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<h2>SL Paper 1</h2><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined for all&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#8712;</mo><mi mathvariant="normal">&#8477;</mi></math>. The line with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>1</mn></math> is the tangent to&nbsp;the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</p>
</div>

<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#8712;</mo><mi mathvariant="normal">&#8477;</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo><mo>(</mo><mn>4</mn><mo>)</mo></math>.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>4</mn><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>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mn>4</mn><mo>)</mo></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>Hence find the equation of the tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mn>4</mn><mo>)</mo><mo>=</mo><mn>6</mn></math>               <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1</strong></em><em><strong> mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>4</mn><mo>)</mo><mo>=</mo><mn>6</mn><mo>×</mo><mn>4</mn><mo>-</mo><mn>1</mn><mo>=</mo><mn>23</mn></math>               <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1</strong></em><em><strong> 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>h</mi><mfenced><mn>4</mn></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mn>4</mn></mfenced></mrow></mfenced></math>                 <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>4</mn></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><msup><mn>4</mn><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>×</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>4</mn></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>4</mn></mfenced><mo>=</mo><mn>23</mn></math>                 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use chain rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo></math>                 <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mrow><mi>g</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced><mo>×</mo><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>x</mi></mrow></mfenced><mo>'</mo><mo>×</mo><mi>f</mi><mo>'</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>x</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mn>4</mn></mfenced><mo>=</mo><mfenced><mrow><mn>2</mn><mo>×</mo><mn>4</mn><mo>-</mo><mn>3</mn></mrow></mfenced><mi>f</mi><mo>'</mo><mfenced><mrow><msup><mn>4</mn><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>×</mo><mn>4</mn></mrow></mfenced></math>                 <em><strong>A1</strong></em></p>
<p>         <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>30</mn></math> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>23</mn><mo>=</mo><mn>30</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>30</mn><mi>x</mi><mo>-</mo><mn>97</mn></math>                 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3</strong></em><em><strong> 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>Let&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {p^x} + q">
  <mi>g</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>p</mi>
      <mi>x</mi>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>q</mi>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x{\text{, }}p{\text{, }}q \in \mathbb{R}{\text{, }}p > 1">
  <mi>x</mi>
  <mrow>
    <mtext>,&nbsp;</mtext>
  </mrow>
  <mi>p</mi>
  <mrow>
    <mtext>,&nbsp;</mtext>
  </mrow>
  <mi>q</mi>
  <mo>∈<!-- ∈ --></mo>
  <mrow>
    <mi mathvariant="double-struck">R</mi>
  </mrow>
  <mrow>
    <mtext>,&nbsp;</mtext>
  </mrow>
  <mi>p</mi>
  <mo>&gt;</mo>
  <mn>1</mn>
</math></span>. The point&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( {0{\text{, }}a} \right)">
  <mrow>
    <mtext>A</mtext>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>0</mn>
      <mrow>
        <mtext>,&nbsp;</mtext>
      </mrow>
      <mi>a</mi>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>&nbsp;lies on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
  <mi>g</mi>
</math></span>.</p>
<p>Let&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {g^{ - 1}}\left( x \right)">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo>−<!-- − --></mo>
        <mn>1</mn>
      </mrow>
    </msup>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
</math></span>.&nbsp;The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
  <mrow>
    <mtext>B</mtext>
  </mrow>
</math></span> lies on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> and is the reflection of point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
  <mrow>
    <mtext>A</mtext>
  </mrow>
</math></span> in the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x">
  <mi>y</mi>
  <mo>=</mo>
  <mi>x</mi>
</math></span>.</p>
</div>

<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
  <mrow>
    <msub>
      <mi>L</mi>
      <mn>1</mn>
    </msub>
  </mrow>
</math></span> is tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
  <mrow>
    <mtext>B</mtext>
  </mrow>
</math></span>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}"> <mrow> <mtext>B</mtext> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( a \right) = \frac{1}{{{\text{ln}}\,p}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> </math></span>, find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> <strong>in terms of</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span> is tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> and has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left( {{\text{ln}}\,p} \right)x + q + 1"> <mi>y</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span>.</p>
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span> passes through the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 2{\text{, }} - 2} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mrow> <mtext>, </mtext> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p>The gradient of the normal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}"> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span>.</p>
<p> </p>
<p>Find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\left( {a{\text{, }}0} \right)"> <mrow> <mtext>B</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>  (accept  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\left( {q + 1{\text{, }}0} \right)"> <mrow> <mtext>B</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> <mrow> <mtext>, </mtext> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>)           <em><strong>A2</strong></em><em><strong>   N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> There are many approaches to this part, and the steps may be done in any order. Please check working and award marks in line with the markscheme, noting that candidates may work with the equation of the line before finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<p> </p>
<p><strong>FINDING <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span></strong></p>
