeR8T57Vtml1uuI2frJc3RwxTjQlfEO2Yfq8dF5mcMuApsg418Ok6oRUsG+ATgKqgl1bNTg3weOGr/owMzvXOFqYIegBCw6MDIJj//lxd+ZPVQ5M+n9Q10OO9jh89LP379paKe3di+XR4lvfpKHgF2g5mupHDh2okG9dpQ0Kqk9bNyheYHDPgKbAOM2DJG6HkIrFanV8Df0UwZ2X5siVSdumC1pMyKnBfgOlINZwnLRqSDGisXrlCJk+aqPXJJe2MCxpolvKyi9K3dy/RpaYBi2Cl0ijhT30og4kNUDx0YJ8MHTxIaqsf63nVdNCYpDVv+jjqnex6jvLTgCfHhjMAKHnhAKDoOjyBcCeRJUkx339vjpw6eVzTvVRXPVZAApS6WkgxrwbAQYvYsG6tTshE0/GaUNL6QAdAffknP9IlvjkmwADzWJz3U00nyDq9f98eGTigv4KXAiSA47XCUBaDpHW08gUizwx4CqSjDOi6XHBHCTGAh7QufKGXXbwg06a8LW9PnCB/+9d/pT6LyZMm6IRK8o0hpKPodeR1p305LeyTDet0yeqaqkfOlBX4auI8z/uoMBeyPAJ53K5dKU/5eqDhAeUZegGY6HWWN8CM1tosXx88oOY1UvSQ9fqp+z34GL93Kb8803ed3f4GPAY8Xc50nc3UXURfHeF8nTNZMpiPgpZBjrLyskuBkPf5xzqfz1jOAIDwghxBv+urHTJ08BuSi08HevAG5sI7t2/pMgkXzp8N3jVag+P5CriB6Yx6sWQ1q5neuXVT241zALf5dTqfP/JqnBvwlFiHd5FQzism78R3RrCiFSBA8V2wXbTgA7cCKc/VNDCYkTpf41EhHvhcAMW9u3eqZlF5/14q0CFJv0CPQAJ8OQP69dEF2PANEayA5hKHFu3jzZE3rl2Rfn16KSgD1Ph0HOC0arvFoWdlikReGfAUSUd2onC1wZ6ZRxzYuC92/boPzE6EK6PxcB0BS/uhiXRFO2o9Wlvk2BFCnfvq2jqAB0CU9PnUH3MYIdOnThwPwp3bsllH0eN+Xx8Ai8CG61ev6EJzXPNBBJSJomXXk/dfXreZAU+RdagBUJcJMSdUnSBmX8Omn7ToPJ5rVy4r8PgynSEEvGD3tNEeOHfuzGmNOnv4oFK1MK7HEe76DoH/BzpkImBZgh3bv9BMB9DhfCoYoB1eI8pN5zQRbs5Cck0Nsn7tatmwbk1qeQT/rDj18u9n2yKSVQY8RdSZ7QgCG7Bd08de2DKBFODp7HZ3YdouqgxzX0NDnVwuuyivvfqK3L55I/HzyRiAFoKG9OhhpS6TsGXTZzpPBzNinPdRIAmyTDfU12lmgtUrP5YJ48dp/bgeN7NBnOdZma7h7Q5vZwOeAu04A5lYgrDDB0yWdu9q4PHmO/WXtDbLpQvn5fvf/Xs1Z6FlJH137mmsr1XgGTVyhHy++TM1FaLleJNhNpo+rBoQBGCgt2XzZzJuzCjd9/X022y07FqRyyUDniLv4CyC0gZ3x/Z9VwOPBwOizwhx7t+3j5RfvqRaBmCRtH8Bivq6GiEMfO3qVc4Xg4+KqLMY9CgD4ABAaE0uY/VQzToNoHHNg2XSuln5juXVF96eBjxF1qEGNIkFbkcNws4GHqUfAIDTKBpVkJO2pnfPHurb0QgyH8YcwQteKwLAAA3AgXlIhIP7c7SNZl+IEZygGg/Rak9aZNvnm2XwGwOl6vGjlJ8nrrmuo/rD6OSxbDPgyePOiRAcNrDyq+86G3i8iUrDt5sbdW7Og8oKzZl25JuvQ3NhWMkz2ifDnCNoAkBoTfPmvqcJRNnPRTPhHrJfHzt6WIYMGihVVY/0GD71IGc8m188+8L6w4DHGOGFMV+RAWvnA0+TppXx2ghLRA8fOlgO7t8nTSmfTquWiQMcaDk4+gGaD+bNlVkz31ETGaDFM5LyBfQI4+7ZvZtOENV0OgY6idsxabsXZHkDHgOegmTcPAStzgYe1RqIOmuoF9a/6fHaq3Jw/149dkk/HegwWZW6RPWrC79ulhXLP5RJE8arVsJ9Htii7k+/fubUSU2pU1lxT2l5nw+TTU3jMTnzFL8Y8BhDPMUQeSjQC6V+HQ08XnDjO0ELwWHfrKHODzT32ldfbtd5MVyjrLYTJrZ2zGyUc2XbNJqNG9bL2NFtUWeeThyNx5dle/1quabCITsBx/h5qI+2SZCnrVD60erZBTLRgKcLGtmEeeTXdzEM9o4GHjdPxzn+EeaADgEAU96eJJ9t3BCYxYI1bDzwZOE1BZ5gEijAsvmzT3WCKCY7wC1pHxAsQJ2IpOve7VVNlCpPWl0wQpZ6JH2OlU/eN3nfZgY8RdipNugTC9GOGKgdDTxe68BfA2hgrhozaqSsXPFRaslonycuDnAAXvyh++UXW+XNEcN1GQd/PmkbEEhw984tefknP5Yb1686DSfQtqhvUnpWvoRkkQFPCXW2AVKnCsOOBh40Hg86ZBWYO2e2rm6qiToJW0ZL8UlIY/StmuqaGmTf3t0alKDLEpBVO0iDk1TwP3xwX30658+e1uAG1ahi+peSPsvKF5mcMuApsg6NIYBsEHdQnwdf9WgQ/AECQpSXLl6oKXMwRfkvf788dpy297nOPE3ueW/2u8Ky1fV1tZp/TYU89LOY2Pw1DwgAz9eHDqgvhjBs6osWhcYTp14AHrTQssh4zdwh6EEjXFdoxaUZ57lWpoP4NZ9kgwFPEXZqPjFYEdcFIYwwd6Yq57BHSyE79ZXyspQmoZpLTOEOTafF4LtpkZqqx7J08SKd2Ilpy2lBbkmBKIHswMDVi7xpp08e1zxugE4uwADAoHmxyipr/BzYt1eXRyBsOk74dlR97XoJySIDnhLq7CIGgRchtBDEgIPPCs0Mf7ScxQs/kEsXzzstIIjwiiPolRY+mCDXGRrGmlUrZcb0aalF03gG7xpH0Dc3N6oGRgDA2TOndC0cfDI+fU3SNgNkAZ2Rw4epj4h3UvMdk1Abk+eGS/p8K19EssqAp4g604AllsmoowSYT0+DZnHqxDEZOKC/DBzQT779J38k3V55WbM7T586RaoePYilYSDInVnM5Tr79JMNMvGt8VJbXZUKJuC6+n1i9LUHBjJlv/yTH8ntm9cVDAEL6py0HdB2+vbuJds+36KABn3VqvwS4DHqlPSZVr5I5ZMBT5F2rAmBxII1qZBD8HJPc5PTLOpra6Sxvs5pPBfOqVmKyZ4KFllMbWpeU7+I8wlhTvti6+cKZEwURcshu4ADJT8X51m/TPg5HhRvXr8mvXq8Jteuljv/TAA4aGtR7wtAUQa6gA7JQwnBBmyYFOrrE0XHrpuMeYYHDHiMKZ5hCgOtSKGcqc0SR7Xp3JnWVJYBBDoCnrxrw4YMksePHrgZ/+LmxmR6ZvgcgKX/YELpnVs3ZUC/vnL86OGcNByi3QDOuppqnTtEhgPVcDS0O7nGFK6r7Ze43DHgKXEGMJDJCWQyCc5cgMfd4+a84Cc5/M0hefXln0jVI6fpoFV4n0+mZ4bPoYHhY+JPAMFLP/6RnD19SjWyutrqxO/JPQAZms6SxQtV60Kjg35bbjgbP+E+sP2Y/GDAE7OhTEAnFlylNghzAR4JtB4CAEj2+cbr/aU6WErAgY7LXICpLao9KY95jCWvoXP460MKHAAamkrU/enXobXwg3nywbz3la4Lngj8OjZBNHF7prdvSR8b8BjwlPQA6MAPiqTA43xErRoscO7MKRk+dEhqyeow0DhN5lng8X4a9bkE6+A8rLwvgwa+Lgf27XFh3gBO4I+J6mfv0/Ga04L578vsd2fq8gteMwPcoMNxFD27brKlXR4w4DHmaJc5OlAol8IzkgIPgIHp6uKFc/KTH/1Arl65rGChmk4MDUUBgrVz1BzXqHN+WLKaqDMPEEnanfBtfEzcu+rjFTJ50kSNpsPXk4SOlTWZEskDBjzGJJFMYgAUS/DmAjyXLpyTPr16CNFnKcAhmCAG8BBNRznMdDXVVTJh/DhNHoom5Of7JOlbnzpnw/q1Mm3K2w50AMeGOg2fTkLLyppcycoDBjzGIFkZxEAnFujQhkmBh1xnzPe5fOmC86Fo2h03Tydun2CSq6urkQnjx8ra1StDdJLzNSAG6Lw5YpjOPWJCK1FtZJyur6uJ3Q5x627lkvdR0bSZAU8Jd76BSlZh6nwwbh4LWgTHaCXeB5IuBKKAx5vGEPAV9+7opE4WT1PaIWe9f0Y6fReI4PhVo9eCurw98S35+KPlWi/vG/L+n2dohPrcvx/1wb+ze+cOGTZksE5YDdeBff7ZaNk1kyOJeMCAxxgmEcOEBFex34dwRygjoBHkZHN+cL9Cz3E+/f2jgIcEnwhwEmyS5eDUieOqTUA/jmCnHKHVmMR0UmlTo7wzbaowvwZzWHp9oo65x5vrThw7IkMGDRRWD9X3jjHBNIq+XTfZ0i4PGPAYc7TLHCUEMpnagDkshCLjcCek+L/9/rdk44Z1Kc0n/Z4o4AEwmBTar09vOX70iEaz+eWnvfaRTjN8DPhRp4a6GgUqIs6ol9dYwmXj7HtNiwmr3bu9Io8e3Nf35RmZgDUOTStj8iQWDxjwGKPEYpQSBCHMTyxzgIaCSey3f/PfyaZPN6rQzySYo4CnsuKuZiTYumWzJhbVdmdZa6LGYqwA6oGCZ6PloO0wyZNjriXtR+45duSwvN6/r1TcuxtMPnWLxcXRwJI+z8qbrEnxgAGPMUOKGUoQXLK9u5rAgrVl6mqq5L/9/n+RT9avazdiLB148Ld4AQ5AjB87WjZv+lTNW2ENBye+XzsnvT4+YzV0fNTZ+nVrZNKEt1Khz1rPGD4Yv6ic0mpu1KWqB/Tro2sHcY6/L5MJWNPrZscmO3LmAQMeY56cmackgKpVBXJt9WP57//1vwgZowGYTBpKOvDgP2ENHXwp48aM0rkxOucm5oqfquEE/h+AgAizr77cLn169cxJw3Fmulo1pzFn6Iff/57OHSJijSUUjA9MFnQZDxjwGLN1GbMVJFDlDjwEE2CqmzZlsnz04VI1ZQFCgI/PEpCt7X1wA6CD9kTUGQEAzNnx0WvZ7k+/pv6k5iYheWi/Pr3kctlFrQd1Mg3H5EA6v3TqsQGPMVynMlhBgk2YJ3IHnoa6Wpk7Z5aGOqO9MNETs5j7R4cn6z1BQMGeXV9p/jWWSeA8Zrek/Qbg+TDus6dPqiZGXfxzktKz8mE+sf1E/GDAYwyTiGEKHkiS9ze+j5qqR/Lf/+vvy4Z1a1O+ENrNawoeTDBnfbh0sZCRYMmiBTLznemp8qky+HQyBBNwPdwXHOsyCYe/1uwGjx9WKmiFnxsun2lfnxlMTCUcvHfP7nLi+FE1t1F370NKf3YmWnYuOe9Ym7XTZgY87TRMCQpYGyRP84ITzM7pXltTJb//rd/TcGrO44TnT5v52f2Yz9BE5r8/V8aPHaNLVrvIOKfpxGlfDwT4izCnEXX2xusD5NbN60+BUhxaCjpoWK3NOgdp5PChsm/PbgUdTIBxaFiZp3nC2qOD2sOAp4Ma0oCq6ASZ1wIAGpaf/v1v/WcFHoSPN085gGiU82fPyMjhw2TIoDfkT//4j+Q3/u2/luFDBwtLXxPRFifljAcyF1nWoql0CCS4e/tmStNJIvioP/V89PCBTlj98ottSsdFtT2tXSWha2VNZjw3DxjwGBM9NxMVKej6EGOA4+Txo/Jbv/FvZfrUyXLv7u1USLTXKqqrHun5JYsWyu/9x9/V8hV3bwvLFNRUP9aJn1HtTDABE1Yx12Gq69u7l2auBjziBCNkos+EVTJWb9m8STUoXRKbxeJirO+TiZ6dM3nRITxgwGOM1CGMVKTgQ9sQ9YVG8/WhA3L82BG5d+eW840E817QiFjeYMumzzRsGlPbtSvlqgF6AU+Z9HZWk13Ir4NJjD+hzv379pErl8uCQATmA0XzqdfQeA7P5XjK2xPlk/VrFbg49mDKNr0+dhzdxtZGHdRGBjwd1JBFLHhtsGXmEfXpBJNEDx3YL0MHvyFVjx/J4oULFDyi2g0NBwDwgADAkbG6V4/uqvGgnTgaLrIuih600I4oh9+JCatoYJxro5X5XaJo23Vrtw7lAQMeY6gOZagSAmCAB2GPw54MACxZjcBftmSRXCkvi9QoMKkRYu3B5/69u9LjtW5y+iTJQz1ftupkVa85tZ3319u2mjg0yDQ9d85sWbp4kUbdca9Oek3RbLsnGz27Zu3UaTxgwGPM1WnMVeSCDk3i6OFvZEC/vvLoYaUKeUAE4Ll+1ZnasrWtaiHSqmYwNJ3X+/eTrw8e0GOXScCBji5dHWOpaW+6Y8lqzH2N9bXq1+F8LhNOs9XdrpnceC4eMOAxBnouBipycAm3jXfwe3/NhXNnpWf3blJx947zxQQh1mgaly9dDGktjsdcSp0285oDhEYhVHvwG6/L3t07RaTVLUvgnf/4YtrxxyjQBOXwAWGq+3DpEpn41njNJwcI+mewH34X27dx/0J5wIDHGPCFMmABAZeaxAI/CjP/e/fsIdeulquGgtnMAUuLThy9duVyRkGPOQw6/AEyouFGDBuigQkaHADwBBM+o/qFXHDc431N69as1vDthvo6jY6Lut+u29h/YTxgwGPM98KYr4BAhzYigzQ+mZs3rmmo841rV1XLUEAiYkyDBdoHHq8pYaKjLIDBXB9dJiG4l/PaH9KaEbjCfQXoAGScI3Jt9JsjpKGhTkjVQ+BCuKzt2zjPKx4w4DGGzCuGzGMwAmBu37qh6WvOnTmtwEFKG6+9RAGPAk4QxVZTUyUT3xona1avVHMYi7uFTWrtLZMQ7itAh+fv2P6FDBo4QCep8gzKMPcoXNb2bZznFQ8Y8BhD5hVD5hHwqOmLeTwKLi1y++YN6dn9Nblw7oxgzgo77CnbHvAAInoNDaW5UQFi2pS3ZdmSxTpZFLBQDcXP1WnHH+Pr4wEMWvv37pYRw4ZKTdVjNbvRl34Oj/Wrje285QEDHmPOvGXOFwxCasoiL1tLk9yvuKdRZ5pgE1NZBnDIBDxaTpiH45z/HM+b+54s/GCeghe+IQco0aY1AMcHCxBIQF0AwsqKe6n5O9aXNp4LggcMeIxRC4JRXwAIqZBvaRKyQjNPZ9+eXQo46tSPATw+4WdLAFQAzAfz5srMd6ap81/n1wTv5cAnOy8CWoAP2tbRI9/onJ8HlRV6jvPWj9nbz9onj9rHgCePOuMFCFcbjO33v9d0Bg7oL3t371KwcOvguLk36W2XrvFwP2ABABFIwPya92a/m1oHR564jAQAUzqtTMfM7SHi7dKF8zp36O6dWxpcgPZj2abb78dMbWnnXnB75Tvw8FXIl6eaLDpAMIfDWZ+X+VSoBMIFIRPnqzXTM51wclFIvGuutHwbIeigk+lZSc759/F0/XESGuGy4fvD++EySfahwZ/6PQ893/6OnqOFzwUH/aQJb2mos/erUD+yE/g2CdfX+XGaNZyafGuU0aSfrc3yyfp1MnnSRKWZqX8y0gvm4fAMpdXUIOVll+TVl19KZax2YOfCs8N1Ce9zr6cPAIavJdn3dMLjUenim/L+qZhj1NPw/ebrl6Q+6WW91udppl9PcuxTDHlayKEk92vZQCt27ebC7b32m/h9Q+3r6wQfpeiErsepp/M7unfyssK/c7v387xgzHEPixLyfF+f9Pt8gEz4uufXHHBHfir2TTGZML3C4WMq7V8u1cg50nXzI4JFuIIZ4+FnJd2nbt6R6+uZlAYD1ncG7+ffUTsrgzknir57x2adiAhjRpWPuu4HXNyv8mz00BZ8IsxcVtBMpw04+A8JL3TSy0QehxZl0/7UpJ2tGi1GrrM1q1ZqAlDqHUXLD1wWgmN+D4OT+z7ZsE5GjRyuc3YISKBfMNdF0eOdqFNdbY1uAZ0+vXpI+eVLsT4s4Cue4flK+zLwNyUFCqUThHx70FAatAt8lgOv+foAzlFtEXXdBXq0arviNwsHfkTd29516veUcE8qd0JtghD27Ubd/Lhq79mZzofHdko+PEf78wxoMo4872Z6rj+nsokwf2mVDevW6PIfZ8+c0vfSsR1KeKu0U7K7WT/WMBErL9KOOfy6DHi8IKDD/ODzjZDLFjo0kG6DBs+Fjr+HRqwn4im0EqW/FmerXwQB8FCeesGQ/otamSsBs1NendXNjYJZ5nm+bn39lSlbmnV+SNL6eBp+C63Ghnr9s+/P57qlPrS/789c6HjhTH0Q9AACAwRNZ93a1SmnPWHLUfS9kFq0YL5cLS/TepGx+s0Rw6S2plrBwwOkn3+TjSZl0bp4T5ZiIHnoxfPn9Jh3znYv1/wHDft8OACCmOWglwtvwO/KA9Kq41HfIfgKTgo80OH9EMLUKc77ZHtfXzdo0YccZysfdY36wKvqzwu+8qPuyXTd8VdrykwLXcdnycDW9yVtzv1kQ1fafKzmMJZof/8H+Jn/FdUHnr+57/jRw7o+1f/zf/+OnD0N+BC5+bRWSNDLyhUfyayZ72jyW5YHmTZlsny1Y3sOsCNdp/Hwgi5stVG6d3vVMX0CQZzOCDAjjDl86BC3RspzMifCatXHH6UyEVPf9GdmO1aG0S9QBjTmmLWyY/s2ZXbuS0qPe2AehAyJJr/Y+nmi+qTXlecjoCru3ZFhQwbl9JUWpsk7bv9iq/7ZD1/LZZ+1bpgLQ5/m4uOAH7yAqq1+LG+NG6vpawCOBfPnaVvSBnwMxP2CpvzyZUvkavll2b9vj86vIY8bwoJrDF7dxuBjBi7BCLQ/ms6JY0dV4NAncerjnsU7tkh9bbWMGfWm1AGAoSzWSdrdCT9o1cjcObO0LvipvGaQiFYwFlk+Yv7776X6IQmNcFn6kT7kHeEH36/hMkn26aN3pk2VB/dd8AbHSe73YMCWe5cuXignjx/TfY6T0uPZ9DttffzoEfli6xbHR/BTDmZAXwdonjl1Qt6d8U5kmwEstKs3HzOtgOXgf+e3fkO+OXTgmfvh+Qvnz8q//+3fkt07v5LZ786Qbq+8rNMIckGe2BqPfzkaTRlB1cIg91SMfRqUyiNMf/j977mOj3GfTsTLVC5YW2XggH7y6EHlc9PjnUiBz6RCmF3V1UzPzXIODQU6tBVfBzt3bFfzUYqZstyb/p5tTNEiZ06flC2bP2szgag5JH7bQ5tZ+tSLturbu6f7IkpQn/T6weRfbPtc/+ynX096zBfUsCGDA9NVQ2J68BVtxr/68UOlxRcZQpVz1MdppQFAPfPuJPFs+z9pdXnWli1ZIuvWrNE1de5X3E3xsAqjEM3094XX/Vih/yvu3ZXJkyZoIAEL0bWZtoKF3J6pT1r/Km85zQehzNLXaNN+XKU/P/I4mCNEuyGo+OrG7OIEEua2tOdnOWasNDXUyY3rVzXKL5ex89TznrQon8IPKY0ny/OfujdDOeozYfxYqXr0QPvEaXcJ3i+wPsBHtBfgevrkCZdpQtCAEvJ/ADzwx6kTx+XzzZv0Y1W1EP8Rl+E92ntP7TM+LBvq5NTJE+p/9Lze3j2p86pNOyC+fu2K/MV3/kzBZddXO9y4CT6akR28P8E5AI6fdK1tmQPyxAYewlD5P7h/T/9k4OULIsn/3p3bqqb977/5X1KpdJLdH34WX8hEAvXr00tu3bgulRV3E9UlTMvvz5k1UxkBIcGXqT8fZ/uw0r0L9WKfCYVoPbwnbUXyyTh0fBm+kPm6ph4A2OqVHytdaLln0Q/x2+/+vTvaRixI9spLP3nu9kII4zPhz36SumQqix/ljdcHKK07t28lpqe8WVmhjnrS4Py33/+WzJn1ri70Fqd+D+9XSvhfWUE7P5BpU6bKX37nz4V8bbQhTlhWI830DuFz9B38CdDDA5jVfue3flO+/GJr0H/x+07pVlbIwweMvwq5ffO6al9sWQo7Ka9CQ/n0wX0FCxKR3r19S/nNX9NnxuQvTIe8IyabqZPffm7e4tmkMCLDN23oZE7C9kqrO0AN7xOyTn2TvJ+W1fF9T/vunWlTZP/ePfrOjNOksofyrCYLX8IPjG3qxbimj70siVtH2og/soFpAlgOkspX+JRJ1WWXLsif/NEfyj/9J78mn336iVpccGnwAQDobvp0o3z7T/440JTQ+PFBJ//FBh7MFfz5Onp3xnTdYu+L+0c1e/+9OVr+W7/3HxPfn/6c2e/OlJnvTJc//p9/oMkUZ82cEbsu6bQ4ht4Pvvdd9QdAl+NM5do7R7swp8PVY4a89OMf6cDh/HuzZyWi5Z/haeLMfu3VV5QG7ch1rvlycbaUf/+92dpWqNSeTpx7M5XhfjQn/s9LC/q0+R/94R9oHdnP9Mxs56jD9KlTtK3hid/9D/9ew53pR7SebPfqtRkzZVboP3vmu9KnZy/5Z//k12Xs6NGpNuc5fKBE0ps5Q3kIIUUE3L/91/9S/v5//41wnAs/eN6iH6dOnqTCgfeiLkl51dedcQw4//VfVGnaxwAAIABJREFU/aXa66HFOX897hYa7j0nyN/+9V/lXB//PNoYbfU73/4THVO5vp+nx5Z3ZAXXeXPn5PCOM3S80Qe869/97d9oLj3WSSKkPvycOPtt8mWGJpVFg/D35TaWkAnw2wwZM2qkfOfbfxqP50Pye8b0qbr8Bu3+rd/7T/JzP/PT+o4AC9o7fmZ880wj+Hf/5l8paKIBqTUhOe7E9/Hga+CvD8NG6qNf4m6ftGh0EWasH//wB86uHffeDOVoDExbb7zeX308oHHiOoXo0sBLFi1IpcBH7U1Kz6ujmGHWrl4pu3fuCPwBgQ8kAU3amXeCJv6ATZ9+4jpZI7cCJ2QCeqj19XU1OjOeTMwcJ32/cHn8Els/36x/9VEkqUuGsnzxjX5zpL5vLn1Je+FD3LhhvS5LMGjg60rLO5TDdc+439IqEvqfPX1GgeetcePk6pUr6vTVMRBMCchII/Revv+qqx7L2NFvysoVy1V44b9ypplk7e/9aPADvh3aChOnjgM1yySnRx3x8SBQSSWkpsHApBL1funXqRcL5iFUcxk7YXr0P++ovsgggCh8PfH+kxYZP3aMahLwQ9L7aSfaH7MmWz7AyWhOPVP+yFDfR9FXWoFv6NiRwzq2VXakzGzJ+1JptjbLyRPHFGCTjm/lhboamTB+nPzyP/oFBX4WQ1RNJ9B2yK6xc8eXak3YtfMrbQM1X3Ym8OiLBE5UGil8HGcfm6MiZ1ODfO/v/y7x/ZmegYBn7ZPa6ionlGM4eTPR4RwNv/CD+apuusl9ubyjc/7CkGtWfawmMuimmDNB/VKMKK3KTJ9t3PA08CBsEtBDGFAvVPzePburUE5yf3pZ7OYeeNSmn6Au6bQ4rql6pH4Z6pjpeqZz2kaBT437GBQsSwCIYVrxIZ/wXab7w+eaGhBIT6S5sUlu37gpfXv3lmvlV2TZ4iVy5XJ5ir/oz/B9ft8/g+vUiz/AgG+B5KGsSIqGQj2TvKOnD13vQ3Q+nmH6IeGf5cvF3SKo4Ak+RtByNLIqGAcKHAn60wvkm9evyYzp03JykKfXG5MmfUlbtdfm6fe0d8z9BJs8fvjARWy104fZ7vf9Sj+jNZ055YCHe5LWj/LqI5RWSQFPEO7dXh2ynVd6QZAJ4IBmB19ku4drvgz3O7/oIPnlf/SLqsW5PnV+ndUrV0j/vr3l44+W6z24N9Ag9+zaqSa3HHAnvsYT9RJxrvOCvCxRE3HKR5WBFlEhbGmoqPJR18sunpe6mqpgAa/k9NShFzDizetX1ZaMUI4TtZSpbvoV09yoTIH9NVOZJOc88OPQVOdiAuGS/hza29uWO6Lt0SYYzP5rMP156cd8iXlthqUDvty+TRNsYqvmPc+dORV8pQJk0bnTWppJINqgoIOJ7eTxE/Kk9YksWbhIrl25Gtn2gAx8yLOxiVMnzE+sYIqWj2ZBWLbyagyhkP6+HHthRVAB76cgFgNUM9Giz9x4bNZEqbS/louxhEM6Pd6JwAS0lHNnT0e2Vfr96cfUDZrwaa5jJ50m0V4aIRdznkv4fhXsGtHmTEssDEgwjG+zpEDtaRNRBhgytlV2+LZHe8phbOLor3r0UP2JyitZaGhQAMu1NzWqP4iVd3/1V35JFi34QC1TfhFD+Jo+ZWFDne7Q0qRzz0hoS3vqR1QOyBPbx5NLQ4TvcRV0C2P5lw5fz2Ufmn6w53J/+B46itBUz2Tha3H3lQFhmiCsmi9Ul0oFoZEQyABpvv79JMEsTBSnfrQV78ZgoV4qLJ6TJjQ6gg71h44XhGyj3on38X8cva/376uOfwS/0gp9icbRyNB07t25Kz26vSYnjh2XJy3waot8uHSZlJdlXgguXEdti4B/2EeL+OjDZU7Y6deom9xH/cL3xd8n4ox2cSDqhJ3bzwX4VePxZjX4QIVdQNubfBLwB/XxbcA2/ntlFrJuLEIziEZMUJf0Z+uYDvVN+vWoY8+PtH9qPwAHpZ3wfX1/Ka1AVrj2px/anhFVL38d2aJ/HTekgIrHY5RDFrC0+i/94i/Ip59s0DGlIdaBRuw+nBtTMsN/7NEWPlI2B9zpWo3HCb8OmolMY+tclw4Ufqmlh5N3vmOCti9rLxQ5z75nkiRbZfSA2T3DJ7k/vayvk6ebfj3JsddMVNjkIKjSn8X78XWbYuYIQUNZ+v/I4a81RJlIJeriNQ/9atZZ5QBuZuEWrsPjhw9l5LDh8vXBg9LSRP8/kdYWFoJbLOWXyyP7j7bl+fhKMNnidNZw55YmjSBbv3aNZrKmLh3x4eX7MA6oht/T79N+1IM6K2/RRggu3Ua3l6fjt9SD/sgdWNue6XjBfVgAQNTRPyfXLaHGnm4uNHwd9D0VHJwG6kEkKU3ff7T3M+0fwfvtPYv+jP2Owccd/UUk3ZFvvlZNBxqejo4hBTXXB/C4p6911g/GTo5qa+9l4573DM4MYl/puPe2V87TbO96kvM0MvVKMVhSZg99pUMHxgrT4jhJfbyG5O973jZz7+bqAGgkqUumsvqOwdfZ89YN+rSVr1cmoHZM74STK9ukzk3szXdu3dQvPr7GqAuDydeJre+H8HtQBmDgHOvfvN6/v3z5xRfC/J3mJky39GGLMI+HiZHhe3192XrA4RnQZJkEogfVrBPYzgkyYVY4gTDwWab3S6f/zHH4CzvgtfA7PlM+Qnhpu4TroqATAE/EvZmeBT3f1p53M5WLdS4QxvADfRrrnix19nWDDv3EcSKaoXyJme7NxF/Z6Ifrk9JUfPtneY/2aPJ8XweveWaqZ/h+rns+hG/9PufbaLl20voG7eavp8ojJ3P4dZmpLfzStt/2dWdtEa8t/AcLGhFfZPhLur3ykuDQRstJ2o4MHEwNzLfBia0T5oKJfVzTL9LW9pe+ZgBSF7b6ldjcpBOQCf31QQ2uTi5r9Us//qECD+f8wE5aZysfj1esnbq4nQx4urjBc/g6sUGRWx95Exrb61fLUxOH/RdY0nYlXxtmMfK4bfxkvX7xATh8/al/IbBxE2LP5NF0+vqVGOTaAkjWrl4lJCJlGWwPhK5uThMgTRRzngx0cuv/9Pa34zxqRwOePOoMA6VnhPXzCAtAgT950/r16S2XLp7XUGC0DbShpLQBntFvjtD5NZ62+pcIf8ccHAE8XuMBSLZu2aTzaghV55hr/KkTW+rYs3s3DYDAfOSvJa2zlbfxnZc8YMBjjJmXjJkDCAMGaq8OQmux9d+5fVNBh/BO71RHiHsNI9u7KwAEzlHKM9/k4xXL22zbASCgyVA2HXjCYOHt6GhGu3bu0PQ1zIfR+wIbud/3dere7RXp06unvlOYlr9uWxu7BcsDBjzGvAXLvGng5DUHNAQEPFFrw4cO1klruZir0IqgBaCRJZpJgAAQtDJFJj0DPDqPxgVAaJ1amuTAvj0yZNAbOmE1qt0xtTEJj3ujytp1G8cFxQMGPMawBcWwaWATrrvTeNy8I2axY17b+dWXqjGglYTLxtkn1JfZ+T7fIKDDMwCeTBpTOvB4cPKaDOGnTLojMWOc+hjw2NiMw6cFWcaAx5i7IBk3AwABCABD1eNHMmTQQNn11ZcKEIAEocpJ35NAAgIApk+d7PwuweJ6fl5GOr104AFw8NUwgZOsAf379lHTnwJYjNB7Ax4bm+k8VjTHBjzG3IXKzN4ExaxpJjdiXqutqZKxo0fJ9m2fK1g4s1i8eR2AFm2h9wRJW8m8C03Oa1RaMGHYazPhtuMcvpwlixZq8kun6bTIpQvnNdcdSy94TSejxhSaD0EdCP3u8Vo3DeE2H4+N0zCvFfy+AY8xdKEysWoUmL6C1SzJdcbqshs3rEsJ+ETvJq2qGSH00XRGDh+moc4co81E02rV9EKsNlledlFTwjNn6OWf/FiuX70Seb8HF96DRJBvjRujyRsPHdgXaE7Gq9F9YG1UEG1kwGOMWhCMmtG05jIOEMqMBsGCYiyQheaSy+x1AAw6LBk+auQIqal282s470EhW1upWU1aZfHCD4SVGVn8jag0NB6v6WS7n2vUHa2J+rOvKXSCzApR99p1G8sFwwMGPMasBcOsaeADGAAU/PHDsIIrvhyEPKHKSd8LzWbH9m0adUZEHCY8QAcTWibTWjp9Dy4fLl0sXx/cL4RDnz93RsEjTn185Bt0AB+AjC3vSd3Sn2fHNnYLlgcMeIx5C4V5vdaBEPZaDcDA6pys8ujP8z6AUdR7QYMy3EfAwJFvDumkTdLXE9Hmr7VHL1wf1vPZvesr2bN7p6bT+YP/8f/qyosH9u1NgVdUfdLBBfoKPAY6kX0Z1bZ2Pc/knAFPnnVI2le9DZi2/kEjQAsAbFxAQaMsXbxIlwDOJWoNzUJBp7VZTp86IQMH9NeoMzVxxUgTDzi5DMYtuj7Kp5+s1/k+/+Hf/7bMmT1L1q1doxoUy//mYvqzvm/re2uLImsLA54i69AiBi4AAZMXQhzAWLniI3l74gQ1ryHckwongAcfCksSv/bqy3LrxnXVpDwYRdHT+gT+F6LqMKexui2rtbKkMwEJai5Dq8pxmYuoOth1G78FyQMGPMa4hcK4AIKan560yNrVK2XcmFFSV1utIASAJH0P6JVduqBhy6yMCnhAQxe+i5EG35vCKA/wjRo5XDZv+lTw8ZRfvpQKSFAg64DVbpO+n5W3sZ23PGDAY8yZr8yZCmEOosrwd+C72bhhvWaIJmmnd/oj3KPegzJoKQAG2s61q+VqXrtcdlF9KQoQHtwyAI8HGp7DvtO+HOhMmjBew7ihT1RbODt1nLpF1d2u2zgtKh4w4DGGzleGVn8OeddUW2hV0Nix/QsZNmSwPHpQGQk06e8FKGAOQzu6eeO6ajpnTp2UFhZki+HTwcekkW7BAoDQg9bEt8bL6pUrFBQBR5ZFiDNvJ71+dmxjsWR4wIDHmD1fmd2lpnH+Ecxghw7ul149XtOF2DBvJa23AkVjvSboZGXPUyeOS31tTezll53mAgC61RvRvlg5FKCBNtfRhJYtWSTlZZcS1y/p+1h5G7sFywMGPMa8+cy8KsyftMjRwyTY7KOaDlpLLlFsgAPJQwcNfF0Ic8ZUx7uThRrTW1Q7UBdooNXw/A/mvS/z35+b0nQ4D1gu/GCeXM2wEFwUfbtuY7FkeMCAx5g9X5gdbYG64Nvx2gMCHu2BtDNkdfbCn22cerM+vdJsbZbqx4809xlzbRQkVENx2QIAlHR66scJ+XwAFff8Js3HNnnSRAWtVDnW52lB41msi8+l07NjG2vGAwEPGPDYYMiXwaBgwkz9QNgj6Ek3Q6LMe3duPwMMUfUm4s2bxZgUyuqhLJOAzygT0KTTc5kD/KqgbWls1q1ZLQQToDGppiSt6vsBdLyPJxxckE7Xjm3MlTwPGPDYIMiXQaB+GxXibhnpa1fKpXfPHqrx+EzUSepK0EBTY4MAOm+OGCabPt3oAEKTikaHXwMiTBJVTSeYP7R+7RoZN2a01FZXuUCDACTVBGfAk/jjIEl/WtkiklUGPEXUmQU+edRrOgj7G9euSrdXXhZCncnWnMvMf8CA+94aN1Y+/mh5ykxGUICfs5NVmAXaF+vpQGvb1i0y+I2BCmTQ8BkUoIG2ZhqPjaWs/FTg47ND382AxwZLhzJUgsGlwtr7dXDMB2HNFXfvCFFnJ48fcwI9dnZoFyDgfUWYwaZPnSKrPv5Il4/2AQRoT4BE1HtTH8xybPfu3qmmOrQnghG8CQ8a7j2cb8pMbTaeovjKrrfkADsiPxX7rgRCyDqj9AasE9hPBxIQsdaz+2ty4tjRYA2aVgcQGSZ0pvMMmhJLVaOhABjTp03RCDPm2uQSfo2PCBA7/PUh6du7l1Tev5cCwkwamGk8pcfD6TxoxzF5IDaItBU04DFAjdQW4gxAhDqA4LaNUlP1WPr16SUsfIYpCwBROq0wc7Cfpe3RQlziznr5cOkSeXfGdDc5tMmtUBqnTuEyACOgQx43QAdtppl64YfKEH5twBNT6GTpw3D7234Rt2cbnsTeM+CxgdMhwEPUmArxxnqpevxIRg4fKnt27VShDhjxd1kL4g1ANX81N8qK5R/K3DmzNXM0vhmizwCkpIKs7OIF6dOrh9y7c0uaQxNEoYl/J52eAU/yNk5vQzsukTaMDTdtBQ14chBiNqBcfjO0BtoCUEGrQVjX1lTJ+LGjZdvnW/Q851JRbKHy6W2ovpoQQKGFkLrmzRHDNZqNZ3jNBLBIvz/9mPKcg+7V8svyox98XzNWo/lwf9v1JpFUvZyPR7UtFm9rdks1+AmkDozirWCaXh87LhEhXIrypA1PYu8Z8JQio3TQO6PBIMARyAAPf8KT165e5YAowYqbgAqaDEABOGzdslnGjn5T87FxnFRwcw+mNECjb++eCj5KR5zPKJ2eezbv4wSkB7dFC+ZLednFlI/KByik32/HBiwlywOx4aatoAFPBwnhUmM6FdDiQpMR6ADQmFFvqpZCpBjH+HzctWih5LQMB2I7d2xXTYfsBGQ78NpJkjYGBFnymnxwTAAF0LzGlImemgoJtdYlqpvV/EY5Mhf4ZRE49v8kdbGy0f1vbVTAbdSGJ7H3DHgMeBJrEwgJp1G4YAL8LixZTUJNQCcFNtIaGzSY2AlY7N2zS7MbsPw0QIE/hvNJBdP9e3flxz/8gZCxGhpoZdDQAIgMfe7Nb2SsrqmuUk3p1s3rMmfWTDm4f69cv3ZVKu7d0TrxfknrY+ULWLBm4Bfrz1B/xoabtoIGPMZUiYSofvHrrH6nQQA07783R+bMeld9ImgWKsR1fk12AU1ZBjA0uYeoMyZ11tWQHqcxoJedhhcAngbHgNYbrw+QkyeOqebiQcVrK2z9fX6rYBLMP7p86aJMnjRBJowfK3/zv/5Sl24grQ5mNxKTop35+2wbEkA2lkqTL9rwJPaeAY8NltiDxWsNmNkQ5pjBWKHzvdnvumizIHItTtSZm7jpEnWi0aCZ4Ish6iwTMEQJeA88DysrpH/fPrJ/7x4FHQAlloaSAiNnbvP3YWq7frVc28gFFzhtL6o+dt0AqWR4IDbctBU04DHgiQU8Hgz8XJ36ulpZu2aVprAhhxrXfbg0oBQ16NR386RFAevM6ZM6v+bO7ZtqynL+lmSCC7McGa+Jgtu86TMFG8ADWl6zyl6n1lR0m05aRWN70iKLFy5Qs1tYa4ozDyn7s5K9m9Gy9sprHmjDk9h7BjwGPJEgAdPjg2GeDgCDhkLk2vixY1Tr4dhrCOqcj9GmlMefcuniedVQbt64llpLJ9O8mqiB9/jhA3l74gQN4wZoFAhV23FmvKj7/XUfqefep1k1OhKc+txz/j19edsaKJQ8D8SGm7aCBjwxhGQpMpbXcHh3v+9Djrd9vlnGjBqpc3aaGupUS6EMJjgv8NPbzGsdXoCjoRB19urLP5GySxfUdMc9CHZv0kunET72AQfQYR8Q3LBubZvJLzCxuXo969MJ02Lfv2NqTk9Y4ym/nDK1cT1O/dLp27EBVNHyQBuexN4z4DHgaV/jEZfaBrAAEDAxHTt6WIYNGaRmLRX+QUh11KDifsoDTuzfvHHd5XE7fjSgnUwwqY+oqVHqaqpkytsTg4zVbg6Qq2syeun19/4clsK29Xiery3T29aOi6w9Y8NNW0EDHgOe9oGHtgkmVGL+evzogWoW+FLwefi5N3F8OmhLlIPOw8r78nr/fnL65Ann08khSkwncrY2y4zpU2X5sqUKXvia8B1lSvqZVNgZ8BSZcLRxnn2cP0/7tOFJ7D0Dnudp8GK+VzMSuLBhzFBkip4xfZoc3L/PLRGNFqTLDOBDiQ4v9iY4EnSyTALJQzGTAQgpM1eC9gTEZr87Q2bPmqmaFHN1AB3o4TtKCjTp5Q14DHjSecKO2+GJ2HDTVtCAJ4GwKzXGUyf9kxadVwPoEOGFQPbOe9pDzVqBVhRuH68FOXDCVNcs9+/d0agzkodSNptPKEwr9ZxgAiiRdYQ5M3fIT1ilHgBYLiCW/iytmwYotIiZ2toRNjZunvvjJhPfFeS5NjyJvWfAYwMo4wAizxnggSbxzrSpGt2lPhoEcoyZ+woqQd41zGt1tTW6+Borf6pvKGG7kx0BjQaA+XTjBpk8aaKwxg4g1BmD1TQeA5zO4KuipBkbbtoKGvAkFIBFyTgZ2gDNAeBZMH+ezH//PRXuXtNBKEe1g2pC0qogA+iQPHT9ujXOzxMDuNLpY5ajPps+2yiDBr6eCuNW3xFLMmR4h+c5Z8BjwPM8/FNS97bhSew9A54OFljFwnBoJfPmvqc+lNrqx6nIM+bzxPHp0A6ARV1djaae2bhhnfMN4S8KfDFJ2opnAjosuQCQAUKAo593k4RWnLIGPAY8cfjEytjS1x3+1VtqTIWWgmMeIb/y449k5jvTgzk6bQ779vwoCgKBJgMdgAtQIOfZmlUfK3D5YII4prb05+zbs0uGDn5DE3h6zQvwoY/8tiP7y4DHgKcj+amoacXWc9oKmsZjGk9KeKfMWZ9u1CwADx9UpjSdqIFDCDO+HEfDAc+sme9oYs3G+trEHwWACX9o7tzxpSb9vH//ntYHUIqqz/NeN+Ax4HleHiqZ+9vwJPaeAY8BjwpxtBMGCuas0W+OlMcP20BHBX2GyLXwwEJLIgBANZ2WJpn3/nuyZNHC1JLV4bJx9r1W882hgzJi2BB5cP+eAyLqGVGXOPSjyhjwGPBE8YhdD3gkNty0FTTgMeBRwAFc0CyGDh6kPhSc9vqPmcIG8xo0MNWtXLFcTWws5KYaUA7Of+idO3taevXoLrdv3Qg0HWdWi+tjeh7BYMBjwPM8/FNS97bhSew9A54SBR5AImXOammSgwf26cROJniitajfRHOvOQGkWk+GtgrPxSFoYMP6tTJ29CilweDzgBQ1ECnHM3kO+xfPn1PQYenqMA1fLope0uvQ5R5972BSK8DL3KUrl8tS84N8/ZLSt/IGZEXLA7Hhpq2gAU8GYVq0DBJ6V9VEgmUDDn99ULp3e1UePaxUjcXPl8nWDt405wEJACJybezoN6W+rkaFeLb7069pWLRmH6gXMlX37tlDlyNoaghAMFT39Hs74hjA8eDjwY13W75sibAwHM9wS0IEk2Y7uT4d8U5Gw8CuS3igDU9i7xnwlLAAASAunj8rA/r1kQeV98ULea95ZGNa/+XvhbSPOquuepxTyhqeiZmONXlee/UVrRdZCahDLuHX2eqe6Zr6lFRDaxbWGnpwv0IqK+7q6qonjh1RUCZXHfcyuTYTDTtngr4keSA23LQVNOApUeABMC6cPyuvvPQTuXHtSpvZTefGRE/IBCicltAse3Z9pWa6x48eSkNdbU5JOqnPzetXpV+f3nL29CmlgYD3ANcVA1rXHHrSIhfOndX5Qm+NGyt//Vd/KYPfeF3Nh/PmzpGa6scGOiU6ZrqCBwvyGW14EnvPgKeEBpEX4jjnAZtur7wk169d0S94TGdOe3G+DvbDg8CnyfGpcLwfiGUSEMzVVY9SpirMeOF7M+1TF8478GrStX0GDugnx48eSdGhjK9zJhodec7XJ7zF/LdsySK5Ul6Weh/eP71tOrIeRsu0poLjgdhw01bQgKdEgKdNQ2mSO7duSN/ePaW87JIT7DGWqlYQ0DxtARg0N8mxI4elf9/ecu/O7ZRgjjtoXAYEN0/nfsVd6denl+zdsyuYeBqtccV9Tq7lLKrNACBX3im5+9rwJPaeAU+JAA+CHvC4ddOBzulTJ9R3gvbTohkLov0W+EG8ZnT2zCnp3bO705hyCJdmcOJjwm8y+s0RsuurHapJqCaVpm29iIFswGPA8yL4riCfGRtu2goa8JQK8LQ0S+X9Cunbu5ecPnlcwQaTEeakuM57Ld/aLJcunteos1s3r+tyB3FMa+kDCloN9XUaBUfGagW/JgeOaGfp5bv62IDHgKerea5gn9eGJ7H3DHiKGHjQcLzDnMmc5Do7sG+vaj5EsKn5LJg3w36Y8cPHPnQasKisuKerh166cD4FOvg9wvdm2vc+Ig9eLGnAfJ/VKz925r5gHg+gE352Jlpdcc6AJ7pPu6If7BkF0A+x4aatoAFPkQKPgkrgvGd+DtFi27dt1YXc4mgU3I9pTSPLgomgTQ118s60KRoN5wFNnrTGAgqn0TRoWSLfpk2ZLJ9sWKc+HYR8vgkYA54CEHhFOnbzbSxE1qcNT2LvGfAUKfMCHPhQmAszauQI+WLr52pWa8GcFUPQa/RaKrsBWQWaZPNnn8qqj1eoWcwxY6vmTfMaUTYGJYmorxPLLSz8YH7gY3Jh2dnufRHXDHgMeF4E3xXkM2PDTVtBA54iBR78LvX1tTLxrXGyZvVKBQ4XGOC0C2/6ysbomOMwjaGtsOrn4DcGaiJQ59NxoCOEPMcBspZmaayvE+bCzJo5w/mVpFVzuWWrw4u6ZsBjwPOieK/gntuGJ7H3DHiKCHgABI08C/wlaDofr1iumgYA4v0rZAjIxNweQCjnrwMW2z7fIiOHD9PJoWg+vhygo/8MbRg250GPulGX6VOnuOSjgW+J58QBQV+frtoa8BjwdBWvFfxzYsNNW0EDngxCsxAZATOWA562tXCWL1sqmMwAC6eltC9MuJ/39mHXRLsR2uxS4QySxw8ftPly0gIRMraXtAoBBB501qxaKWQCAJCa2gG+jHReUP8Y8LTPK/nUT1aXPOinNjyJvWfA84IEW0cPGA8cCPb3Zr8rC+a/r+HKrJHDs6KAR3OVBSYzNCKA59CB/RqUQHBCSoPRCLbWlEbU3nvgS8JEx32Y+iaMH6fLLWg0XQG0uQFPHgi0AuCT9vi/pM7Hhpu2ggY8RcLcXuNh8bUpb09UfwrgwQqe9bU1qvlkGwzc74Ut9506eVx69XhNk3b6/GT4iLiWjY6/hqYDmO348gsZN2ZUKocb95OE05fL161viyWLFsi1K5fzvr752o5Wr3jV+4OaAAAXkElEQVTjpaDbqQ1PYu8Z8MQUpPnGGAoUQS4ztAoE/bo1q2TalLdTE0LROjhP3VMaS/C+/nxYUyI0GrA4ffKEztW5cf2qu18437byp78nvU3UrBf4mdCwdn71pZB/jewE3M/f1zv93nw7NuApAYFZoGM/38ZKbLQJFTTgKWDmQ7hjumK7/MOlas4ihJrjtnk2mQWIhku3NIuGV6vPplXBiQXYXn35JQF0nH/IZRFoD2z8ICD7AT4hkVY18e3auUPzwQE66aDn78nnrQFPZr7J5z6zur2gPgsBStxdA54CBR7VIDSgoFG2b/tcRgwb4nwojS4EWjWULO/mo98ACkxogAOrfZJ/rezSBdVMOKfmOk0O2hbplmmAA0wAD/N1zp89o2v8ADqAIP9M9+TzOQOeFyTEsvBsPvNLSdctLtqEyhnwFDCjAy5kdH5zxHBdqIxoMcBCNZ5QSHSmQaEajLSmQpnJu8aCcKdOHFca3iwGvTgTRHk2z+V+Vg+9deO6Rq8Rjh0Fgpnq96LPGfAY8LxoHiyY54cAJe6uAU+BAE/YN4Ig579/7x4V8g8qK1Jg4QEjE9OGaXgtBM3n7p1b6tM58s3XQbJOF9UGDQQw23TwcLRaUiHWLAtddumi1qfi7h019fl7AK9M9cmnc7yPr6dvQ7S9ZUsW69pF2hbBWjz+vfKp/lYXA8oXxgNx0SZUzoCnQIAHoYj/BnMW27OnT2qoM6ATBpT2mI8yQig0wjNY9A1aDyvvy7Ahg2Tfnt2BWc2Z3dqj488DXBp2jRnuSYugMaHpXL1S/gxI+XvyeevBhLZp1mANl6tu6eJFCqia8ieI1IujAebzu1rdDKQ6lAdCgBJ314CnQIAH4AAoEJBlFy+okL9964aCDucVWLK8i7+uJiTW1WlplqrHD2XIoDfkqy+3Kxg5E50POIgenLqWT1OjXLlcJj1ee1UITIChvTbVocyd5d064jm+fah72cXzqgHSNn/6x38k3bu9KgMH9Jepk9/WlVZ57454ptGI5jFrowJoo7hoEypnwNPJAq2jBg6Ag4Zx7Uq5fO/v/07n1yAkEYJkjY7zHM1KgIbS6kDn9f59NeSZDAP6ld8cJOz04dNZ2gYtBy3szq2bunro5UsXNZM1IFgImQnS2ytd4yFKj/+iBfPl/LkzGj3ogd+b5NJp2HEBCMksPG39l2P/hQAl7q4BT54yov8CZzCwD8DcuHZFv7xv37zhlisI8p15oZlt4FDGz+tB05k8aYJs2fRZKqVO+HmUe4ZWyL8BLco/fFApA/r1la8PHkgFIORj3rVn3iVDn7s2du/N+/l3XLp4oYI91/WvwJ3jAM3w3Dh1szLW3nnNA3HRJlTOgCdPhQECXIVhIAQfPrgv3bu9IuVll/TrO+lXN+XRkBrqamT82NGyYd1a1XI4j5CNYmzW5VGBHDjh7929LWhMRNVRz6j7C+26RbWZsC80nn1h9Q0BStxdA558BZ5AwAMUFffuyA+//105eeK4E/4BGCVlNPK2TZ40UUirA1gAJORTU+CJs4pokIi0rqZa0+Bs/2KrmtWIaEtal3wvb8BjwJPvPJo39YuLNqFyBjx5DDzkWcOchWZx8sQxp7HU17m1dWJoKWHGBGjmzpkly5YsCiZ6NmiotEa4xXSWox2RZ230myPl882fqebl8sCZxhNua9s30CopHggBStxdA548AR5v7mILSJCuhpn/JOrcs2tnW5JPb9aKoaF4MxpO8vnvz5UP5s3VhKH+WQwObyYLn+O812KgQRm//fij5bJh3RqnLQXzc7hWbAPNNB4Dj2Lj6U57n7hoEypnwJMnwINwd+vXoD20yoP792T40CEKOg11tc4clqCu0PN521gLh1BgTG0ATFygAHy0fJD25tKFc/LujOkFGS6ddNAZ8BjwJOWZki0fApS4uwY8CYR5ZzIWQOHCdZvl8aOHMnb0m7J75w4NlW6or0tpJnHrQOg0AIN2wqqfhGK7c00agcbzstECcJjrQzQd+9evXtHkoffu3IoNXNno5/s1Ax4Dnnzn0bypX1y0CZUz4MkT4AEkEPC11VUyYthQ+XTjBg1/JhKNeTFcS8po+GGGDh4ktTVVej803NIF0aYxgInZ+mhbZKomY/WtIIybOiWtS6GVN+Ax4Ck0nn1h9Q0BStxdA54XBDwIdv4AjttvUdCZNmWyrF29Us+FwSbKPIa25BkPofnlF9sUdDCv+RVFCSTgWZTzW3+PP+ef4wCvXgibHjRwgJw/ezqop6t3+L5i3DfgMeApRr7ulHeKizahcgY8Lwh4YAC0CQS9F/L4T5YvW6Imt0zAkI1pACnAh0CCnTu+lJHDh8qDyvtu9dEMC8Gl09LoNrIjBDnY2N6+eV169eguRw9/kwK19PuK9diAx4CnWHm7w98rBChxdw14XiDwADqARVNTg7z/3myZ+Q6O+yBtTYyotTADATz8Dx7YJ31795LK+/dSAIavJgrIuO5A0NXpfsVd6d+3jxw/ekTn+sRNyxOuUyHvG/AY8BQy/3Zp3eOiTaicAc8LAh4XMeYCCj5culhBp7GhbRG3KKBIZyzKHz38tS41/ehhpYIIQOIBJb18pmPKMmGVMO7hQwfrsgspIHxB7ZSpnl1xzoDHgKcr+KwonhEClLi7BjxdKFABAhgNAe+XniZ1zZS3JwbaiUu8ibksiiGhQRkfvXalvEzNYndu33LBCMGkUO8n8uU93fScalynfjXVj3VBuK++/CIV0KD3JtTA/HMKdWvAY8BTqLzb5fWOizahcgY8XQQ8fk4MQhywAFw+2/iJDB38hoKOB4gkTMM9jfW1crW8TL7/3b+Xmzeuaag05wG2SFqa+LNZVyHFHAfouKwEm6Lv7aJ2i3yHTqqHAU8M/umktn9RfW7PzbHPQ4ASd9eAp4sGj2oU+GGCHGlkhp40YXxqkqdqFQnqgsaChsKyBL17dpdrV68oCAE6nPfaVXuDieextIFeb3Wa1oTxY+WzjRuC8O3s83zao1ss5w14chRCnqdsWzofb3HRJlTOgKcLB4h38rtQ5zeE1UMBCgIMooAiXaCjNVVW3JMf/eB7ul4MUWiqTbW4jAXp5dOPPdAxQbSmukoAnZUrlmtUG74mfz39vlI5NuAx4CkVXn/u9wwBStxdA57OAh6f/Tlw8BOuDLjs3b1LAwBYctplBmhU01uczgeg1IzW0iz37txWX8zJ40eVro86A4CgRbkwzfAx+9RHpFXDrefOmS3kYKN+3J8UBMPPKdR9D7TaLuqHwxfXIgR+4D/jvdQvF/Rrob6n1dsAtcN5IC7ahMoZ8HQW8ARLQGPOAhQQ5qdPHtfJmHdu33wKFOIwAtFlmsGguUke3K/QyaEH9+9T8AqDSnZarZr8UzWj5iadR7RwwXxNIAqoZb+3uAcsbeKB/fzZMwrq/fr0lj/6wz+Q1159Rfr27qmph1hEz4N7KbeXvXtxj4dE/RsClLi7BjydBDwIMsCGL2jA59KF85pp+taN6wnBwjG40mppEkKlySTAJFEf0RYHeBRsWtrW4CGB6McffSgT3xqn9YwVjNBJbZWIyTupDrQvbcSftiH4Az/a4oUfyKWL5/U8E36pKx8B+VBnq4MJ/7zggbhoEypnwNNJgswJesxrTXLm1AldPfQWUWctTbqmDdeTMA3A8Ogh82uGyKZPN2oGa2ijBflnRdFzq4iSg61B1q1Zrctf49+hTqX+Fa8+ncBECQih/fDBwPpFl8suumjBIOVQqWuHUXxm10sMEEOAEnfXgKeDgQcthIGHMEdAEeJM2pkb164EX8sk/Yw2a3lg8toM23FjRsvmzwCdtvVysoGO15I8LfcV3yybN32q2a8Jnw7TKmWB4dvIz2+ivdFWly5epD4ervu29n1Syu1l715i4JJNTsZFm1A5A55sDZrgGhoJGgiag49eI6tzt1deEnwGgECSweqFHNvqqkcKFCtXfKQZqzUkO4KeLuQmrVoX6qTJQpsaZOeO7TKgX1/NUMAXvT4nglaSehdLWYtqM8FaLLzc6e8RApS4uwY8CcAlWwci6AEf7P/4CO5X3FOH9Lmzp9W0lu3eTNcAhKaGeqU3a+Y7surjj6QR8w9r5MT0MTC5FDrQR8va8eUXMnL4ME2Jw3loCdelNREoZqpvsZ0z4DHgKTae7rT3iYs2oXIGPB0EPHQqyw/U11ZLdfVjTdR57Mg3aq5pZkJnTLDwzAEwoKksmP++LPxgvpruADbVUgJNxZdtbwsI8uee3bu+0kitxw8fCPXR8Guf3cCDUAe2RXt1KpTzBjwGPIXCqy+8niFAibtrwJOjsHUag5srg18Auz9aBUsRDH5joOzbszuRT8ADDUzEPhrU/PffkxnTpzrzXWj+iPoYPGgE9U/3O0DDR9Qd/vqQvPH6AKm4e8cBV3BNGTaNzgtn4hz7o6PrbcBjwNPRPFW09OKiTaicAc9zCLoU+KCJtOCLeSzDhgx2oc6YsRLQBjjwA2Gm479i+YfC+jxoUUSuxYk60/BqJpAG9eH5J44dVU3n9q0bCjoKUPh+zK+TtX8MeAx4kozfki4bApS4uwY8CcAhzFxegGNCwwRWV1MtY0ePks+3bNLjOJFrYXpe44HW55s3qS8GsMHPAxCFy2badxoOAQxMEm3UOSjXrlyWV176iS5d7UCyWYGM+znORMfOOYFrwGPAY2MhJg/ERZtQOQOeHIFHBbmGNTfqomuTJ03QuTF+DggRbkkYl1BnwOyT9Wt1rk5t9WMFBz9ZMW5UHOWo29Url6Vn925SXnaxbYJoEESAUE1St1Isa8ATU+jkOH5KkaeK9p1DgBJ314An5sBRDSekKXgBj2YzY/o0WbPqYxXwaCkKEjE0CgACuvwBmO3btsqkCW89HXWWZdlqFY4amca8ITIlOHPdjWtX5fX+fTV5KHUh/Jp6+Xco2gEQsy/jvL8BjwFPHD6xMi1xseapcgY8SYSVrl/jgAJhXl9bI7NnzdS0KnF8MGEmxaSGltNYX6c+HPKusWQ1GavD5drbB0TCmgvPr6utFnw5LJNw5tTJIJDA+Z/MtJZMkBrwJGuv9vjUzpdAOz4FKfEODHhiAo/TTto0FAT/vLlzZO6cWRrNFgaBOIMNei7lTb0uMf16/37y8AEZq110XBRQAFw8h8g1fDpsKyvuakTd0cPf6DUCEyhD3dCI4tTLyrh2NeApAYEZc+zbmIjghXhY81QpA54EzKdmK5YOaG6UhR/Mk9nvzggmeTaoMz8JgxKCDbicOnFcwYJQZ9bBITggbmCChkszB+dJiwYgoDEdPLBPF4SjLgqWQfRaFJAlqXsplDXgiRA2CcZNKfBLSb/jU5AS78CAJ8sA8qlvvNDGnMU+C6YxvwYzGceaniYLHc+UYR8LGgvJQ/v06inlZZdUGwHY4prs9LkBqDApdMiggeojQssJ5xvj2Um1MV/fUt4a8BjwlDL/J3r3eFjzVCkDnnYAQ0FC3Po17LvMzk2yfu1qeWvcGKmtqVbQ0XQzMQIJHKC4UGdMY5cunJN+fXrJ9SB5qAe3uB3uTHINmnNtxLAhsm3rFg1qALz4J6UX97mlUs6Ax4CnVHj9ud/zKUiJd2DA0w7w0BloMmg9mpSzpUm2btmkoc4EBaTMYQlm/gNeANDd27c0AKDs0gX18wAiUUCh1xXgWt0EUcxrtTUyfuxozTZNChxf1yhaz81oWdqsWGgb8BjwFAsvd/p7xMOap0oZ8LQjRDGFOdMXGkSTfLl9mwwd/IZg1sKnAlh4H0ucJJsqyJ60CKuPEnV26uQJZ17DfBcK026PSdrAhKwDTVJXUyXTprwtq1aucIuWBaHZaFPUR7ftvFt7z7DzbcLWgKetLYwvrC2y8sBTkBLvwIAnEM5esPstoIMGgdZzYP9e9aHcv3fXRZ1pEECLah6+Q/x9/pit12TQjrheGWSsxrejvphgjRcffRa+N30/HMWGb+m92bNk+bKlCo7qi/r/2zvz1yaCKI7/r+JRBKm13iIitlW86m090Db1RFBbiyDijaLW1iqK1gsvEGvV2no2iU++b2diNia7s3X6S+b7QzGbmX1ZP3N89831THRR3GfneCpt8Lp2B4KyAh+UO4Y/93bslt07d8jsmTNk/bo26di1QxCWAkvWq5U12dZmSzZ1zsZNa2K5KDxlXgG8BHQ8aCjo6HFUzf2hQVkwv1FDTlvPxOWcM3RkGFaDvUJhUsbHPmkcnIH+W+qx2I4uS6OELdjs7jooW9s3a5gE2LHPnMUW88Y7A8sQPBFeHAerDt0dlNY1q+VM72nBSeM49w4iH83Xxe8nT/IItg7EJMXtgsJT5vFAbCAu8HLwGUudsQAAmzJVRLBfxrwZp1Uy5NNwCPlf6uksXtgsF86fU5FAWta3ZitgODy0fdNGXXoNGzqvMwV7ac8fWroOnyK0tZ4k8Tfa6KFcl7x49lTrA5ig7KxIhcaI/1+Ka9U64KY1sVwUHis8haizgdeDTghRQzVk9ZvXKhIav0bzYnI/ylu1EKw9sxBg4suYrFq5Qq5fu6IeCobV0HFlFR4IH5ZxQ3TgianoaEeJZ2GDSCoLlzQdrkSZadTWaP4OIpPrPCA4bBVzPrBjvV4Xm8zDehlEHYhJittF2MJjAqCpB6Jvs5GgPHsyrPFr8C86eCsU+Gw3flZWKJsP3yM/bH6dGNf5gZ5TJ/U7+6bs+tZsh3Twm2f7zsja1hYNYV1aUWeGBPGbnNf5v04O5QeOKBvlqfuzfsmhXKcRnuj7cvbIxz8yCL4OuGlNLFewwoOOpjRXo2+50XE4b9+8krkNc+TJ8KNo2M14J2kduxUvnCqNzgmT0NjvgwlpLL9Ou7+y8upw3++CblLt6+2RdW0teqSOihFWrTGezrR2+lzVRkGpbJO8rlEnYpLidhGs8Ng3W4iPejLFvIy8fycNs2fJ0+HH2qlh340OrTi+2f40B35iMcGWzRvlcHdOj9TBb2EJdNaKi2G5q5cvyvKlS2RifCzyvMyQYFZbzF+j0dQoWwpPNl6sXwHzctOaWK5ghUc9BimW5lpGP7yXFcuWSv+tmzqcpR6MhhrIsKDA7KVB5FCErca8waQJ5pa1YUIMb9+MTqz+ODoSeTgaORQCVlRBy2qT+d07BwqPOyvWq8BZxSTF7SJo4VFPpDCp8W8WLpgvN65f02EyuwBAG5SZuLdzAOWNzB6jg8l9xNNBtFDs/di/d08pNk5pnsbMIZTfX/kZz2OH0Abu3NYVdfDC7HPaNPxu5b289tv4KTx+ebJ+1jFPN62J5QpGeP5d+VXUuRecINDWskYePrhfWiqLjj6todjhOeTFZ+z92L5ti+75sIeHptkoT4cNG5/nbF+vbNqwXmPzwD4ErZrwld/Pz34bNoXHL0/WzzrmGZMUt4uAhCdvThoo6tJYeDWXL13U1WsYyoIw4U+PwTEr09IaCzwQCAOG6ZqbGjUkAb6bynE1EJafP77pqde5zoOl4T48J37Dek5pz8R0Pw2cwuOHI+tjABzdtCaWKxjhER0yK+r8CFaMYZ9OU+O8aKUYBCRvIouaUNRpHoaKTjEv418+6+Q/4uBguE33ATns86lskLDX3dUp+/Z0lJ4FtiA4VuAq7+H19DVqCs/0sWW9rTO2MUlxuwhGeNB5oxP/8f2bnDh+TFecwcOw3gm8iiwNAvYGB/oFJxJ8/hhFDs1yPzwZ5IcdhLte1Nwk94YGI49rCsKV5beZN73ho1ywmfT40SPy8sXzTHWDfNP5klEdMXLTmlguCg9Ocy7bOOjaICg8ddRwqiyppvDUd/m6tnPmc6gHMUlxu3AXHjd7zEUCJEACJEACiQQoPIl4mEgCJEACJOCbAIXHN1HaIwESIAESSCRA4UnEw0QSIAESIAHfBCg8vonSHgmQAAmQQCIBCk8iHiaSAAmQAAn4JkDh8U2U9kiABEiABBIJUHgS8TCRBEiABEjANwEKj2+itEcCJEACJJBIgMKTiIeJJEACJEACvglQeHwTpT0SIAESIIFEAhSeRDxMJAESIAES8E2AwuObKO2RAAmQAAkkEvgDyPBEovjrYbMAAAAASUVORK5CYII="></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> parallel lines with positive gradient <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>c</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mi>c</mi></math> <strong>A1</strong></p>