<p>valid attempt to find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>       <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( 0 \right) = a{\text{, }}\,{p^0} + q = a"> <mi>g</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>p</mi> <mn>0</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mi>a</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = q + 1"> <mi>a</mi> <mo>=</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span>       <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>FINDING THE EQUATION OF</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span></p>
<p style="padding-left:30px;"><strong>EITHER</strong></p>
<p style="padding-left:30px;">attempt to substitute tangent gradient and coordinates into equation of straight line        <em><strong>(M1)</strong></em></p>
<p style="padding-left:30px;"><em>eg</em>       <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = f'\left( a \right)\left( {x - a} \right){\text{, }}\,y = f'\left( a \right)\left( {x - \left( {q + 1} \right)} \right)"> <mi>y</mi> <mo>−</mo> <mn>0</mn> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p style="padding-left:30px;">correct equation in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>       <em><strong>(A1)</strong></em></p>
<p style="padding-left:30px;"><em>eg</em>       <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = \frac{1}{{{\text{ln}}\left( p \right)}}\left( {x - a} \right)"> <mi>y</mi> <mo>−</mo> <mn>0</mn> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p style="padding-left:30px;"><strong>OR</strong></p>
<p style="padding-left:30px;">attempt to substitute tangent gradient and coordinates to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span></p>
<p style="padding-left:30px;"><em>eg</em>       <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 = \frac{1}{{{\text{ln}}\left( p \right)}}\left( a \right) + b"> <mn>0</mn> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> </math></span></p>
<p style="padding-left:30px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{{ - a}}{{{\text{ln}}\left( p \right)}}"> <mi>b</mi> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span>       <em><strong>(A1)</strong></em></p>
<p><strong>THEN</strong> (must be in terms of <strong>both</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{{\text{ln}}\,p}}\left( {x - q - 1} \right){\text{, }}\,y = \frac{1}{{{\text{ln}}\,p}}x - \frac{{q + 1}}{{{\text{ln}}\,p}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> <mi>x</mi> <mo>−</mo> <mfrac> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> </math></span>           <em><strong>A1</strong></em><em><strong>   N3</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for final answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1} = \frac{1}{{{\text{ln}}\,p}}\left( {x - q - 1} \right)"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> There are many approaches to this part, and the steps may be done in any order. Please check working and award marks in line with the markscheme, noting that candidates may find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> before finding a value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</p>
<p> </p>
<p><strong>FINDING <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span></strong></p>
<p>valid approach to find the gradient of the tangent      <em><strong>(M1)</strong></em></p>
<p><em>eg</em>      <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_1}{m_2} =  - 1{\text{, }}\,\, - \frac{1}{{\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}}}{\text{, }}\,\, - {\text{ln}}\left( {\frac{1}{3}} \right){\text{, }}\,\, - \frac{1}{{{\text{ln}}\,p}} = \frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}"><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mn>1</mn><mtext>, </mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>−</mo><mfrac><mn>1</mn><mfrac><mn>1</mn><mrow><mtext>ln</mtext><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac bevelled="true"><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mfrac></mfrac><mtext>, </mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>−</mo><mtext>ln</mtext><mrow><mo>(</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mtext>, </mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>−</mo><mfrac><mn>1</mn><mrow><mtext>ln</mtext><mspace width="thinmathspace"></mspace><mi>p</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mtext>ln</mtext><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac bevelled="true"><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mfrac></math></span></p>
<p>correct application of log rule (seen anywhere)       <em><strong>(A1)</strong></em></p>
<p><em>eg</em>       <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}{\left( {\frac{1}{3}} \right)^{ - 1}}{\text{, }}\,\, - \left( {{\text{ln}}\left( 1 \right) - {\text{ln}}\left( 3 \right)} \right)"> <mrow> <mtext>ln</mtext> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>correct equation (seen anywhere)           <em><strong>A1</strong></em></p>
<p><em>eg</em>       <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,p = {\text{ln}}\,3{\text{, }}\,\,p = 3"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo>=</mo> <mn>3</mn> </math></span></p>
<p> </p>
<p><strong>FINDING <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span></strong></p>
<p>correct substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 2{\text{, }} - 2} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mrow> <mtext>, </mtext> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span> into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span> equation        <em><strong>(A1)</strong></em></p>
<p><em>eg</em>       <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 = \left( {{\text{ln}}\,p} \right)\left( { - 2} \right) + q + 1"> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 2\,{\text{ln}}\,p - 3{\text{, }}\,\,q = 2\,{\text{ln}}\,3 - 3"> <mi>q</mi> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo>−</mo> <mn>3</mn> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>q</mi> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>−</mo> <mn>3</mn> </math></span>  (seen anywhere)           <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>FINDING <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span></strong></p>
<p>correct substitution of <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span> into <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span>        <em><strong>(A1)</strong></em></p>
<p><em>eg</em>       <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{{\text{ln}}\,3}}\left( {x - \left( {2\,{\text{ln}}\,3 - 3} \right) - 1} \right)"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{{\text{ln}}\,3}}\left( {x - 2\,{\text{ln}}\,3 + 2} \right){\text{, }}\,\,y = \frac{1}{{{\text{ln}}\,3}}x - \frac{{2\,{\text{ln}}\,3 - 2}}{{{\text{ln}}\,3}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> <mi>x</mi> <mo>−</mo> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> </math></span>           <em><strong>A1</strong></em><em><strong>   N2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for final answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1} = \frac{1}{{{\text{ln}}\,3}}\left( {x - 2\,{\text{ln}}\,3 + 2} \right)"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn></math>, for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#8800;</mo><mn>4</mn></math>, and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn></math>&nbsp;for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#8712;</mo><mi mathvariant="normal">&#8477;</mi></math>.</p>
<p>The following diagram shows the graphs of <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>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graphs of <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> intersect at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>. The coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>3</mn><mo>,</mo><mo>&#160;</mo><mn>0</mn><mo>)</mo></math>.</p>
</div>

<div class="specification">
<p>In the following diagram, the shaded region is enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>,&nbsp;the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&#8712;</mo><mi mathvariant="normal">&#8484;</mi></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>The area of the shaded region can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>(</mo><mi>p</mi><mo>)</mo><mo>+</mo><mn>8</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#8712;</mo><mi mathvariant="normal">&#8484;</mi></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn></math>           <strong><em>(M1)</em></strong></p>
<p><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>15</mn><mo>=</mo><mn>0</mn></math>  OR  <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>1</mn></math>           <strong><em>(A1)</em></strong></p>
<p>valid attempt to solve <strong>their</strong> quadratic           <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mo>±</mo><msqrt><msup><mn>8</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mn>15</mn></mfenced></msqrt></mrow><mrow><mn>2</mn><mfenced><mn>1</mn></mfenced></mrow></mfrac></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mo>±</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>5</mn></mrow></mfenced></math> (may be seen in answer)          <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>2</mn></mrow></mfenced></math>  (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>2</mn></math>)          <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>recognizing two correct regions from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> and from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math>           <strong><em>(R1)</em></strong></p>
<p>triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><munderover><mo>∫</mo><mn>5</mn><mi>k</mi></munderover><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>3</mn><mn>5</mn></munderover><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>+</mo><munderover><mo>∫</mo><mn>5</mn><mi>k</mi></munderover><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>3</mn><mn>5</mn></munderover><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>+</mo><munderover><mo>∫</mo><mn>5</mn><mi>k</mi></munderover><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p>area of triangle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>·</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><msup><mn>5</mn><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mn>3</mn><mfenced><mn>5</mn></mfenced></mrow></mfenced><mo>-</mo><mfenced><mrow><mfrac><msup><mn>3</mn><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mn>3</mn><mfenced><mn>3</mn></mfenced></mrow></mfenced></math>           <strong><em>(A1)</em></strong></p>
<p>correct integration           <strong><em>(A1)</em></strong><strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>x</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.<br><strong>Note:</strong> The first three <em><strong>A</strong></em> marks may be awarded independently of the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p>substitution of <strong>their</strong> limits (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>) into <strong>their</strong> integrated function (in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>)           <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>k</mi><mo>-</mo><mfenced><mrow><mi>ln</mi><mo> </mo><mn>1</mn><mo>+</mo><mn>5</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>x</mi></mrow></mfenced><mn>5</mn><mi>k</mi></msubsup><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>k</mi><mo>-</mo><mn>5</mn></math>          <em><strong>A1</strong></em></p>
<p>adding <strong>their</strong> two areas (in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>) and equating to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>p</mi><mo>+</mo><mn>8</mn></math>           <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>k</mi><mo>-</mo><mn>5</mn><mo>=</mo><mi>ln</mi><mo> </mo><mi>p</mi><mo>+</mo><mn>8</mn></math></p>
<p>equating <strong>their</strong> non-log terms to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> (equation must be in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>)           <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>8</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>11</mn></math>          <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn><mo>-</mo><mn>4</mn><mo>=</mo><mi>p</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>7</mn></math>          <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[10 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;">
<p>Nearly all candidates knew to set up an equation with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> in order to find the intersection of the two graphs, and most were able to solve the resulting quadratic equation. Candidates were not as successful in part (b), however. While some candidates recognized that there were two regions to be added together, very few were able to determine the correct boundaries of these regions, with many candidates integrating one or both functions from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math>to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math>. While a good number of candidates were able to correctly integrate the function(s), without the correct bounds the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> were unattainable.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the probability distribution of a discrete random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
  <mi>A</mi>
</math></span>, in terms of an angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
  <mi>θ<!-- θ --></mi>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_09.10.36.png" alt="M17/5/MATME/SP1/ENG/TZ1/10"></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="\cos \theta  = \frac{3}{4}"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \theta  &gt; 0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>&gt;</mo> <mn>0</mn> </math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \theta "> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{\cos x}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 &lt; x &lt; \frac{\pi }{2}"> <mn>0</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>. The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \theta "> <mi>x</mi> <mo>=</mo> <mi>θ</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> is rotated 360° about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. Find the volume of the solid formed.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>evidence of summing to 1     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum {p = 1} "> <mo>∑</mo> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></span></p>