<p><br><strong>Note:</strong> To award <em><strong>A1</strong></em>, each <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept should be clear, but condone a missing label (<em>eg.</em> <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>If candidates represent the lines using slope fields, but omit the lines, award maximum of <em><strong>M1A0A1</strong></em>.</p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced><mo>,</mo><mo> </mo><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p>to the left of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>, the gradient is negative <em><strong>R1</strong></em></p>
<p>to the right of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>, the gradient is positive <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Accept any correct reasoning using gradient, isoclines or slope field.</p>
<p>If a candidate uses left/right or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>1</mn><mo> </mo><mo>/</mo><mo> </mo><mi>x</mi><mo>></mo><mn>1</mn><mo> </mo></math> without explicitly referring to the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> or a correct region on the diagram, award <em><strong>R0R1</strong></em>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>y</mi></mrow><mrow><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>y</mi></mrow><mrow><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn><mfenced><mrow><mo>></mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> accept correct reasoning <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac></math> that is increasing as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> increases.</p>
<p><strong><br>THEN</strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> is a local minimum <em><strong>AG</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>integrating factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mo>∫</mo><mtext>d</mtext><mi>x</mi></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><msup><mtext>e</mtext><mi>x</mi></msup><mo>+</mo><mi>y</mi><msup><mtext>e</mtext><mi>x</mi></msup><mo>=</mo><mi>x</mi><msup><mtext>e</mtext><mi>x</mi></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><msup><mtext>e</mtext><mi>x</mi></msup><mo>=</mo><mo>∫</mo><mi>x</mi><msup><mtext>e</mtext><mi>x</mi></msup><mtext> d</mtext><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi><msup><mtext>e</mtext><mi>x</mi></msup><mtext>-</mtext><mo>∫</mo><msup><mtext>e</mtext><mi>x</mi></msup><mtext> d</mtext><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi><msup><mtext>e</mtext><mi>x</mi></msup><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for the correct RHS.</p>
<p><br>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> gives</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>e</mtext><mo>=</mo><mtext>e</mtext><mo>-</mo><mtext>e</mtext><mo>+</mo><mi>c</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mtext>e</mtext></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[8 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><strong>EITHER</strong></p>
<p>attempt to solve for the intersection <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></msup><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempt to find the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></msup><mi>-</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></msup><mo>></mo><mn>0</mn></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></msup><mo>≠</mo><mn>0</mn></math> or equivalent reasoning.</p>
<p><br>therefore the curve does not intersect the isocline <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is an (oblique) asymptote to the curve <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Do not accept “the curve is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math>"</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is the isocline for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p>therefore the curve does not intersect the isocline <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>The initial point is above <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math>, so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo><</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo><</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>y</mi><mo>></mo><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p>therefore the curve does not intersect the isocline <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><img style="display: block;margin-left:auto;margin-right:auto;" 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"></p>
<p>concave up curve with minimum at approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>asymptote of curve is isocline <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Only award <em><strong>FT</strong></em> from (b) if the above conditions are satisfied.</p>
<p><em><strong><br>[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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the differential equation</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {\frac{y}{x}} \right),{\text{ }}x > 0.">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>></mo>
<mn>0.</mn>
</math></span></p>
<p>Use the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = vx">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mi>x</mi>
</math></span> to show that the general solution of this differential equation is</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\int {\frac{{{\text{d}}v}}{{f(v) - v}} = \ln x + } {\text{ Constant.}}">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
<mo>+</mo>
</mrow>
<mrow>
<mtext> Constant.</mtext>
</mrow>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, solve the differential equation</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{x^2} + 3xy + {y^2}}}{{{x^2}}},{\text{ }}x > 0,">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
<mo>,</mo>
</math></span></p>
<p>given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1">
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> when <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>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g(x)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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="y = vx \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = v + x\frac{{{\text{d}}v}}{{{\text{d}}x}}">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mi>x</mi>
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p>the differential equation becomes</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v + x\frac{{{\text{d}}v}}{{{\text{d}}x}} = f(v)">
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>v</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="\int {\frac{{{\text{d}}v}}{{f(v) - v}} = \int {\frac{{{\text{d}}v}}{x}} } ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p>integrating, Constant <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}v}}{{f(v) - v}} = \ln x + } {\text{ Constant}}">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
<mo>+</mo>
</mrow>
<mrow>
<mtext> Constant</mtext>
</mrow>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(v) = 1 + 3v + {v^2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mn>3</mn>
<mi>v</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\int {\frac{{{\text{d}}v}}{{f(v) - v}} = } } \right)\,\,\,\int {\frac{{{\text{d}}v}}{{1 + 3v + {v^2} - v}} = \ln x + C} ">
<mrow>
<mo>(</mo>
<mrow>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mo>=</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>3</mn>
<mi>v</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}v}}{{{{(1 + v)}^2}}} = (\ln x + C)} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>A1 </em></strong>is for correct factorization.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{{1 + v}}\,\,\,( = \ln x + C)">
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v + x\frac{{{\text{d}}v}}{{{\text{d}}x}} = 1 + 3v + {v^2}">
<mi>v</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mn>3</mn>
<mi>v</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}v}}{{1 + 2v + {v^2}}} = \int {\frac{1}{x}{\text{d}}x} } ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>v</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}v}}{{{{(1 + v)}^2}}}\,\,\,\left( { = \int {\frac{1}{x}{\text{d}}x} } \right)} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>A1 </em></strong>is for correct factorization.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{{1 + v}} = \ln x( + C)">
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
<mo stretchy="false">(</mo>
<mo>+</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1A1</em></strong></p>
<p><strong>THEN</strong></p>
<p>substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1">
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 1">
<mi>v</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> when <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> (<strong><em>M1)</em></strong></p>
<p>therefore <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = - \frac{1}{2}">
<mi>C</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> This <strong><em>A1 </em></strong>can be awarded anywhere in their solution.</p>
<p> </p>
<p>substituting for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>,</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{{\left( {1 + \frac{y}{x}} \right)}} = \ln x - \frac{1}{2}">
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award for correct substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{y}{x}">
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span> into their expression.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \frac{y}{x} = \frac{1}{{\frac{1}{2} - \ln x}}">
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award for any rearrangement of a correct expression that has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> in the numerator.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x\left( {\frac{1}{{\left( {\frac{1}{2} - \ln x} \right)}} - 1} \right)\,\,\,{\text{(or equivalent)}}">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>(or equivalent)</mtext>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = x\left( {\frac{{1 + 2\ln x}}{{1 - 2\ln x}}} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong><em>[10 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use L’Hôpital’s rule to determine the value of</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{{\text{e}}^{ - 3x}}^{^2} + 3\,{\text{cos}}\left( {2x} \right) - 4}}{{3{x^2}}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</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>Hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{\int_0^x {\left( {{{\text{e}}^{ - 3t}}^{^2} + 3\,{\text{cos}}\left( {2t} \right) - 4} \right)} \,{\text{d}}t}}{{\int_0^x {3{t^2}} \,{\text{d}}t}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mi>x</mi>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mi>x</mi>
</msubsup>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{{{\text{e}}^{ - 3x}}^{^2} + 3\,{\text{cos}}\,2x - 4}}{{3{x^2}}} = \left( {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</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=" = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{ - {\text{6}}x{{\text{e}}^{ - 3x}}^{^2} - 6\,{\text{sin}}\,2x}}{{6x}} = \left( {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right)">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mo>−</mo>
<mrow>
<mtext>6</mtext>
</mrow>
<mi>x</mi>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>−</mo>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>6</mn>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{ - {\text{6}}{{\text{e}}^{ - 3x}}^{^2} + 36{x^2}{{\text{e}}^{ - 3x}}^{^2} - 12\,{\text{cos}}\,2x}}{6}">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mo>−</mo>
<mrow>
<mtext>6</mtext>
</mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mn>36</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>−</mo>
<mn>12</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>= −3 <em><strong>A1</strong></em></p>
<p> </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="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{\int_0^x {\left( {{{\text{e}}^{ - 3t}}^{^2} + 3\,{\text{cos}}\,2t - 4} \right)} \,{\text{d}}t}}{{\int_0^x {3{t^2}} \,{\text{d}}t}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mi>x</mi>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>t</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mi>x</mi>
</msubsup>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{0}{0}">
<mfrac>
<mn>0</mn>
<mn>0</mn>
</mfrac>
</math></span></p>
<p>applying l’Hôpital´s rule <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{{{\text{e}}^{ - 3x}}^{^2} + 3\,{\text{cos}}\,2x - 4}}{{3{x^2}}}">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p>= −3 <em><strong>A1</strong></em> </p>
<p> </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>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{x}{{{x^2} + 1}}y = x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1">
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</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="\sqrt {{x^2} + 1} ">
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</math></span> is an integrating factor for this differential equation.</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>Solve the differential equation giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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>integrating factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^{\int {\frac{x}{{{x^2} + 1}}{\text{d}}x} }}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>∫</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{x}{{{x^2} + 1}}{\text{d}}x = \frac{1}{2}\ln ({x^2} + 1)} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^2} + 1">
<mi>u</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span> for example or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{f'(x)}}{{f(x)}}{\text{d}}x = \ln f(x)} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</math></span>.</p>
<p> </p>
<p>integrating factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^{\frac{1}{2}\ln ({x^2} + 1)}}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^{\ln \left( {\sqrt {{x^2} + 1} } \right)}}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>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="{{\text{e}}^{\ln \sqrt u }}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</msup>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^2} + 1">
<mi>u</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {{x^2} + 1} ">
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}x}}\left( {y\sqrt {{x^2} + 1} } \right) = \frac{{{\text{d}}y}}{{{\text{d}}x}}\sqrt {{x^2} + 1} + \frac{x}{{\sqrt {{x^2} + 1} }}y">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