<p>correct equation     <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta  + 2\cos 2\theta  = 1"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct equation in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta "> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </math></span>     <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta  + 2(2{\cos ^2}\theta  - 1) = 1,{\text{ }}4{\cos ^2}\theta  + \cos \theta  - 3 = 0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mrow> <msup> <mi>cos</mi> <mn>2</mn> </msup> </mrow> <mi>θ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>4</mn> <mrow> <msup> <mi>cos</mi> <mn>2</mn> </msup> </mrow> <mi>θ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>evidence of valid approach to solve quadratic     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>factorizing equation set equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0,{\text{ }}\frac{{ - 1 \pm \sqrt {1 - 4 \times 4 \times ( - 3)} }}{8}"> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>±</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mn>8</mn> </mfrac> </math></span></p>
<p>correct working, clearly leading to required answer     <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(4\cos \theta  - 3)(\cos \theta  + 1),{\text{ }}\frac{{ - 1 \pm 7}}{8}"> <mo stretchy="false">(</mo> <mn>4</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>±</mo> <mn>7</mn> </mrow> <mn>8</mn> </mfrac> </math></span></p>
<p>correct reason for rejecting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta  \ne  - 1"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> </math></span>     <strong><em>R1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta "> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </math></span> is a probability (value must lie between 0 and 1), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta  &gt; 0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>&gt;</mo> <mn>0</mn> </math></span></p>
<p> </p>
<p><strong>Note:</strong>     Award <strong><em>R0 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta  \ne  - 1"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> </math></span> without a reason.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta  = \frac{3}{4}"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>    <em><strong>AG  N0</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>sketch of right triangle with sides 3 and 4, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\sin ^2}x + {\cos ^2}x = 1"> <mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mi>cos</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct working     </p>
<p><strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>missing side <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt 7 ,{\text{ }}\frac{{\frac{{\sqrt 7 }}{4}}}{{\frac{3}{4}}}"> <mo>=</mo> <msqrt> <mn>7</mn> </msqrt> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mfrac> <mrow> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \theta  = \frac{{\sqrt 7 }}{3}"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </math></span>     <strong><em>A1</em></strong>     <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula involving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^2}"> <mrow> <msup> <mi>f</mi> <mn>2</mn> </msup> </mrow> </math></span>     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \int_\theta ^{\frac{\pi }{4}} {{f^2},{\text{ }}\int {{{\left( {\frac{1}{{\cos x}}} \right)}^2}} } "> <mi>π</mi> <msubsup> <mo>∫</mo> <mi>θ</mi> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <msup> <mi>f</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mrow> </math></span></p>
<p>correct substitution of both limits and function     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \int_\theta ^{\frac{\pi }{4}} {{{\left( {\frac{1}{{\cos x}}} \right)}^2}{\text{d}}x} "> <mi>π</mi> <msubsup> <mo>∫</mo> <mi>θ</mi> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p>correct integration     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x"> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </math></span></p>
<p>substituting <strong>their </strong>limits into <strong>their </strong>integrated function and subtracting     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \frac{\pi }{4} - \tan \theta "> <mi>tan</mi> <mo>⁡</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </math></span></p>
<p> </p>
<p><strong>Note:</strong>     Award <strong><em>M0 </em></strong>if they substitute into original or differentiated function.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \frac{\pi }{4} = 1"> <mi>tan</mi> <mo>⁡</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mn>1</mn> </math></span>    <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - \tan \theta "> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \pi  - \frac{{\pi \sqrt 7 }}{3}"> <mi>V</mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mfrac> <mrow> <mi>π</mi> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </math></span>     <strong><em>A1</em></strong>     <strong><em>N3</em></strong></p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
  <mi>θ<!-- θ --></mi>
</math></span> be an <strong>obtuse</strong> angle such that&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta&nbsp; = \frac{3}{5}">
  <mrow>
    <mtext>sin</mtext>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mi>θ<!-- θ --></mi>
  <mo>=</mo>
  <mfrac>
    <mn>3</mn>
    <mn>5</mn>
  </mfrac>
</math></span>.</p>
</div>

<div class="specification">
<p>Let&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^x}\,{\text{sin}}\,x - \frac{{3x}}{4}">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mi>x</mi>
    </msup>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>sin</mtext>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mi>x</mi>
  <mo>−<!-- − --></mo>
  <mfrac>
    <mrow>
      <mn>3</mn>
      <mi>x</mi>
    </mrow>
    <mn>4</mn>
  </mfrac>
</math></span>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta "> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span> passes through the origin and has a gradient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta "> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>. Find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The following diagram shows the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>  for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 3. Line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M"> <mi>M</mi> </math></span> is a tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> at point P.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M"> <mi>M</mi> </math></span> is parallel to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span>, find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of P.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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>evidence of valid approach       <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   sketch of triangle with sides 3 and 5, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{s}}^2}\,\theta  = 1 - {\text{si}}{{\text{n}}^2}\,\theta "> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span></p>
<p>correct working       <em><strong>(A1)</strong></em></p>
<p><em>eg</em>  missing side is 4 (may be seen in sketch), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta  = \frac{4}{5}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta  =  - \frac{4}{5}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta  =  - \frac{3}{4}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>       <em><strong>A2 N4</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution of either gradient <strong>or</strong> origin into equation of line        <em><strong>(A1)</strong></em></p>
<p>(do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx + b"> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </math></span>)</p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x\,{\text{tan}}\,\theta "> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>,   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = m\left( {x - 0} \right)"> <mi>y</mi> <mo>−</mo> <mn>0</mn> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>,   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx"> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y =  - \frac{3}{4}x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mi>x</mi> </math></span>     <em><strong>A2 N4</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L =  - \frac{3}{4}x"> <mi>L</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mi>x</mi> </math></span>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to equate <strong>their</strong> gradients       <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f' = {\text{tan}}\,\theta "> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>,   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f' =  - \frac{3}{4}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}\,{\text{cos}}\,x + {{\text{e}}^x}\,{\text{sin}}\,x - \frac{3}{4} =  - \frac{3}{4}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>,   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}\,\left( {{\text{cos}}\,x + {\text{sin}}\,x} \right) - \frac{3}{4} =  - \frac{3}{4}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span></p>
<p>correct equation without <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </math></span>        <em><strong>(A1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x =  - {\text{cos}}\,x"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x + {\text{sin}}\,x = 0"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - {\text{sin}}\,x}}{{{\text{cos}}\,x}} = 1"> <mfrac> <mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct working       <em><strong>(A1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta  =  - 1"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 135^\circ "> <mi>x</mi> <mo>=</mo> <msup> <mn>135</mn> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span> (do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="135^\circ "> <msup> <mn>135</mn> <mo>∘</mo> </msup> </math></span>)       <em><strong>A1 N1</strong></em></p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong></em> if additional answers are given.</p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </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">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A school café sells three flavours of smoothies: mango (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M">
  <mi>M</mi>
</math></span>), kiwi fruit (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="K">
  <mi>K</mi>
</math></span>) and banana (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
  <mi>B</mi>
</math></span>).<br>85 students were surveyed about which of these three flavours they like.</p>
<p style="padding-left: 210px;">35 students liked mango, 37 liked banana, and 26 liked kiwi fruit<br>2 liked all three flavours<br>20 liked both mango and banana<br>14 liked mango and kiwi fruit<br>3 liked banana and kiwi fruit</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the given information, complete the following Venn diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of surveyed students who did not like any of the three flavours.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A student is chosen at random from the surveyed students.</p>
<p>Find the probability that this student likes kiwi fruit smoothies given that they like mango smoothies.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><em><strong><img 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">     (A1)(A1) (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 18, 12 and 1 in correct place on Venn diagram, <em><strong>(A1)</strong></em> for 3, 16 and 11 in correct place on Venn diagram.</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>85 − (3 + 16 + 11 + 18 + 12 + 1 + 2)<em><strong>      (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for subtracting the sum of their values from 85.</p>
<p>22      <em><strong>(A1)</strong></em><strong>(ft) </strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from their Venn diagram in part (a).<br>If any numbers that are being subtracted are negative award <em><strong>(M1)(A0)</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{14}}{{35}}\,\,\,\left( {\frac{2}{5}{\text{,}}\,\,0.4{\text{,}}\,\,40{\text{% }}} \right)">
  <mfrac>
    <mrow>
      <mn>14</mn>
    </mrow>
    <mrow>
      <mn>35</mn>
    </mrow>
  </mfrac>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>2</mn>
        <mn>5</mn>
      </mfrac>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>0.4</mn>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>40</mn>
      <mrow>
        <mtext>% </mtext>
      </mrow>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span><em><strong>     (A1)</strong></em><strong>(ft)</strong><em><strong>(A1)</strong></em><strong>(ft)</strong><strong> </strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for correct numerator; <em><strong>(A1)</strong></em> for correct denominator. Follow through from their Venn diagram.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows a circular horizontal board divided into six equal sectors. The sectors are labelled white (W), yellow (Y) and blue (B).</p>
<p style="text-align: center;"><img 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"></p>
<p>A pointer is pinned to the centre of the board. The pointer is to be spun and when it stops the colour of the sector on which the pointer stops is recorded. The pointer is equally likely to stop on any of the six sectors.</p>
<p>Eva will spin the pointer twice. The following tree diagram shows all the possible outcomes.</p>