<mi>y</mi>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{x^2} + 1} \left( {\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{x}{{{x^2} + 1}}y} \right)">
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for attempting to express in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{x^2} + 1} \times {\text{(LHS of de)}}">
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mo>×</mo>
<mrow>
<mtext>(LHS of de)</mtext>
</mrow>
</math></span>.</p>
<p> </p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{x^2} + 1} ">
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</math></span> is an integrating factor for this differential equation <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></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="\sqrt {{x^2} + 1} \frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{x}{{\sqrt {{x^2} + 1} }}y = x\sqrt {{x^2} + 1} ">
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</math></span> (or equivalent) <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}x}}\left( {y\sqrt {{x^2} + 1} } \right) = x\sqrt {{x^2} + 1} ">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>x</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\sqrt {{x^2} + 1} = \int {x\sqrt {{x^2} + 1} {\text{d}}x{\text{ }}\left( {y = \frac{1}{{\sqrt {{x^2} + 1} }}\int {x\sqrt {{x^2} + 1} {\text{d}}x} } \right)} ">
<mi>y</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mi>x</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
<mo>∫</mo>
<mrow>
<mi>x</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}{({x^2} + 1)^{\frac{3}{2}}} + C">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for using an appropriate substitution.</p>
<p> </p>
<p><strong>Note: </strong>Condone the absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>.</p>
<p> </p>
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0,{\text{ }}y = 1 \Rightarrow C = \frac{2}{3}">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>C</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for attempting to find their value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{3}({x^2} + 1) + \frac{2}{{3\sqrt {{x^2} + 1} }}{\text{ }}\left( {y = \frac{{{{({x^2} + 1)}^{\frac{3}{2}}} + 2}}{{3\sqrt {{x^2} + 1} }}} \right)">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mn>3</mn>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>3</mn>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore the behaviour and key features of cubic polynomials of the form</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></math>.</p>
<p> </p>
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mn>2</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> is a parameter, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>The graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn></math> are shown in the following diagrams.</p>
<p style="text-align: left;"><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn></math></p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>On separate axes, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> showing the value of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept and the coordinates of any points with zero gradient, for</p>
</div>
<div class="specification">
<p>Hence, or otherwise, find the set of values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has</p>
</div>
<div class="specification">
<p>Given that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has one local maximum point and one local minimum point, show that</p>
</div>
<div class="specification">
<p>Hence, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math>, find the set of values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn></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>Write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></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>a point of inflexion with zero gradient.</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>one local maximum point and one local minimum point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>no points where the gradient is equal to zero.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local maximum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math>.</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>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local minimum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly three <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo> </mo><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find all conditions on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has exactly one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept, explaining your reasoning.</p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn></math>: positive cubic with correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept labelled <em><strong>A1</strong></em></p>
<p>local maximum point correctly labelled <em><strong>A1</strong></em></p>
<p>local minimum point correctly labelled <em><strong>A1</strong></em></p>
<p> </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><img style="display:block;margin-left:auto;margin-right:auto;" 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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn></math>: positive cubic with correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept labelled <em><strong>A1</strong></em></p>
<p>local maximum point correctly labelled <em><strong>A1</strong></em></p>
<p>local minimum point correctly labelled <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept the following exact answers:<br> Local maximum point coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><msqrt><mn>2</mn></msqrt><mo>,</mo><mo> </mo><mn>2</mn><mo>+</mo><mn>4</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math>.<br> Local minimum point coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msqrt><mn>2</mn></msqrt><mo>,</mo><mo> </mo><mn>2</mn><mo>-</mo><mn>4</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo> </mo><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi></math> (an expression).</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"><mi>c</mi><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></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>considers the number of solutions to their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 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"><mi>c</mi><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The <em><strong>(M1)</strong></em> in part (c)(ii) can be awarded for work shown in either (ii) or (iii). </p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to solve their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>±</mo><msqrt><mi>c</mi></msqrt></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> if either <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><msqrt><mi>c</mi></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mi>c</mi></msqrt></math> is subsequently considered.<br> Award the above <em><strong>(M1)(A1)</strong></em> if this work is seen in part (c).</p>
<p> </p>
<p>correctly evaluates <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><msqrt><mi>c</mi></msqrt></mrow></mfenced></math> <em><strong>A1 </strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><msqrt><mi>c</mi></msqrt></mrow></mfenced><mo>=</mo><mo>-</mo><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>3</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mi>c</mi><msqrt><mi>c</mi></msqrt><mo>+</mo><mn>3</mn><mi>c</mi><msqrt><mi>c</mi></msqrt><mo>+</mo><mn>2</mn></mrow></mfenced></math></p>
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local maximum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p> </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> </p>
<p>correctly evaluates <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><msqrt><mi>c</mi></msqrt></mfenced></math> <em><strong>A1 </strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><msqrt><mi>c</mi></msqrt></mfenced><mo>=</mo><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>-</mo><mn>3</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo> </mo><mfenced><mrow><mo>=</mo><mi>c</mi><msqrt><mi>c</mi></msqrt><mo>-</mo><mn>3</mn><mi>c</mi><msqrt><mi>c</mi></msqrt><mo>+</mo><mn>2</mn></mrow></mfenced></math></p>
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local minimum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> will have one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept if</p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo>></mo><mn>0</mn></math> (or equivalent reasoning) <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>the minimum point is above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award<em><strong> R1</strong></em> for a rigorous approach that does not (only) refer to sketched graphs.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>c</mi><mo><</mo><mn>1</mn></math> <em><strong>A1 </strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo><</mo><mn>1</mn></math>. The <em><strong>A1</strong></em> is independent of the <em><strong>R1</strong></em>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> will have two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts if</p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> (or equivalent reasoning) <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>evidence from the graph in part(a)(i) <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1 </strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> will have three <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts if</p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo><</mo><mn>0</mn></math> (or equivalent reasoning) <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>reasoning from the results in both parts (e)(i) and (e)(ii) <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>1</mn></math> <em><strong>A1 </strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>case 1:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>≤</mo><mn>0</mn></math> (independent of the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>) <em><strong>A1 </strong></em></p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> does not have two solutions (has no solutions or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> solution) <em><strong> R1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>≥</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mo>∈</mo><mo>~</mo></msup></math> <em><strong> R1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> has no local maximum or local minimum points, hence any vertical translation of this graph (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math>) will also have no local maximum or local minimum points <em><strong> R1</strong></em></p>
<p><strong><br>THEN</strong></p>
<p>therefore there is only one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award at most <em><strong>A0R1</strong></em> if only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo><</mo><mn>0</mn></math> is considered.</p>
<p> </p>
<p><br>case 2</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><msqrt><mi>c</mi></msqrt><mo>,</mo><mo> </mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mi>d</mi></mrow></mfenced></math> is a local maximum point and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msqrt><mi>c</mi></msqrt><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mi>d</mi></mrow></mfenced></math> is a local minimum point <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate seen for either the maximum or the minimum.</p>
<p> </p>
<p>considers the positions of the local maximum point and/or the local minimum point <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>EITHER</strong><br>considers both points above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis or both points below the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis<br><br><br><strong>OR</strong></p>
<p>considers either the local minimum point only above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis OR the local maximum point only below the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis<br><br><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>></mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></math> (both points above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis) <em><strong>A1 </strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo><</mo><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></math> (both points above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis) <em><strong>A1 </strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award at most <em><strong>(A1)(M1)A0A0</strong></em> for case 2 if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math> is not clearly stated.</p>
<p> </p>
<p><em><strong>[6 marks]</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;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to examine various polygons for which the numerical value of the area is the same as the numerical value of the perimeter. For example, a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> rectangle has an area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math> and a perimeter of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math>.</strong></p>
<p> </p>
<p>For each polygon in this question, let the numerical value of its area be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and let the numerical value of its perimeter be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
</div>
<div class="specification">
<p>An <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon can be divided into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> congruent isosceles triangles. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> be the length of each of the two equal sides of one such isosceles triangle and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> be the length of the third side. The included angle between the two equal sides has magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></math>.</p>
<p>Part of such an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Consider a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
</div>
<div class="specification">
<p>The Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>5</mn></msup></mrow><mn>15</mn></mfrac><mo>+</mo><mo>…</mo></math></p>
</div>
<div class="specification">
<p>Consider a right-angled triangle with side lengths <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≥</mo><mi>b</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the side length, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>></mo><mn>0</mn></math>, of a square such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</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>Write down, in terms 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>n</mi></math>, an expression for the area, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub></math>, of one of these isosceles triangles.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</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>Use the results from parts (b) and (c) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Interpret your answer to part (e)(i) geometrically.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>8</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>4</mn></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the result of part (f) or otherwise, determine the three side lengths of the only two right-angled triangles for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo> </mo><mi>P</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the area and perimeter of these two right-angled triangles.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.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>A</mi><mo>=</mo><msup><mi>s</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>4</mn><mi>s</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><msup><mi>s</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mi>s</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mrow><mi>s</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>s</mi><mo>=</mo><mn>4</mn><mfenced><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1M1A0</strong></em> if both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mn>0</mn></math> are stated as final answers.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mrow><mn>2</mn><mo> </mo></mrow></msup><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1 </strong></em>for a correct alternative form expressed in terms 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>n</mi></math> only.</p>
<p> For example, using Pythagoras’ theorem, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mrow><mn>2</mn><mo> </mo></mrow></msup><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mn>2</mn><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</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>METHOD 1</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mtext>opp</mtext><mtext>hyp</mtext></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mstyle displaystyle="true"><mfrac><mi>y</mi><mn>2</mn></mfrac></mstyle><mi>x</mi></mfrac><mo>=</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>uses Pythagoras’ theorem <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mi>y</mi><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mi>y</mi><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>uses the cosine rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p>uses the sine rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>y</mi><mrow><mi>sin</mi><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></mstyle></mrow></mfrac><mo>=</mo><mfrac><mi>x</mi><mrow><mi>sin</mi><mstyle displaystyle="true"><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mstyle></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</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"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><mi>n</mi><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mi>n</mi><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for equating correct expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo> </mo><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math> (seen anywhere in part (d) or in part (b)) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math></p>
<p>attempts to either factorise or divide their expression <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac><mo>,</mo><mo> </mo><mfenced><mrow><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>≠</mo><mn>0</mn></mrow></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> (or equivalent) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>n</mi><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mi>n</mi><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mfenced><mrow><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Other approaches are possible. For example, award<em><strong> A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>tan</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math> and <em><strong>M1</strong></em> for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
<p><strong><br>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> (or equivalent) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>n</mi><msub><mi>A</mi><mi>T</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mn>2</mn></msup><mfenced><mrow><mi>sin</mi><mfrac><mstyle displaystyle="true"><mn>2</mn><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mn>2</mn></msup><mfenced><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><msup><mfenced><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mfenced><mn>3</mn></msup></mstyle><mn>3</mn></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><mn>2</mn><msup><mfenced><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mfenced><mn>5</mn></msup></mstyle><mn>15</mn></mfrac><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mn>4</mn><mi>n</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>+</mo><mfrac><msup><mi mathvariant="normal">π</mi><mn>3</mn></msup><mrow><mn>3</mn><msup><mi>n</mi><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi mathvariant="normal">π</mi><mn>5</mn></msup></mrow><mrow><mn>15</mn><msup><mi>n</mi><mn>5</mn></msup></mrow></mfrac><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></mrow></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mfenced><mrow><mi mathvariant="normal">π</mi><mo>+</mo><mfrac><msup><mi mathvariant="normal">π</mi><mn>3</mn></msup><mrow><mn>3</mn><msup><mi>n</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi mathvariant="normal">π</mi><mn>5</mn></msup></mrow><mrow><mn>15</mn><msup><mi>n</mi><mn>4</mn></msup></mrow></mfrac><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mo>=</mo><mn>4</mn><mi mathvariant="normal">π</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>M1A1A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder></math> is not stated anywhere.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><mi>P</mi><mo>→</mo><mn>4</mn><mi mathvariant="normal">π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>→</mo><mn>4</mn><mi mathvariant="normal">π</mi></math>)</p>
<p>the polygon becomes a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong></em> for alternative responses such as:<br>the polygon becomes a circle of area <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">π</mi></math> OR<br>the polygon becomes a circle of perimeter <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">π</mi></math> OR<br>the polygon becomes a circle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi mathvariant="normal">π</mi></math>.<br>Award <em><strong>R0</strong></em> for polygon becomes a circle.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> <em><strong>(A1)(A1)</strong></em></p>
<p>equates their expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for isolating <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>2</mn><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math>. This step may be seen later.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></mrow></mfenced><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>+</mo><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo>-</mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to expand their RHS of either <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mo>…</mo></math>.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>a</mi><mi>b</mi><mo>≠</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mo>-</mo><mn>4</mn><mi>a</mi><mo>=</mo><mn>4</mn><mi>b</mi><mo>-</mo><mn>8</mn></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo>-</mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mi>b</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>2</mn><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>a</mi><mfenced><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>b</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>8</mn><mi>b</mi><mo>-</mo><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>b</mi></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>b</mi></mrow></mfrac></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>b</mi><mo>-</mo><mn>8</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>b</mi><mo>-</mo><mn>16</mn><mo>+</mo><mn>8</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>8</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>4</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>A1A1M1M1M0A0A0</strong></em> for attempting to verify.<br>For example, verifying that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mfrac><mn>16</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mi>b</mi><mo>+</mo><mn>4</mn></math> gains <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> marks.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using an appropriate method <em><strong>(M1)</strong></em></p>
<p><em>eg</em> substituting values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> or using divisibility properties</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>12</mn><mo>,</mo><mo> </mo><mn>13</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>6</mn><mo>,</mo><mo> </mo><mn>8</mn><mo>,</mo><mo> </mo><mn>10</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> for either one set of three correct side lengths or two sets of two correct side lengths.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>30</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>24</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>A1FT</strong></em>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">g.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.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore the behaviour and some key features of the function</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><mo>(</mo><mi>a</mi><mo>-</mo><mi>x</mi><msup><mo>)</mo><mi>n</mi></msup><mo> </mo></math><strong>, where</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> <strong>and</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math><strong>.</strong></p>
<p>In parts (a) and (b), <strong>only</strong> consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn></math>.</p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>(</mo><mn>2</mn><mo>-</mo><mi>x</mi><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mn>2</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>n</mi><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>n</mi><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>By using the result from part (f) and considering the sign of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced></math>, show that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></math> is</p>
</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>y</mi><mo>=</mo><msub><mi>f</mi><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo></math>, stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.</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>Use your graphic display calculator to explore the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> for</p>
<p>• the odd values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math>;</p>
<p>• the even values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>.</p>
<p>Hence, copy and complete the following table.</p>
<p><img src="data:image/png;base64,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"></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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>.</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>State the three solutions to the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></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>Show that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></mrow></mfenced></math> on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></math> is always above the horizontal axis.</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>Hence, or otherwise, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>a local minimum point for even values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>a point of inflexion with zero gradient for odd values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>-</mo><mi>k</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>State the conditions on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> such that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>=</mo><mi>k</mi></math> has four solutions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img 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"></p>
<p>inverted parabola extended below the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn></math> <em><strong>A1</strong></em><br><br><br><strong>Note:</strong> Accept a graph passing through the origin as an indication of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.<br><br></p>
<p>local maximum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p><br><strong>Note:</strong> Coordinates must be stated to gain the final <em><strong>A1</strong></em>.<br> Do not accept decimal approximations.</p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1</strong></em><em><strong>A1A1A1A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for each correct value.</p>
<p style="padding-left:30px;">For a table not sufficiently or clearly labelled, assume that their values are in the same order as the table in the question paper and award marks accordingly.</p>
<p><br><em><strong>[6 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>attempts to use the product rule <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> and <em><strong>A1</strong></em> for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</p>
<p><br><strong>EITHER</strong></p>
<p>attempts to factorise <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced></math> (involving at least one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>) <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>-</mo><mi>x</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced></math> as the difference of two products with each product containing at least one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Award the final <em><strong>(M1)A1</strong></em> for obtaining any of the following forms: </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mfenced><mfrac><mrow><mi>a</mi><mo>-</mo><mi>x</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>x</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac></mfenced><mo>;</mo><mo> </mo><mo> </mo><mo> </mo><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></mrow><mrow><mi>x</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>;</mo></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>-</mo><mi>x</mi><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>;</mo></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>-</mo><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup></mrow></mfenced></math></p>