<p style="text-align: center;"><img 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YgKsK7qlGGvYE95MrAuAg+Sj/1Lb+PWjJU4oFF1VUL5KOJ3FaNowmVkqu+FQjEAD8Ycx9C0Izi8YiwCulJK/kxhpLXIfDABJuBRAhTTZVr4PkSeOYVtk4Z4tGxPFWZpn58zZ47LTTaeaotcyrH+syeX1nE7mAAT8DgBT9nnPd4wGRTII3wZCJGb4EsEaDn/VEzeXttVaZl4gJB9/vjx46isrMSqVaswbdo03HfffV3t5DOvE2CF73URcAWYgO8SIPv8Bx98gH379gmNSE9Px7hx43y3QTKvOSt8mQuYm8cE3EHA0j4/f/58hIWFuaMoztOFBHjhlQthclZMwJJAcHCw5SW7P1++fNnutJ5KSPZ5WhGblpaGPXv24NSpUy70tPFUK/y3HFb4/it7brkHCEhRafen2Zb2+du3b7N9vj8gvfwMm3S8LAAunglIlQDZ50+ePNkZyIzt81KVlP31YoVvPytOyQT8ggDZ5w8fPiwoeopvw/Z5+Yid/fDlI0tuiY8RMLTV4UTJTswfGYzgkeu9Hled7PNbtmwBxbehg+zzFN+GJ2N9rGPZqC7b8G3A4VtMwD0E7qLtkzysS8oHFv4KCX86j7ee6Bnayz1l98yV7PP5+fn429/+JvjPs32+JyO5XGGFLxdJcjt8hIAB+rr9mJf0KV4oKEXqmCB44zWb7fM+0l1cXE224bsYKGfHBGwS0H+Cna+sQNOi32NfXLDHlT3b521KR/Y3vTG4kD1UbiATsE7gDi4e+0/sxE/wjKEYC8l2H/wM5u98Hxf17t0Km+3z1iXib1fZpONvEuf2eo+A4QucK/k7nox6AU89OxPzLiyDvun3+GXcUqShAMeWj3F5qF3RPv/VV18hKSkJbJ/3nvilUDIrfClIgevgHwT0Wly8+CCiVsRidNA9QpsDwmciM/MUJq//Txx7MRfznnB+Mw1z+/zQoUMxb948jm/jHz2sz1aywu8TESdgAq4hYPiyBQ3fAhHdshuIx8c9j0jk4uI1PeCEwje3z9Nm4OxS2Q00fwDACp+7ARPwEAHlv4Qg4oHraGi5AQOGWEzYBuOJYf3bO4ns88eOHUNpaalgtuH4Nh4SqA8WwwrfB4XGVfZRAoGj8Hz8EJR8dAFtSeFQCy4TBuivfY6Lkc9j3OOOmXMs7fO0bSDHn/fRvuGharOXjodAczFMABiCnyz5FSZUvYH1BX8XNs02tH2E/KLLWLghDk/Y8W0k+3xJSQnGjh0rLJYi+zyN7Gl/2P4pe9qQuxIlx3IwN3QNKnTu9RbiXuBdAnZ0Me9WkEtnAnIioFS/jJw/bcIPz6bgyeBghMx7D/+0+A0sG/2wzWaSfX7v3r24//770djYiKKiIhw8eNDpyVhDcz5+uvFtFC9NR8Ftm1XgmzIgwAuvZCBEboJ8CZB9/q233hK2DSS3StoIfNCgQS5u8B005yUhfF0ozjRmYVIgjwNdDFgy2bENXzKi4IowgS4CZJ/Pzs4WLpCi37hxYz9NNl158hkTYIXPfYAJSISA6D//xhtv4OmnnwbHn5eIYGRUDX53k5EwuSm+ScBd9nnfpMG1dicBHuG7ky7nzQRsELC0z9+4ccMN9nkbFeBbfkeAFb7fiZwb7G0C5vb5xYsXs33e2wLxo/JZ4fuRsLmp3iNgbp+nbQPZPu89WfhzyWzD92fpc9vdTsDcPt/a2ir4z1OMm3Hjxrm9bC6ACVgS4BG+JRH+zARcQMDcPp+amgq2z7sAKmfhNAFeeOU0Qs6ACXQRsLTPP/fcc9L2n9dVIHPEZGzXim1QQZNbgZPJIyyCu4n3+a8vE2CF78vS47pLggCZbSorK0H+82Sfnz59OptsJCEZroQlAVb4lkT4MxOwk8DVq1eFwGVpaWnYs2cPNBoNwsLC7HyakzEBzxNgG77nmXOJPk6gvr4ehw8fFkb1bJ/3cWH6WfV5hO9nAufm9o8AuVXW1dV1xrch/3nJ2+etNDU4ONjKVfsuXb582b6EnEqyBFjhS1Y0XDEpEGD7vBSkwHVwFQFW+K4iyfnIigDb52UlTm6MiQAvvOKuwATMCJB9PiMjQ9hBKjAwUPCfJzu9bCdjDW34ZPd8jAwORnDwy1hb2iTsxGWGhE9lRIBH+DISJjelfwTkYp93vPXfoL60Av/vmRcxJgho+yQP65J+j39587+wOWaI49nxE5InwApf8iLiCrqLgKV9nnaTioyMdFdxksvX8I8Pcfa+aPxEfY+pbtdRufb/w39gEz7eHINAydWYK+QsATbpOEuQn/c5AmSfp/1hBw8eDIpvQ5uCU3wbf1L2JDTl48+aKXsAhhtoaRiGhTMiWdn7XK+2r8Lsh28fJ04lAwLm/vOrVq3i+DadMjVAf/EvOJZfjM9X7MCGPjZU73yMT3yOACt8nxMZV9gRApb2eQpLzPvDmhPsMOMkFX4B4AE8icn4R/TLeCKAX/7NKcnlnG34cpEkt6MbAbLPv/vuu4LphuLb+Jt9vhsMez4Y2lD3+z1Ys7YQX7ywG2f2xUHNOt8ecj6VhkXqU+LiyvZFgMISi/Z5nU7nt/b5vjj1uK8MwujE17Dn9efwbVUdmvWGHkn4gu8TYJOO78uQWwCAwhIfP35ciG/D9vn+domBeHzc84jE5/3NgJ+TOAFW+BIXEFevdwJkn//ggw+wb98+IRHb53tnZd8dA/TXPsflCaMRxjZ8+5D5WCo26fiYwPpX3TtozpsFhUJh5V8sMvMqur3CG5rzEKuYhbzmO/0rzs1PkX2+oKAAEyZMwOnTp7FhwwYhTDFtG3jfffe5uXS5ZH8XraW/xMipa1FY1wYy4Bja/oycHf8Xv1w+ie33chGzRTtY4VsAkffHmcht+g5Go7HzX3vregRVLUPMslJck7jZlu3zruydP0DgyCiM+6IQa+PGIiT4GSz8wy3EHfgN5oX7ypIrA/S1OzFRocbEnOpeQkJcR0VmNBTqVJRcu9sHQNPAKDYPzcJ3wfJzH4/7wG026fiAkNxZRaXqx0iZ+xKyJ/8RZZkvIDlsoDuL61febJ/vF7Y+HlIiIDwRb11I7COdlG8rERD1C+Rkn0F0+uvYP7EIK6MeNqsw/SAUYtP275FcnI64YeKKYrMkfnbKI3w/E3ivzVWFIiTI2hdCHCGtQYXO7BWA9kJVK6DOrIBOzNSgRe2hTEwUTUcTM5FX0dzLyEt8yPpfss+Xl5cjLi5OiEFP2wZWVVUJQc0GDRpk/SG+6ocEHkbUovXIjqlD+sq3UWvuXaSvwf6V2WhK3ojX4x7lPXppdbUf9hBushkBg/a/kft7HdJL0xAT6ER3MHyBkgUavFQRjG2t38NobMetN59EVcIMLCv5QrARmxVr85SU/f3334/Y2FjB84a8b8aPHy9csz4PYW1ugq/5OiubncT8ZkA0FuWkI6YyGyv315gGGN+gdv/rSEc6TuyKwzCha9+FtrYQmRPVprmsnvNX5tlaP9ehuSKvlzxMJiC1+eDIAF3FGqgV0cisuN6VpTBgsrjWdddtZ058w91WJ864DwIUIoBGv44fR5ESfl+3idsB6vFYkXcRbVdv4rbjGZqeMEBX+Tuk5o1E1toUjFXRm4ISASMSsafoJZxM/R0qzd8O+iiHJl5pnuGvf/0ryPNmzJgxOHTokBAKwXz+gc+75mLkyKKPbmJ2WzTtjEYlmXZqv4a+9m2sTAeyc36BKMHj6C6ulaxA1EunEbStFu00j3XrNwh3aP5Kh8a85YhJqOrMQ5wDC39pF2r19yB0/FRotOUoq/naVL+vUVNWDi1aUd9y3TTwMUBXcxqF0CA22rNvq6zwzbqNL5zS6HfhwoX453/+535Ut+ekrfHWBRRnANtnrMT+2m/6kSc9YurUPcxCSgQMD0VEty+A/UVQMDMKakbBzWgRFQU7o0VVNHnLBxPoTuBhRK34LYqTv0D6yjQsWflbIHs9Fok2fd1Z7E4tQUTWaiwbq+owbQSMQvKeXUg8uQW7K81G390z7vqkq8Hb66oxbe/rnXkoVeOwfM8uZDRlY807zUDos5it+aZLuRuuo6UemJP0IzQc+RCXBKtox/cFiVMQ7cxbdVfN7D5jhW83KmkkzM3NFezaLovsGDACcSmzocF72PFOXZc9vj/N1W7F5IcGdH+DCE9Bf95FzIsfPnw4xM3C1Wo1EhIShE1KaDKXDybQSUD5KOJe34jkpsMowBLkLIpGgHDTNKLWqhEZMqS7HTtAjfCIVhSWfdpH3xfzGIlxo/7FSh5AQ1Mr9MoQjJ89GuUm5W649CGONGgwb8kLiGi4hKs0x6D7FGWFQGLsUx6PSsoKv7O3SP+ERrbkf758+XKXVlYZFIJIlQuy1OSiqd2amaEG2yY5v6EGTdbGx8ejuroaNImbnZ0t/PjRGwC9+fDBBJQPPgzqaaqxT1sJAFeL7ZOHdhuQKAaMREq51g5wd9HWcgm2UmrrW9BmGNhh1ik/hbOXbkN/9RL+kTgFP4p+DrMjqgRTj6GtBfVeMOdQI1nh2yFqKSQhhUZb72VlZbl8cZHQAbVRvYw4AjA8/LE+EAxCdKwGqoY6nNea+zqLftKuX8RFi6xKS0sFk89HH30kLMIicw8tyuKDCVgnYMWkaVqT0rptUh+j7XsQFBIKW+MiVWQIgpSAUjDrXEPT1RbUlFXh8XA1ApRDEBIJ1LdcQfPZU2jwgjmHmLDCt94zJHf15MmTwr6qGo3GtXXTN6I09wjKY36OWWOtTSCZRizaEzh07LMODwh9I0o2b8P2zuGOEoExr2LvtCqkrjmIakHpG6BvLsUmwZa6ErPc5N9Pe82Snf/UqVMCl6lTp2LLli1s53dtL/Hx3JQIjJ6CRNUFnDv/ZXePMX01ciaGIjavsfv1Hi22lUcrmhqACFLs9Jyg3L9HYe425BYOw+zxIVCCBkUT0FCYg62F13oZXPUo1PUXjHxInsCVK1eMAIw3btzoZ12/MzblzhTyoHy6/9MYM3JPGGtav+/Mu70p16jBTGNu03emazeNTeU7jEkq07MxGcb8j8uNBzQqoyrjjPGm+OStJuOZ3AxjjFiGKsmYXVxjbG0XE7j/7+3bt43FxcXG6dOnC//Onj3r/kK5BOkQuHnGmKFC934p1O5749XiJUaVKsm44+NWo9Alb10wFmdojIjZYay5RVdM3xNNrrFJSGD5+abxQm6yRR4NxtykUWZ5UGHtxptnVhtV9D1QrTaeudnxBej4XnW/5mlw5P7Gh8QJpKenC0pM4tWUXPVI2c+fP984ZswYgR/9GPAhcwK9Knxq901j05lcY0aMyjToGWVMyi42G+xYKnjLz9byoAHTGWOT8INhxtZaPdovGHM1KiM6f1DM0nvolDdAcf1Lk0tzJH97igZJ9mo++keAJruJY1paGvbs2SNshsKrdfvHkp/ybQKs8CUsP5qAJJt0UVGRYL+XcFV9omq8C5ZPiIkr6UYCrPDdCNfZrGnykXzQk5KSnM2KnzcjQB5PlnH0R48e7XLvJ7Mi+ZQJSIIAK3xJiKFnJSh8Aq2opYBhHOO9Jx9XXSHOhw8f7twpi/a/ZXOPq+hyPlIjwApfahIBhEVEtLnHgQMH4LIVtRJsp5SqdPXqVWGeRLTzk/sruXzywQTkRIAVvgSlSatpSQGtWbNGgrWTd5XIzl9ZWYk33ngDNNqnFb20yIsPJiAHAqzwJSZF8iihWDG0kIhNC94TDtn56+rqhPANVIvFixfjueeeY/Oa90TCJbuAACt8F0B0ZRYLFizAzJkz4fIVta6spJ/lRT/Cb731ljDypwn0OXPm8I+xn/UBuTSXQytISJIUBIwOVvYSEgog2PLNwzdQmGYO3yAtGXFt7CPAI3z7OLk9FdmOSZFcuXJFcMV0e4FcQL8JkLmHYhvRXMvQoUMxb948tvP3myY/6EkCrPA9SdtGWRQJ85lnnhHC/9pIxrckRoBi8ufn5+Nvf/sbVq1ahWnTprGdX2Iy4up0EWCTThcLr53Rsn+yE5Oy4MO3CJAHz8GDB4XV0I2Njaf0rRIAACAASURBVMK+uxym2bdk6E+15RG+l6VN5gHyuefwCV4WhIuK5/ANLgLJ2biFACt8t2C1P1Oa/AsMDBS28LP/KU4pdQIcvkHqEvLP+rHC96LcRZ97Dp/gRSF4oGiy8x8/fpzDN3iANRdhmwDb8G3zcdtdGgHSRO3OnTt5ks9tlKWRMdn5ya2T3G5bW1sFbyyy89MPPh9MwJMEWOF7krZZWUePHhX8u3nZvhkUmZ9S5NPU1FTcuHEDarVaWFFNP/r0BsAHE/AEATbpeIKyRRkUJyc+Pp7DJ1hw8beP9JbH4Rv8TerebS8rfC/wp/AJ5IJJSp8PJkAEOHwD9wNPEGCTjicom5Uhhk9gZW8GhU85fAP3AY8Q4BG+RzB3FEI+2rRlISl9sufywQR6I0DmHg7f0Bsdvt5fAqzw+0uuH8/RBN2oUaN4y8J+sPPnRzh8gz9L37VtZ5OOa3n2mhttpUd2Wgp9zAcT6CCgQ3PFcRzLmYvQzAroesHC4Rt6AcOXHSbACt9hZI4/QK/ntD/thg0b2OfecXwyfeIOmvOWY+OhfCxNL8BtO1pJWy7SLmjk1kmrs8k8SG+NNJjggwnYQ4BNOvZQcjINLbLR6XS8ZaGTHGX5uKERedMmYV1kERq3TUKgA42kgcQHH3yAffv2CU+lp6dj9OjRPKhwgKG/JeURvpslTmYcipu+fPlyN5fE2fsbgfvuu0/YLKe0tBSk7Cl8AwXiI6cAchDggwlYEmCFb0nEhZ9pBEav3FlZWTzqciFXzqonAQ7f0JMJX+lJgBV+TyYuu0JudWR35S0LXYaUM+qDAIdv6AOQn99mG76bOgCFT3jkkUeECbZBgwa5qRTO1ucJOGHDt6ft9JbJ4RvsIeUfaXiE7yY57969G8XFxWBl7ybAnK1dBMjOT+YesvNTxM7Tp08Ldn7elcsufLJLxArfDSIVtyzk8AlugMtZ9psAmRdJ6Z86dUrIY/DgwaANeDhMc7+R+tyDrPBdLDLyjli3bp3wxXJx1pwdE3AJAXrrpDDNt2/fxogRIwTHAgrox2GaXYJX0pmwwnexePbv3y98mWg0xQcTkDIBMvfQWyiZe+bNm4f8/HyMHTtWcOsk2z8f8iPACt+FMqUVj/Tl4fAJLoQq26wM0FWsgXrASKSUa6HdPhkPKWYhr/mOV1rM4Ru8gt3jhbKXjouQ04iIFr0