<p> </p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mi>x</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mi>n</mi></msup></math> <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>a</mi><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup></math> <em><strong>A1</strong></em></p>
<p>attempts to use the chain rule <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>a</mi><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>AG</strong></em></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><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>a</mi></math> <em>A2</em></strong></p>
<p><strong>Note: </strong>Award <em><strong>A1</strong></em> for either two correct solutions or for obtaining <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></math><br> </strong> Award<em><strong> A0 </strong></em>otherwise.</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>attempts to find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mo>=</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></mrow></mfenced><mi>n</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mi>n</mi></msup><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mi>n</mi></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow></mfenced><mo>,</mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mi>n</mi></msup></mfenced><mn>2</mn></msup></mrow></mfenced></math><strong> <em>A1</em></strong></p>
<p><br><strong>EITHER</strong></p>
<p>since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>></mo><mn>0</mn></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>n</mi><mo>></mo><mn>1</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Accept any logically equivalent conditions/statements on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.<br> Award <em><strong>R0</strong></em> if any conditions/statements specified involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> or both are incorrect.</p>
<p> </p>
<p><strong>OR</strong></p>
<p>(since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>), <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mn>2</mn></mfrac></math> raised to an even power (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi></math>) (or equivalent reasoning) is always positive (and so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The condition <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> is given in the question. Hence some candidates will assume <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and not state it. In these instances, award <em><strong>R1</strong></em> for a convincing argument.<br> Accept any logically equivalent conditions/statements on on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.<br> Award <em><strong>R0</strong></em> if any conditions/statements specified involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> or both are incorrect.</p>
<p><br><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></mrow></mfenced></math> is always above the horizontal axis<strong> <em>AG</em></strong></p>
<p><br><strong>Note:</strong> Do not award <em><strong>(M1)A0R1</strong></em>.</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><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>=</mo><mi>n</mi><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>n</mi><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><msup><mfenced><mfrac><mrow><mn>3</mn><mi>a</mi></mrow><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math><strong> <em>A1</em></strong></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><msup><mfenced><mfrac><mrow><mn>3</mn><mi>a</mi></mrow><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>></mo><mn>0</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo><mo> </mo><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> are all <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>A0R1</strong></em>.<br> Accept equivalent reasoning on correct alternative expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced></math> and accept any logically equivalent conditions/statements on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<p> Exceptions to the above are condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math> and condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>0</mn></math>.</p>
<p> An alternative form for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>n</mi></mrow></mfenced><msup><mfenced><mn>3</mn></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<p><br><strong>THEN</strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math><strong> <em>A1</em></strong></p>
<p>(since <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub></math> is continuous and there are no stationary points between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></math>)</p>
<p>the gradient (of the curve) must be positive between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></math> <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Do not award <em><strong>A0R1</strong></em>.</p>
<p><br>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mi>n</mi><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> even:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mo>=</mo><mo>-</mo><mi>n</mi></mrow></mfenced><mo><</mo><mn>0</mn></math> (and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mo> </mo><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> are both <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo><</mo><mn>0</mn></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math> (seen anywhere)<strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Candidates can give arguments based on the sign of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> to obtain the <em><strong>R</strong></em> mark.<br> For example, award<em><strong> R1</strong></em> for the following:<br> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is even, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn></math> is odd and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo><</mo><mn>0</mn><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>.<br> Do not award <em><strong>R0A1</strong></em>.<br> The second <strong><em>A1</em></strong> is independent of the other two marks.<br> The<em><strong> A</strong></em> marks can be awarded for correct descriptions expressed in words.<br> Candidates can state <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 a point of zero gradient from part (d) or show, state or explain (words or diagram) that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math>. The last <em><strong>A </strong></em>mark can be awarded for a clearly labelled diagram showing changes in the sign of the gradient.<br> The last <em><strong>A1</strong></em> can be awarded for use of a specific case (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>).</p>
<p><br>hence <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 a local minimum point<strong> <em>AG</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> odd:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mo>=</mo><mi>n</mi></mrow></mfenced><mo><</mo><mn>0</mn></math>, (and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mo> </mo><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> are both <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>0</mn></math>) so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>></mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Candidates can give arguments based on the sign of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> to obtain the <em><strong>R</strong></em> mark.<br> For example, award<em><strong> R1</strong></em> for the following:<br> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is odd, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn></math> is even and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>></mo><mn>0</mn><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn></mrow></mfenced></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math> (seen anywhere)<strong> <em>A1</em></strong></p>
<p><br><strong>Note:</strong> The <em><strong>A1</strong></em> is independent of the <em><strong>R1</strong></em>.<br> Candidates can state <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> as a point of zero gradient from part (d) or show, state or explain (words or diagram) that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math>. The last <em><strong>A</strong></em> mark can be awarded for a clearly labelled diagram showing changes in the sign of the gradient.<br> The last <em><strong>A1</strong></em> can be awarded for use of a specific case (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math>).</p>
<p> </p>
<p>hence <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 a point of inflexion with zero gradient<strong> <em>AG</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>considers the parity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong> (M1)</strong></em></p>
<p><br><strong>Note:</strong> Award<em><strong> M1</strong> </em>for stating at least one specific even value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> must be even (for four solutions) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The above 2 marks are independent of the 3 marks below.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi></mrow></msup></math> <em><strong>A1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for the correct lower endpoint, <em><strong>A1</strong></em> for the correct upper endpoint and <em><strong>A1</strong></em> for strict inequality signs.</p>
<p> The third <em><strong>A1</strong></em> (strict inequality signs) can only be awarded if <em><strong>A1</strong></em><em><strong>A1</strong></em> has been awarded.<br> For example, award <em><strong>A1A1A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi></mrow></msup></math>. Award <em><strong>A1A0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>0</mn></math>.</p>
<p> Award <em><strong>A1A0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></math>.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will be exploring the strategies required to solve a system of linear differential equations.</strong></p>
<p> </p>
<p>Consider the system of linear differential equations of the form:</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>x</mi><mo>-</mo><mi>y</mi></math> and <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><mi>a</mi><mi>x</mi><mo>+</mo><mi>y</mi></math>,</p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>,</mo><mo> </mo><mi>t</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is a parameter.</p>
<p>First consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>4</mn></math>.</p>
</div>
<div class="specification">
<p>From previous cases, we might conjecture that a solution to this differential equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation <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><mi>y</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is a constant.</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>Show that <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>x</mi><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math>.</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>Solve the differential equation in part (a)(ii) to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</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>By differentiating <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><mi>x</mi><mo>+</mo><mi>y</mi></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is a constant.</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>Hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iv.</div>
</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><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></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 two values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let the two values found in part (c)(ii) be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math>.</p>
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math> is a solution to the differential equation in (c)(i),where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> is a constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>=</mo><mi>y</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mi>y</mi></mfrac><mo>=</mo><mo>∫</mo><mo>d</mo><mtext>t</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>y</mi><mo>=</mo><mi>t</mi><mo>+</mo><mi>c</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mfenced open="|" close="|"><mi>y</mi></mfenced><mo>=</mo><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em><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"><mi>ln</mi><mo> </mo><mi>y</mi></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>rearranging to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>y</mi><mo>=</mo><mn>0</mn></math> AND multiplying by integrating factor <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mi>A</mi></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></p>
<p> </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>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> into differential equation in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></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>integrating factor (IF) is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>∫</mo><mo>-</mo><mn>1</mn><mo>d</mo><mi>t</mi></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>A</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>A</mi><mi>t</mi><mo>+</mo><mi>D</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfenced><mrow><mo>-</mo><mi>A</mi><mi>t</mi><mo>+</mo><mi>D</mi></mrow></mfenced><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The first constant must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, and the second can be any constant for the final <em><strong>A1</strong></em> to be awarded. Accept a change of constant applied at the end.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><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><mfrac><mrow><mo>d</mo><mi>y</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></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mfenced><mrow><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[3 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"><mfrac><mrow><mo>d</mo><mi>Y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>Y</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>Y</mi></mrow><mi>Y</mi></mfrac><mo>=</mo><mo>∫</mo><mn>2</mn><mo>d</mo><mi>t</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced open="|" close="|"><mi>Y</mi></mfenced><mo>=</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi>c</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>Y</mi><mo>=</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>AG</strong></em></p>
<p> </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><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><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>∫</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mtext> </mtext><mo>d</mo><mi>t</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> The first constant must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, and the second can be any constant for the final <em><strong>A1</strong></em> to be awarded. Accept a change of constant applied at the end.</p>
<p> </p>
<p><em><strong>[2 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>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><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> and their (iii) into <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><mi>x</mi><mo>+</mo><mi>y</mi></math> <em><strong>M1(M1)</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> A1</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> AG</strong></em></p>
<p><strong>Note:</strong> Follow through from incorrect part (iii) cannot be awarded if it does not lead to the <em><strong>AG</strong></em>.</p>
<p><br><strong>METHOD 2</strong></p>
<p><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>x</mi><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi></math></p>
<p><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>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>∫</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mtext> </mtext><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>+</mo><mi>D</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi><mo>+</mo><mi>D</mi><msup><mtext>e</mtext><mi>t</mi></msup></math></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><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>⇒</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>=</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi><mo>-</mo><mi>D</mi><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi><mo>⇒</mo><mi>D</mi><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.iv.</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>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mi>y</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><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></math> seen anywhere <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>attempt to eliminate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>4</mn><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mrow><mi>y</mi><mo>-</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>3</mn><mi>y</mi></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>rewriting LHS in terms 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> <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>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mfenced><mrow><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>-</mo><mn>3</mn><mi>y</mi></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></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"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>F</mi><mi>λ</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>F</mi><msup><mi>λ</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup></math><em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><msup><mi>λ</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mi>F</mi><mi>λ</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></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>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>≠</mo><mn>0</mn></math>)<em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math> (either order)<em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[4 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"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math></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>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced></math><em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>6</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>2</mn><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math></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><mi>F</mi><msub><mi>λ</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msub><mi>λ</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mstyle displaystyle="true"><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mstyle></mfrac><mo>=</mo><mi>F</mi><msup><msub><mi>λ</mi><mn>1</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><msub><mi>λ</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mi>F</mi><msup><msub><mi>λ</mi><mn>1</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><msub><mi>λ</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>F</mi><msub><mi>λ</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msub><mi>λ</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></mrow></mfenced></math><em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mfenced><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mfenced><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn></mrow></mfenced></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.iii.</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">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iv.</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">c.iii.</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 defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{arcsin}}\left( {2x} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{2} \leqslant x \leqslant \frac{1}{2}">
<mo>−<!-- − --></mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By finding a suitable number of derivatives of <span class="mjpage"><math alttext="f" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> </math></span>, find the first two non-zero terms in the Maclaurin series for <span class="mjpage"><math alttext="f" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find <span class="mjpage"><math alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{{\text{arcsin}}\left( {2x} \right) - 2x}}{{{{\left( {2x} \right)}^3}}}" xmlns="http://www.w3.org/1998/Math/MathML"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mrow> <mrow> <mtext>arcsin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math alttext="f\left( x \right) = {\text{arcsin}}\left( {2x} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>arcsin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math alttext="f'\left( x \right) = \frac{2}{{\sqrt {1 - 4{x^2}} }}" xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>2</mn> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <em><strong>M1A</strong></em><em><strong>1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for <span class="mjpage"><math alttext="f'\left( x \right) = \frac{1}{{\sqrt {1 - 4{x^2}} }}" xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math alttext="f''\left( x \right) = \frac{{8x}}{{{{\left( {1 - 4{x^2}} \right)}^{\frac{3}{2}}}}}" xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math alttext="f'''\left( x \right) = \frac{{8{{\left( {1 - 4{x^2}} \right)}^{\frac{3}{2}}} - 8x\left( {\frac{3}{2}\left( { - 8x} \right){{\left( {1 - 4{x^2}} \right)}^{\frac{1}{2}}}} \right)}}{{{{\left( {1 - 4{x^2}} \right)}^3}}}\,\,\,\left( { = \frac{{8{{\left( {1 - 4{x^2}} \right)}^{\frac{3}{2}}} + 96{x^2}{{\left( {1 - 4{x^2}} \right)}^{\frac{1}{2}}}}}{{{{\left( {1 - 4{x^2}} \right)}^3}}}} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>f</mi> <mo>‴</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>−</mo> <mn>8</mn> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>8</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>+</mo> <mn>96</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math alttext="f'''\left( x \right) = 8{\left( {1 - 4{x^2}} \right)^{ - \frac{3}{2}}} + 8x\left( { - \frac{3}{2}{{\left( {1 - 4{x^2}} \right)}^{ - \frac{5}{2}}}} \right)\left( { - 8x} \right)\,\,\,\left( { = 8{{\left( {1 - 4{x^2}} \right)}^{ - \frac{3}{2}}} + 96{x^2}{{\left( {1 - 4{x^2}} \right)}^{ - \frac{5}{2}}}} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>f</mi> <mo>‴</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>8</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>8</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>8</mn> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>+</mo> <mn>96</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p>substitute <span class="mjpage"><math alttext="x = 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> into <span class="mjpage"><math alttext="f" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> </math></span> or any of its derivatives <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math alttext="f\left( 0 \right) = 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math alttext="f'\left( 0 \right) = 2" xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> </math></span> and <span class="mjpage"><math alttext="f''\left( 0 \right) = 0" xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math alttext="f'''\left( 0 \right) = 8" xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>f</mi> <mo>‴</mo> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>8</mn> </math></span></p>
<p>the Maclaurin series is</p>
<p><span class="mjpage"><math alttext="f\left( x \right) = 2x + \frac{{8{x^3}}}{6} + \ldots \,\left( { = 2x + \frac{{4{x^3}}}{3} + \ldots } \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mn>8</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mn>6</mn> </mfrac> <mo>+</mo> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mn>3</mn> </mfrac> <mo>+</mo> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[8 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><span class="mjpage"><math alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{{\text{arcsin}}\left( {2x} \right) - 2x}}{{{{\left( {2x} \right)}^3}}} = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{2x + \frac{{4{x^3}}}{3} + \ldots - 2x}}{{8{x^3}}}" xmlns="http://www.w3.org/1998/Math/MathML"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mrow> <mrow> <mtext>arcsin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mn>3</mn> </mfrac> <mo>+</mo> <mo>…</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>8</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math alttext=" = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\frac{4}{3} + \ldots {\text{ terms with }}x}}{8}" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> <mo>+</mo> <mo>…</mo> <mrow> <mtext> terms with </mtext> </mrow> <mi>x</mi> </mrow> <mn>8</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math alttext=" = \frac{1}{6}" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Condone the omission of +… in their working.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{{\text{arcsin}}\left( {2x} \right) - 2x}}{{{{\left( {2x} \right)}^3}}} = \frac{0}{0}" xmlns="http://www.w3.org/1998/Math/MathML"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mrow> <mrow> <mtext>arcsin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>0</mn> <mn>0</mn> </mfrac> </math></span> indeterminate form, using L’Hôpital’s rule</p>
<p><span class="mjpage"><math alttext=" = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\frac{2}{{\sqrt {1 - 4{x^2}} }} - 2}}{{24{x^2}}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mrow> <mfrac> <mn>2</mn> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mn>24</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math alttext=" = \frac{0}{0}" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mfrac> <mn>0</mn> <mn>0</mn> </mfrac> </math></span> indeterminate form, using L’Hôpital’s rule again</p>
<p><span class="mjpage"><math alttext=" = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\frac{{8x}}{{{{\left( {1 - 4{x^2}} \right)}^{\frac{3}{2}}}}}}}{{48x}}\left( { = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{1}{{6{{\left( {1 - 4{x^2}} \right)}^{\frac{3}{2}}}}}} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mrow> <mfrac> <mrow> <mn>8</mn> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> <mrow> <mn>48</mn> <mi>x</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mn>1</mn> <mrow> <mn>6</mn> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> only if their previous expression is in indeterminate form.</p>
<p><span class="mjpage"><math alttext=" = \frac{1}{6}" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>FT</strong></em> for use of their derivatives from part (a).</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Using L’Hôpital’s rule, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{tan}}\,3x - 3\,{\text{tan}}\,x}}{{{\text{sin}}\,3x - 3\,{\text{sin}}\,x}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</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="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{tan}}\,3x - 3\,{\text{tan}}\,x}}{{{\text{sin}}\,3x - 3\,{\text{sin}}\,x}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{3}}\,{\text{se}}{{\text{c}}^2}\,3x - 3\,{\text{se}}{{\text{c}}^2}\,x}}{{{\text{3}}\,{\text{cos}}\,3x - 3\,{\text{cos}}\,x}}} \right)\,\,\,\,\,\left( { = \mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{se}}{{\text{c}}^2}\,3x - {\text{se}}{{\text{c}}^2}\,x}}{{{\text{cos}}\,3x - {\text{cos}}\,x}}} \right)\,} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>3</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>3</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt at differentiation using l'Hopital's rule, <em><strong>A1</strong></em> for numerator, <em><strong>A1</strong></em> for denominator.</p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p>using l’Hopital’s rule again</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{18}}\,{\text{se}}{{\text{c}}^2}\,3x\,{\text{tan}}\,3x - 6\,{\text{se}}{{\text{c}}^2}\,x\,{\text{tan}}\,x}}{{ - 9\,{\text{sin}}\,3x + 3\,{\text{sin}}\,x}}} \right)\,\,\left( { = \mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{6}}\,{\text{se}}{{\text{c}}^2}\,3x\,{\text{tan}}\,3x - 2\,{\text{se}}{{\text{c}}^2}\,x\,{\text{tan}}\,x}}{{ - 3\,{\text{sin}}\,3x + {\text{sin}}\,x}}} \right)} \right)">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>18</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mo>−</mo>
<mn>9</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>6</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{108}}\,{\text{se}}{{\text{c}}^2}\,3x\,{\text{ta}}{{\text{n}}^2}\,3x + 54\,{\text{se}}{{\text{c}}^4}\,3x - 12\,{\text{se}}{{\text{c}}^2}\,x\,{\text{ta}}{{\text{n}}^2}\,x - 6\,{\text{se}}{{\text{c}}^4}\,x}}{{{\text{ - 27}}\,{\text{cos}}\,3x + 3\,{\text{cos}}\,x}}} \right)">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>108</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>54</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>4</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>12</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>4</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext> - 27</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Not all terms in numerator need to be written in final fraction. Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="54\,{\text{se}}{{\text{c}}^4}\,3x + \ldots - 6\,{\text{se}}{{\text{c}}^4}\,x \ldots - ">
<mn>54</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>4</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mo>…</mo>
<mo>−</mo>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>4</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>…</mo>
<mo>−</mo>
</math></span>. However, if the terms are written, they<br>must be correct to award A1.</p>
<p>attempt to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<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=" = \frac{{48}}{{ - 24}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>48</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>24</mn>
</mrow>
</mfrac>
</math></span></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}x}}\left( {{\text{18}}\,{\text{se}}{{\text{c}}^2}\,3x\,{\text{tan}}\,3x - 6\,{\text{se}}{{\text{c}}^2}\,x\,{\text{tan}}\,x} \right)\left| {_{x = 0} = 48} \right.">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>18</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<msub>
<mi></mi>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>48</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}x}}\left( { - 9\,{\text{sin}}\,3x + 3\,{\text{sin}}\,x} \right)\left| {_{x = 0} = - 24} \right.">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>9</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<msub>
<mi></mi>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mo>−</mo>
<mn>24</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{tan}}\,3x - 3\,{\text{tan}}\,x}}{{{\text{sin}}\,3x - 3\,{\text{sin}}\,x}}} \right)} \right) = - 2">
<mrow>
<mo>(</mo>
<mrow>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</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=" = \mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{\frac{3}{{{\text{co}}{{\text{s}}^2}\,3x}} - \frac{3}{{{\text{co}}{{\text{s}}^2}\,x}}}}{{{\text{3}}\,{\text{cos}}\,3x - 3\,{\text{cos}}\,x}}} \right)">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mfrac>
<mn>3</mn>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mrow>
<mrow>
<mtext>3</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</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=" = \mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{co}}{{\text{s}}^2}\,x - {\text{co}}{{\text{s}}^2}\,3x}}{{{\text{co}}{{\text{s}}^2}\,3x\,{\text{co}}{{\text{s}}^2}\,x\left( {{\text{cos}}\,3x - {\text{cos}}\,x} \right)}}} \right)">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</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=" = \mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{cos}}\,x + {\text{cos}}\,3x}}{{ - {\text{co}}{{\text{s}}^2}\,3x\,{\text{co}}{{\text{s}}^2}\,x}}} \right)">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mrow>
<mo>−</mo>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>attempt to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<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=" = \frac{2}{{ - 1}}">
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 2">
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[9 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 simple model to predict the population of the world is set up as follows. At time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> years the population of the world is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, which can be assumed to be a continuous variable. The rate of increase of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> due to births is 0.056<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and the rate of decrease of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> due to deaths is 0.035<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</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="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0.021x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.021</mn>
<mi>x</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>Find a prediction for the number of years it will take for the population of the world to double.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0.056x - 0.035x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.056</mn>
<mi>x</mi>
<mo>−</mo>
<mn>0.035</mn>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0.021x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.021</mn>
<mi>x</mi>
</math></span> <em><strong>AG</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>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0.021x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.021</mn>
<mi>x</mi>
</math></span></p>
<p>attempt to separate variables <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{x}} {\text{d}}x = \int {0.021} \,{\text{d}}t">
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mn>0.021</mn>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,x = 0.021t( + c)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>0.021</mn>
<mi>t</mi>
<mo stretchy="false">(</mo>
<mo>+</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = A{{\text{e}}^{0.021t}}">
<mi>x</mi>
<mo>=</mo>
<mi>A</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>0.021</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 2A = A{{\text{e}}^{0.021t}}">
<mo stretchy="false">⇒</mo>
<mn>2</mn>
<mi>A</mi>
<mo>=</mo>
<mi>A</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>0.021</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> This <em><strong>A1</strong> </em>is independent of the following marks.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0{\text{,}}\,\,x = {x_0} \Rightarrow c = {\text{ln}}\,{x_0}">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mi>c</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{ln}}\,2{x_0} = 0.021t + {\text{ln}}\,{x_0}">
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mrow>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>0.021</mn>
<mi>t</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> This <em><strong>A1</strong> </em>is independent of the following marks.</p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{ln}}\,2 = 0.021t">
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>=</mo>
<mn>0.021</mn>
<mi>t</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow t = 33">
<mo stretchy="false">⇒</mo>
<mi>t</mi>
<mo>=</mo>
<mn>33</mn>
</math></span> years <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> If a candidate writes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 33.007">
<mi>t</mi>
<mo>=</mo>
<mn>33.007</mn>
</math></span>, so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 34">
<mi>t</mi>
<mo>=</mo>
<mn>34</mn>
</math></span> then award the final <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="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0.021x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.021</mn>
<mi>x</mi>
</math></span></p>
<p>attempt to separate variables <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_A^{2A} {\frac{1}{x}{\text{d}}x = \int_0^t {0.021\,{\text{d}}u} } ">
<msubsup>
<mo>∫</mo>
<mi>A</mi>
<mrow>
<mn>2</mn>
<mi>A</mi>
</mrow>
</msubsup>
<mrow>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mi>t</mi>
</msubsup>
<mrow>
<mn>0.021</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
</mrow>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct integrals and <em><strong>A1</strong></em> for correct limits seen anywhere. Do not penalize use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> in place of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {{\text{ln}}\,x} \right]_A^{2A} = \left[ {0.021u} \right]_0^t">
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
<mi>A</mi>
<mrow>
<mn>2</mn>
<mi>A</mi>
</mrow>
</msubsup>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mn>0.021</mn>
<mi>u</mi>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mi>t</mi>
</msubsup>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{ln}}\,2 = 0.021t">
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>=</mo>
<mn>0.021</mn>
<mi>t</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow t = 33">
<mo stretchy="false">⇒</mo>
<mi>t</mi>
<mo>=</mo>
<mn>33</mn>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to determine the value of</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^2}x}}{{x\ln (1 + x)}}.">
<munder>
<mrow>
<mo form="prefix">lim</mo>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mi>x</mi>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>.</mo>
</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>attempt to use l’Hôpital’s rule, <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{limit}} = \mathop {\lim }\limits_{x \to 0} \frac{{2\sin x\cos x}}{{\ln (1 + x) + \frac{x}{{1 + x}}}}">
<mrow>
<mtext>limit</mtext>
</mrow>
<mo>=</mo>
<munder>
<mrow>
<mo form="prefix">lim</mo>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span>or<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin 2x}}{{\ln (1 + x) + \frac{x}{{1 + x}}}}">
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for numerator <strong><em>A1 </em></strong>for denominator.</p>
<p> </p>
<p>this gives 0/0 so use the rule again <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\cos }^2}x - 2{{\sin }^2}x}}{{\frac{1}{{1 + x}} + \frac{{1 + x - x}}{{{{(1 + x)}^2}}}}}">
<mo>=</mo>
<munder>
<mrow>
<mo form="prefix">lim</mo>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span>or<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\cos 2x}}{{\frac{{2 + x}}{{{{(1 + x)}^2}}}}}">
<mfrac>
<mrow>
<mn>2</mn>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for numerator <strong><em>A1 </em></strong>for denominator.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1">
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> This <strong><em>A1 </em></strong>is dependent on all previous marks being awarded, except when the first application of L’Hopital’s does not lead to 0/0, when it should be awarded for the correct limit of their derived function.</p>
<p> </p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to find</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><mfrac><mrow><mi>cos</mi><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>1</mn></mrow><mrow><msup><mtext>e</mtext><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mi>x</mi></mrow></mfrac></math>.</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>attempt to use l’Hôpital’s rule <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><mfrac><mrow><mo>-</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><msup><mtext>e</mtext><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for the numerator and <em><strong>A1</strong></em> for the denominator.</p>
<p><br>substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> into their expression <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac></math> hence use l’Hôpital’s rule again</p>
<p><br><strong>Note:</strong> If the first use of l’Hôpital’s rule results in an expression which is not in indeterminate form, do not award any further marks.</p>
<p><br>attempt to use product rule in numerator <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><mfrac><mrow><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><msup><mtext>e</mtext><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><br><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>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} + \left( {\frac{{2x}}{{1 + {x^2}}}} \right)y = {x^2}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>, given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</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="1 + {x^2}">
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> is an integrating factor for this differential equation.</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>Hence solve this differential equation. Give the answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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>attempting to find an integrating factor <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{2x}}{{1 + {x^2}}}{\text{d}}x = \ln (1 + {x^2})} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p>IF is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{\ln (1 + {x^2})}}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</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=" = 1 + {x^2}">
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>multiply by the integrating factor</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1 + {x^2})\frac{{{\text{d}}y}}{{{\text{d}}x}} + 2xy = {x^2}(1 + {x^2})">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p>left hand side is equal to the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1 + {x^2})y">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mi>y</mi>
</math></span></p>
<p><strong><em>A3</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1 + {x^2})\frac{{{\text{d}}y}}{{{\text{d}}x}} + 2xy = (1 + {x^2}){x^2}">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}x}}\left[ {(1 + {x^2})y} \right] = {x^2} + {x^4}">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mi>y</mi>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1 + {x^2})y = \left( {\int {{x^2} + {x^4}{\text{d}}x = } } \right){\text{ }}\frac{{{x^3}}}{3} + \frac{{{x^5}}}{5}( + c)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mi>y</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mo>+</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{1 + {x^2}}}\left( {\frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} + c} \right)">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0,{\text{ }}y = 2 \Rightarrow c = 2">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">⇒</mo>
<mi>c</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{1 + {x^2}}}\left( {\frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} + 2} \right)">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br>