cOHAAkZGRLsqVs2EC3iFApsl3330XNLkbExODOXPmcL/2jihcWiorfBfhpNW05IpJsU74YAJyIUADGQ7fIBdpAqzwXSBL8nJISEjgLQtdwFKuWQQHBzvVtMuXLzv1vCsepkldCt9QWVmJVatWCSN/djt2BVnP5cEK3wWs4+LisHjxYl5R6wKWnIX0CdCbLM1VpaWlYc+ePUK/ZycF6cuNaiizSVv74ou7UjQUqGro0KGs7F0JlfOSNAEO3yBp8disnIxG+BRffAk2Vn2NPxeUAhlnHA43a5OUlZs0sUWLV65cucJbFlrhw5f8gwCHb/AdOctohD8QYcm5+P3bW5GlUXlEAm+88YYQPoH3p/UIbpkVosPFM7sxf2QwyL4fPHUtCuvaYPDBVvYWvoEcGdwbptnk2mrVnfU6KjKjoVCoEZvXaMGVBoezoFCvQYXuttl57/QNzXmIVUQjs+J6h4T0zXg/ZzMOecmNtr/dREYKv78I+vecGD5h2rRp/cuAn/JjAnfReuIwTmIa/vPCZVxuqUbxC19iS9xi7Kr7xqe5mIdvoE1/3Bu+QYnA6ClIVH2Opqv67twM19FS/z1iYtRoaGpFt7uGFpw9chaqxCmIDrRPBSrDklFmrMG2SUMAGKCrzsfc9L+hvXupkv9kX2sl3wzPVpBeYcXwCTS64YMJOETAcA2XHvp3pE5+AgH0oDIIY1LTkRl5AQeK63vd29ahMryc2DJ8A3mxuSV8Q4Aa4RGtKCz7tBs3w6UPcaThBSxZMhkoPI0andnoXd+KpgY1EmOfcmiHMS8jdUnxrPD7gXHnzp1ISkoSYt3343F+xN8JKP8VP/nJI909JpSDERJBo0d5HWL4hurq6s7wDeTVRs4ONHBy+lCGYPzs8dDWt6CtU6ffwaWzp9CQGAuNJhaJKEdZjbgDmAG6mtMo1D6G8OHCz62pCtdQd3IH5qoVUCgUUKjnIuf95s43gy6Tzv+BrmIdRkzeCi2OIiX8Pqg7N6C/C21tITInqjvyUMQiM68CzfrOijndXGczYIXvIEExfEJKSoqDT3JyJtAXgX/ChDGPdYz6+0rqg/fF8A0UsbOxsRH333+/sJLXOTv/QISOnwpN+SmcvWQKSyGYbOoQEa5GQOBTiE28F/Ut1012/Ltoa7kErWYqxocO7KKofR/v1T2Gtc3tMBq/R2vRY3hPswL7ay1NbEoETspC45nVUGEmcpu+Q6uwF/FdXCtZgaiXTiNoWy3ajUYYb/0G4VXLELOsFNckovNZ4XeJvM8zGpFs2LABNMJnU06fuDiBIwR05/F+wxQsnDK8+8jfkTx8JC3Z+WlF+o0bN4QaT506VYjY2d8wzcqgEESa2/EFk81ozB4fAiUGITp2AhqOfIhLgtLV42rTNWhmP4tQc+2nmofX1sYhLIAu3gNVzGwkahrx/t/snEjXncXu1BJEZK3GsrGqDhkGjELynl1IPLkFuytNk71elpF5k71cFekXf/ToUcGMQyMVPpiA6wh8g7rcdzFkQxJGCwrHdTlLOSfRzl9VVYUpU6YISp/MPbSi1yFzjzCKh8mObzLZRIgjeCUChociQnwD0H2KssLvERkyxI4f1ts9J3ytAhXNROqe+fYyx2A1Gw9cZIVvJ2RaXUiBpGhJOR9MwHUEDNDXvYPif/oFFox+2HXZ+lBO9LZM+z7T6t309HQhfMPPfvYzB1pAo3gNINjxv0ZNWRUizEbwytBnMVtzTfDkMbS1oB4axEa7Y9vRWmyfPNRkvzfNBZiioTrQGLcm/YFbc5dR5hs3bhSUPccOkZFQJdAUQ2s58v4+Dqvn/VC2tntHMNPbM/1zzK5vNopv/jegPgSzU8icYzqEid1hWFf2EWaE02TuIrvdMcUs7PtLNv0CJIeZzQ3Y96DHUnUy8ViJbivIffHFyaOADt6y0G3C88uMDW2V2HtsIGb9XFT2BuibSpD/F2nYe70pFEcHVuIo/u/v/heONDyKkKB7zKrfMbEbUV+Nv/z9Wsdkrtld50/F9QAXcO78l90XeemrkTMx1MriL+dL7U8OMhrh0+z5FrQat/SHQ6/P0EiDVtSKSr/XhHyDCdhNwAD9xT9ha9pqFH72LXJyzB98Dq+fecH8Ap/bQ8A0ik9J2QRVxpkeI3hhYrchBenlNAo3G/3bk3ePNKY3ClwSFmHd1t/GwMDxWLp3Asakrseu4W90TNzqG1Gy6TWkYwlqZoV1vXH0yM9zF2Q0wncPNFL2tP8nh09wD19/zNXQ+iesfHmpoOx7tD/yeYx7XLomgR71lcyFexAUEgoVoqwvqBImdqMAVajF6L9/DVCGxiJz9U2khD+AB2b8EZcM92BY/BZUFk1AW2YUBpAv/4MzcXzoq6g5vARREpmM99ngac7EF7c3tjh5C2RnZ+MPf/gDu2H273vBTzEBJiAhAj6r8N3NkNzCeMtCd1Pm/JkAE/AkATbp9EI7NzcX5BPM+9P2AogvMwEm4HMEZDRp6zr2tOKPQrvSghA+mAATYAJyISDPEb6hDXUnjmDn/GcQHPwc1jqwrJlMORkZGcjKymK7vVx6ObeDCTABgYDsRviGtg+xd90KvInZ2JxQiM/eMoWgtVPgJ0+eFMIn0Mo/PpgAE2ACciIgr0lb/SfY+coK/P2FbchKfRZBDr6/UPiERx55RAjq5OjCDzl1Cm4LE2AC8iQgoxH+N6g7uBUHQn6FM/1Q9iReCp9QXFwMVvby7OzcKibg7wQcHANLF5fhYik27PwfxD/zLY4sJNt9MEbO340zF3V2VZq2LPzqq684fIJdtDgRE2ACvkhAJgr/Dv5x7n3UPxmBEU9NxrK3PsLlz04iE4fxi7R81PWx4wyFTxC3LPRFIXKdmQATYAL2EJCJwtfj2sXLeCDqeUwfHWTafOCHSMpchMjP3sSGYxe7BzSyILN//34hfAJtzMAHE2ACTECuBOSh8A030NLQM8Kg8vEfIz4SuHhR27k3paUgxS0LZ86caXmLPzMBJsAEZEVAHgrftAH0tw0t+LLH3pH34oknVFZjjZPP/cKFC3HgwAH2uZdVt+bGMAEmYI2APBQ+BmH08xPxwMV6/L3tblc79VpcvPhDxI971GpoUtqykMMndOHiMybABORNQCYKX4nAn8zH1gkfYfX6I2iiSVpDGz7J/yMaFv4SrzzRM9wshU+gLQsXLVokbwlz65gAE2ACJgIyW3h1EWcObkPqznJ8i0gkbn4NaT8fY3UBFo3sFy9eLOylyb2BCTABJuAPBOSl8O2UGO1eRSEUDh48aOcTnIwJMAEm4PsE/E7hk8/94MGDceXKFd7Fyvf7L7eACTABBwjIxIZvf4tpy0IKn8BbFtrPjFMyASYgDwJ+pfApfAJN1k6bNk0e0uNWMAEmwAQcIOA3Jh1xy8KioiIh/LEDjDgpE2ACTEAWBPxmhL9z504kJSWxspdFt+VGMAEm0B8CMgqP3HvzxfAJvGVh74z4DhNgAvInIPsRPplyNmzYABrh33ffffKXKLeQCTABJtALAdkrfAqfQFEwx40b1wsCvswEmAAT8A8Csp60pS0L4+PjcerUKd7Fyj/6M7eSCTABGwRkrfAXLFgguGCS0ueDCTABJuDvBGRr0qHwCXSwsvf3Ls7tZwJMQCQgyxE+hU+YOnUqSOnzilpR1PyXCTABfycgyxE+hU9ITU1lZe/vvZvbzwSYQDcCslP4586dE8In8JaF3eTMH5gAE2ACkJVJRwyfQFsWRkZGsniZABNgAkzAjICsRvi5ubm8ZaGZcPmUCTABJmBOQDahFSgKZkFBATh8grl4+ZwJMAEm0EVAFiN8MuVkZGQgKyuLwyd0yZbPmAATYALdCMhC4dN2hRQ+QaPRdGscf2ACTIAJMIEuAj4/aUvhEx555BHcuHGDwyd0yZXPmAATYAI9CPj8CH/jxo3CloWDBg3q0Ti+wASYABNgAl0EfFrh05aFX331FYdP6JInnzEBJsAEeiXgsyYdMXwCb1nYq2z5BhNgAkygGwGfVfhbtmwRQifQtoV8MAEmwASYQN8EfNIPn7cs7FuwnIIJMAEmYEnA52z45HO/cOFCUPgE3rLQUpz8mQkwgd4J3EFz3iwoFAor/2KRmVeBZr3B9LgpbWwemsVLvWfsM3d8TuHTloVxcXEcK8dnuhhXlAlIjcBM5DZ9B6PR2PmvvXU9gqqWIWZZKa7JSMFbkvcphU/hE/bu3YtFixZZtoM/MwEmwAT6TUCp+jFS5r4E5P0RZZfu9DsfqT/oUwpfDJ/APvdS71ZcPybgowRUoQgJusdK5a+jIjMaCvUaVOjMXgF0FchUK6DOrIBOfMqgRe2hTEwUTUcTM5FX0Qy9eN+Lf31G4dPuVUOHDuXwCV7sLFw0E5ArAYP2v5H7ex3SS9MQE+iEWjR8gZIFGrxUEYxtrd/DaGzHrTefRFXCDCwr+QJmPxVeQekTXjrkcz9jxgxcuXLFK5C4UCbABKRJgJw4HHfeOIqU8KNI6dEkDTL+/SZuj1UhoMc9ey4YoKv8HVLzRiKrKQVjVR1vCgEjErGnqAkjEn6HyilZmOTMD4o91bCRxomfMhu5uvgWbVlYXFzMWxa6mCtnxwR8kQApeXrjHzt2LChwouNHz0lb460LKM4Ats9Yif213ziepfDE16gpK4e2h1lIiYDhoYjQlqOs5ut+5u2axySv8Cl8Ak3WTps2zTUt5lyYABPwSQL0pk9OG/fffz9aW1tBq+zj4+Nd05aAEYhLmQ0N3sOOd+q67PH9yV27FZMfGtDN9XNAeArK+5OXi5+RtEmHfsnXrVsnCNbx1zYXk+LsmIDLCejQXPFnfFpXglVfzUXdtkkIdHkZvp8hLbQ8fPgwKisrkZqa6rbIuMqgEESqgAZnkWly0XQyGWESHE5LsEpdtHfu3AkKnUCx7vlgAvIiQAt7lmPjoXwsTS/AbXk1zunW0GDv3LlzwpqbDRs2YMqUKcJudqQP3OWlZ2hrQb02ComxT1n54Q3A8PDH+mjXIETHaqBqqMN57V2ztAboa3diomIW8pq96/IpWYUvhk9ISek5tWJGkk+ZgI8SGIiw5Fz8/u2tyNKofLQNrq82mW3IPj9hwgQcP34c6enpKC0tFbzz3PqWr29Eae4RlMf8HLPGWgu1PhCh46dCoz2BQ8c+63Cx1DeiZPM2bNeKHJQIjHkVe6dVIXXNQVQLSt8AfXMpNq38LZC9ErPCBoqJvfJXkgqfft3pV51G+G4VsleQc6FMgAlYEqCNjMg+P3jw4E77/Pbt2zFu3DjLpC74TF4693WzsSseXIaPw9NQc3gJogKsq0Vl2CvYU54MrIvAg+Rj/9LbuDVjJQ6Y/2ArH0X8rmIUTbiMTPW9UCgG4MGY4xiadgSHV4ztp/ePC5psykKS0TJpM/Lz58+DBM4HE5A1AUMj8qZNwrrIIjT6oQ3f0j7/4osvus1kI+t+ZGfjJDdpK/7Snzp1ys4mcDImwAR8iQC9wdfV1SE7O1uo9uLFi0E71/HbvPulKDmFT4JftWoV/8q7X/ZcAhPwKAGyz5OnDa2riYmJEezz7jHZeLRZPlWYdWOVl5pAkzV0uMy31kvt4GKZABPoIiC+tXvGPt9VLp/1JCCZET79+tMvv6j0e1aVrzABJuBLBMzt8/TWfuPGDX5z97IAJaPwSdnToorhw4d7GQkXzwSYQH8JWNrnya2S7fP9pen65ySh8GmBBYVPoI7BBxNgAr5HwNI+T27VkZGRvtcQmdfY626ZNCKgRRYUF4NX1Mq8t3HzzAgYoKtYhxGTt6Jz3Q4oqFcBkr28OMeskn2ekn2eFkalpaVhz549wspYfkvvE5vXEnhd4dNiC51OhzVr1ngNAhfMBJiAYwTorZxWwpLXDdnnyevGXSEPHKsZp7ZFwKsKn8w4CQkJQowM9sG1JSa+xwS8T4Dexj/44APs27dPqAzZ50ePHs3+894Xjd018JrCp87zs5/9DLToQqPR2F1hTsgEfJFAcHCwU9W+fPmyU8878zDZ5999910h9AGN5OfMmcP2eWeAevFZr03a0sYFZLNnZe9F6XPRHiPgTYXd30bSGzjtRyHa58llmu3z/aUpjee8ovBpooe2LCS/XD6YABOQFgFL+zz7z0tLPs7UxismnQULFgg7WPGKWmdEx88yAdcRYPu861hKOSePh1agV0Q6PKbs9c2oKDmMnLmxyKy4LmVZcN38lYChDXUnjmDn/GcQHPwc1lZ6rp+SfZ6i05Jr9OnTp4Ww5ORmSTFu2JFCfh3SoyYd6lziloUeQUmhZ3+6HlUPXUDB4XuRMdcjpXIhTMBuAoa2D7F33Qq8idnYnFCIz956wiMx09k+b7eIZJXQowp///79QvgEjy2wUo5A8rvvYF5zHtoOd7iSyUp63BjfJqD/BLvmZeLvL+zAn1OfRZAH3rct7fO3b9/mkbxv9yKHau8xhS9uWVhVVeVQBTkxE5AngW9Qd3ArDoT8CmfcrOyt2ed5cyF59qq+WuWBMQVAHW7hwoU4cOAAjyb6kgjf9wsChoul2LDzfxD/zLc4spBs98EYOX83zlzUuaz9lvZ5UvKifd5lhXBGPkXAIwr/6NGjQowNDqbkU32DK+s2Anfwj3Pvo/7JCIx4ajKWvfURLn92Epk4jF+k5aNOb3CqZLLPi/vDUtgS2j2OlL3zplQD9Bf/gtKSnZgfuR6VOufq6VQj+eF+EXC7SUfsfLxlYb/kww/JkoAe1y5exgNRCzB9dBCEUVfAD5GUuQglk7dhw7FYlM4L77juQPvdbZ83XCxA0uazCPy4HBWYhwQH6sZJpUHA7Qo/IyMDWVlZHFhJGvLmWkiBgOEGWhquAxHdK6N8/MeIjwS2XdRCj3AEdr9t9ROZS2nVOrlW0kHxbdxln1c+MQ8l+T/FxfwUVGyzWh2+KHECblX4tBSbwydIvAdw9TxPQDkYIRFD8G1DC740AIHdDKv34oknVH26ZpJ9/vDhw4Kip/g2rjHZeB4Fl+hZAm5T+NQhKXzClStXPNsiLo0JSJ7AIIx+fiIeKKnH39vu4gn1PR011mtx8eIPET/u0V7NOWQiPXbsGNauXSvEnydTKYcllrzAJVPBbmMLV9aKtiwsLi7mYEuuhMp5yYSAEoE/mY+tEz7C6vVH0ESTtIY2fJL/RzQs/CVeeWJgj3aSfZ5CklA48REjRoD852lLUFb2PVDxBRsE3KLwKXwCjUSmTZtmo2hP3LqOisxoDAhPQTlqsX3yUChi89DMzgWegM9l2CKgDEZcTiH2/rAKcU+GIDhkIUr/KQUHlo3pNOeQfZ7MonFxccjOzsa8efNQXV0thCXhsAe24PK93gi4PHgadVLesrA33HydCfRNwNI+P3/+fBe4VPZdrn0p7giTtpO3PY6CjzcgpvsEhH1ZcCqvEXD5CH/nzp1ISkqSUAf1GlsumAk4RIDeirds2YLBgwcLz7nOf96hanBiGRNw6aStt8MnOLOrkC9uUCHjfulXTSP7fH5+Pr766ithsMTxbfxK/B5trMsUPplyNmzY4NXwCay0Pdp3uDAnCJj7zw8dOlSwz1NIYj6YgDsJuMyGTws/zp8/77ZFH+6EwHkzAU8RMLfP02TsK6+84jvmT10l1v4oCYXfirQeQOTrpf1aFSzmwH89S8AlCp+2LKQNTdgn2LPC49J8h4DoP0/By2iOizYCZ5dK35GfXGrqEpPOxo0bsWrVKgl24Lto+yQP65K2oPzbB/Bk4lbsWf0ynghw+Vy1XPoDt8PFBCzt8xQenF0qXQyZs7ObgNMKn/yE6fDYloV2N80AfX0Z3kccDlxYBJh2Fnp568P4eHOMXXFK7C6KEzIBMwKifZ4WHz799NNsnzdjw6feJeCUSYfskVOnThUWhwwfPty7LbEs3fC/8ZezP8D4nzxiWqZugK5yA370H8Cb7D9sSYs/u4CAaJ9PS0vD5s2bfcs+b9F+ZzzeKCt2oLAAKpGPTo3waQRDy7slp+wJrvJf8ZOfmFO+iy9bLuPRhUsQzYtFzMHwuZMEyD7/1ltvobKyUrDP37hxQ4LmTccayQrbMV6+krrfCp9sk9TRyX4v+UN/EZXHClH0eSLyN3QtXZd8vbmCkiZA3wEKeUAHTcTSd4Ht85IWmd9Xrl8Kn2yUy5cvR1FRkcQ7uMmMk5QPwZPsSeDFf4xB3BP2RBr3+77BAKwQsLTPU/x5v/GfpwBve9chKacc3yISibuzsTouvDP2jxVcfEliBPrlrpKbmysEdHJ+yzR301AiMOZ1XLh8CdWlW5GIo1j68nqUtt51d8Gcv8wIkH2etg28//770djYKAx2Dh486D/KHt+g/k8fArP2dXyfil/Al6vnY2vldZlJWt7NcVjhkxmHFlnRCN93jnsQNPrn2LQnE5Hf1uKTZtdtFO07DLim/SFA/Z12bSPnBDrIPr9mzRrfWSzVn0Zbecbwj8+gG/sixgRR7P57EDTmFSTEAyXvnwd/m6wAk+glh0w69Dorblnoi7bKji3kinBRosLgakmHgLl9fvHixX5vn1c+/iy6+UAI2zQOw8INkeziLJ1u22dNHFL4tHemT29ZSDsKXR6FMWFsw++zZ/hhAnP7PG0b6Ff2ebvlbYD+4l9wLL8Yn6/YgQ2jH7b7SU7ofQJ2m3QofAJtWUgran3hMLSWYtHIl7G28BO00YYnhlZU5hzEl79cgCnilnK+0BCn62iAvnYnJirUmJhTDb3V/Do2ilGoU1Fyra/5jTtozpvVtZGMoRF5saGIzWuEr+4rY26fb21tFezztEes30zGWu0T1i5eR+XaCXhychLWF55G9fvV+Aft1sWHzxCwW+GTyxltWegr8T+UgU/gmXH/B4VrZ2BsSDBGLjyCb+J+gwPzfuhnXgVKBET9AjnZo1GZ/jr2135j0TnpB6EQm7Z/j+S96YgbZtpf1SKVHD+a2+cDAwMF+zytK5G+M4K3pDEEMZs/wOWWapRujgcKl+LllX9CK+t8bwnE4XLtMunQloV0SC98go32BvwQ8976CPNsJPGfWw8jatF6ZL8Xh/SVb2PiiWWIEuMJ6Wuwf2U2mpL3oiiu982z5cKKzDZ1dXWd/vNsn++HZJVBGJ34Gva0t2Dytjo061+Gmhcz9gOk5x/pc4RPr7vr1q0T7Jmerx6X6DICAdFYlJOOmMpsrNxfYzLtfIPa/a8jHek4sSsOw4TecBfa2kJkTlRDoVBAoYhFZl4Fmh16ddehuSKvlzxMJiH1GlToxKGhAbqKNVAropFZYebmp6tAptriWj+BUD+muE+0/ebx48eF/kyRKzUajcTXkvSzwW5/bCAeH/c8It1eDhfgSgJ9Kvz9+/cL4RP4NdeV2L2Rl6Vp52voa9/GynQgO+cXphH/XVwrWYGol04jaFst2o1GGG/9BuFVyxCzrBTXRP1ss/o6NOYtR0xCVWce7a3rEVS1DOEv7UKt/h6Ejp8KjbYcZTVfm3L6GjVl5dCiFfUt101zAQboak6jEBrERg+yWaKtmzT3RP7ztG0g2+dtkXL0ngH6a5/j8oTRCBPfFh3NgtN7noDRxvHXv/7VOGbMGOPt27dtpOJbPkWgvcVYnDzKiJg5xqSYx40x2R8bb4kNuHnGmKFSGTW5F4zt4jX6K1yPMmac+cpoNH5nbMqdaYQm19hEidovGHM1j3c9I6QdZUwubrGShylvy2eEz1HGOUlxRpWYr/Er45mMKKMq44zxpnld7Dynvpueni7030OHDhlv3Lhh55OcrCeBzKvyWQAAIABJREFU743X/muZcUTsGmNBrVaQa7v2tPG1l1KNbzf2Rzo9S+ArniHQ6wifbJ0LFy706paFnv/584MSlY8i7vWNSG46jAIsQc6iaNMktmlErVUjMmSIKcKoiUeAGuERrSgs+7SPRTZiHiMxbtS/WMkDaGhqhV4ZgvGzR6P8yIe4ZAAMlz7EkQYN5i15ARENl3CVzEe6T1FWCCTGPmW3nzf1WfKfp52kaLvNKVOmgOLPU5wbX3E2kGYP/AECR0Zh3BeFWBs3FiHBz2DhH24h7sBvMC/cV1yc2VuN+lavk7a0hJyOf/u3f5NmH+RawWg09ouC8sGHMQSAauzTVjaDqcX2yUOx3UrOqj4NtnfR1nIJWivPipe09S1oM0zqMOuknMLZSz9F0NVL+EfiFPwoWo3ZEfmCqSdmeAvqoUGmHeYcss9TpEqK3sr+8yJpV/5VIiA8EW9dSHRlph7OSzRpnkE0eatNLMLKKPM1BGbeasXy9VbrVeEfOnQIZP+kZeR8+BOBmchtKkBy2MBeGn0Hbb3cEZbch4RChUu9plBFhiBICShDn8VsTS6arragpqwKj4cnIUA5BCGRwJGWK2huOYWGxEU2Q1mToifbvPnxySefdHrgmF/nc/kSsH/gw95qvZp0Zs6cCfJiqK+vl29P4ZaZEVAiMHoKElUXcO78l90XUemrkTPRnsVVtvJoRVMDEBGu7jAhCcr9exTmbkNu4TDMHh8CJQYhOnYCGgpzsLXwml3mHFobMmbMGMHr5uzZs8JbDykA/uc/DMw6cd+nnvRW0zejIi8TEwVvNwUUEzORV9Hc4SEnLFhUQ51ZYWYmFRdAWvFg6+bV1ncze0vRq8KnWDkHDhwQ7PhkG+XDDwgEjsfSvRNwMnU9dlVrO5S+vhElm15DOpZgy6yw7nZ5a0gCo/GLrLEWeZxHXtoybA9PN8uDlLsGOFyAwwhFiBCUS4mA4aGIqDyMgqYJfXrnkF2e1oaQnX769OnCyJ7s97RuhPusNeHwNUA07YgLEd3kraY/j7wlM5BQFYxtrd/DaPwerduCUZUQg5doxbswjzUe2sLTqBHdk4V5q1pAewktbeKK9w4PNiROsfm2a69ke1X4lEFkZKQwAXb06FF78+N0Pk3gHgyL34LKogloy4zCABqZPDgTx4e+iprDS7oWa9lsYyBGJO+0yONXaJqwC03mC74gvg0AKrPO3GHqUQERoRhup7sfDU4oDAK9kVJIhNOnTwv+9uSOSWYfPphAdwIPI2rFb1Gc/AXSV6ZhycrfAtnrsUi06evOYndqCSKyVmPZWFXHICdgFJL37ELiyS3Y3WdIaAN01Yex7v0J2LtlAcaqOiKMqsb+B/YUzUNTeg7eaYbJPblLuRvaaN4qDklzPseRsy0dA65+OC90b6vFp76cgcidjVwzm5qa+krK95mAZAhQv92zZw/Nahs3b97M/VcykpFORdqvFhuTVTAiZoex5pboiNxuvHlmtVEF0Q3ZrL6C+7DK5Cpsyz25w6W403XZPIumXKNGzLube3JHfqqME8aPc2eayjDVRbXaeOamWD+zzPpxanOET78N9NqclZUlhEW2+K3gj0xAsgSo31JcnNu3b2PEiBFC/12wYIHgtinZSnPFPErAHm+1jtXmtOJcAcWAkUgpt+WDZqq+4Tpa6ltttMW0wLCbe7IeV5tuIzH2GUSPn4oIwdRzR/B6c5U5hyrUp8KnRLT8nFba0tJ0PpiALxEgcw/Z+cncM2/ePOTn52Ps2LFCX2Y7vy9J0tN1JW+176xO/rdum2R7bYjgkKC2UWFxrcvADrMOrT259leUFd6P8OEBUAaFIBKX0NLahLNH7HNesFFYt1t2KXx6YunSpUJ4ZHLV5IMJ+CIBsvPTtoS0FzNtU0hrTdjO74uSdGedxbklZ7zVOhwSVA11OK8VJ1+pzgbor15CAx4TFDtd6VDuJ5C7bh8KI6ZifOhAIPApxCZeQ+HWHBQ29O284AgNuxX+8OHDhfDIFCaZDybgywTobZXWl9B2hRQWmbYvpJ3c2AXZl6Xqwro77a2mRODYOch6vgqpaw6iWlD6BugbC5GWkI/w7JWYJa5zEZT7vThcUAqY1qgAARgePgyVBYfRZObQ4IoW2q3wqTAxPLIYLtkVFeA8mIC3CJCdn8IukFsnhWGgcAzk1knhGdjc4y2pSKFcF3irkVfPb4tRNOEyMtX3QqEYgAf/4zNMKKrEiZVjzfbkML0NIMps3YnJ1ANV17oVF2FR0ESvI3nRphEJCQk4deoUxydxBByn9QkCpOwpfDKFaqDd3ShUA/0w8MEE5EDAYYVPjS4oKMD58+cFn2c5QOA2MAFLAjRXRRO9aWlp2LNnT6fjgmU6/swEfImAQyYdsWEUdoFG+mzzFInwX7kRoDkrcuskO79arRbeasnOT28AfDABXyXQrxE+NZaUPYVPJvsnub7xwQTkTIBs+pZbIz733HPc9+UsdBm2rd8Kn1hs2bJF8HKgkRAfTMBfCNDb7VtvvSXY+anvv/jii2zn9xfh+3g7nVL4NOqhPULJr5m3QPTxnsDVd5gAxek5fPiwYOffvHkzXnnlFf4eOEyRH/AkAacUPlWUXDT37duHP/zhD/x660nJcVmSIUADn5MnTwobsDz99NPCil5a5MUHE5AaAacVPjWIJrOeeeaZTj99qTWS68MEPEWAJnUpfMNXX30l+PhPmzaNB0Kegs/l9EnAJQpf3HnoypUrIO8GPpiAvxMgO/+xY8ewdu1awa1zzpw5bOf3904hgfb3yy3Tst60MIV2HuKwC5Zk+LO/EjAP30AMxPAN9EPABxPwFgGXjPDFylP4WXqFFUMwiNf5LxPwdwJk5//ggw+E+S5ikZ6ejtGjR7O5x987hofb71KFT6sTSdlz2AUPS5GL8ykCHL7Bp8Qlq8oO2EARo1x0UOTBe++9V4g1/vzzz7soV86GCciLQHBwMOj7QS7NtICR4vUMGTJEWNFL3yE+mIC7CLjEhm9eOTHsAi9BN6fC50ygJwGy85uHb6C3Yw7f0JMTX3EdAZeadMRq0cQURdTksAsiEf7LBPomwOEb+mbEKZwj4BaFT1WisAt00EYTfDABJuAYATL10CpeCtPM4RscY8epeyfgcpOOWNTy5cuF8LLshiYS4b9MwH4CkZGRQvhxcoDQ6XQYPHiwMIji75P9DDllTwJuU/gUQXPnzp2CTZJeVflgAkzAkoAOzRXHcSxnLkIzK6CzvA0Ii7VohH/79m2MGDFCMJWS+zPPkVmBxZf6JOA2k45YMk1CjRo1SlhmLl7jv0yACdxBc94SbKz6Gn+m/UwzzqBx2yTY46PD4Ru49/SXgNsVPoVdoFWGJSUlHHahv1Li5+RLwNCIvGmTsC6yyG6FL8Ig8w6HbxBp8F97CLjNpCMWTmEXaG9QDrsgEuG/TMA1BDh8g2s4+lMublf4BFMMtUCjfD6YABNwLQEaVJGdn9ygp0yZIsybxcXFCXZ+nj9zLWtfz80jCp8g/frXv8aMGTNAJh4+mAATcD0BcpTQaDSCdxzF6jl+/LiwmpcGWvy9cz1vX8zRYwqfwiYfOnRI2CTCF0FxnZmALxGgDVi2b98u7EbX2toquHXu3bsXFO+KD/8l4DGFT4jFsAu0SxYfTIAJuJ8Ah29wP2NfKsGjCp9eOWnUsW7dOrBt0Ze6CdfV1wmQnZ/m0sjOP336dGRnZ4Ps/DT44u+ir0vX/vp7VOFTtWjEkZSUJCzKsr+anJIJMAFXEKBBF5l7SktLQYFyT58+Ldj5CwoK2M7vCsASz8PjCp94pKSkCB2O4oXwwQSYgHcIcPgG73D3ZqleUfg0yjhw4IAwwuDXSW+Kn8v2HgEDdBVroB4wEinlWmi3T8ZDilnIa77j8SqJbp0cvsHj6D1eoNtX2tpqEYddsEWH7zEB7xHg8A3eY+/Okr2q8MWwC0VFRYJt350N5byZABNwnACHb3CcmZSf8KrCJzDkJXD06FEcPHhQypy4bkzAKQK0raEzx+XLl5153OlnaXBG8flpcpe2ZJw/fz4P0pym6vkMvK7wqckU7nXatGmdIRg8j4FLZAJMwB4CNOf2wQcfYN++fUJyWtE7evRo0LwcH9InIAmFT6v/HnnkEdy4cUOI/y19bFxDJsAEyM5P4RtoVy4KkEgjf5oA5kO6BLzipWOJg8IuFBcXc9gFSzD8mQlImACHb5CwcHqpmiQUPtWNTDo0QcRhF3qRFF+WGQEdmt/fiblqBRQKBRQTM3GoVguDD7aSwzf4jtAkYdIRcZHCT0hIAO3jya+GIhX+Kz8Cd3Ht+EG8c//zWPB8GAIMWlTvWoW4FTewoqYIK6Me9ukmk52/rq5OCN9ADVm8eDGee+45tvNLQKqSUvjEg7wAyKa/Zs0aCeDhKjABNxAw/G9U/uUH+EnMI+h8xXZi5ys31NBlWdJqevLuITs/xex/8cUXeTDnMrqOZ9TZ3xx/1D1PUERNivPBYRfcw5dzlQAB5b8ixlzZU5WUQxASqZZA5VxbBQ7f4FqezuYmOYUvhl1YuHAhR/FzVrr8vI8RGIxpP3ocAT5Wa3uqy+Eb7KHk/jSSU/jUZBoVUOhWWpDFBxPwCwK6T1FWPxWLNWZmHhk2nAZ0FKa5uroa8+bNQ35+vvBdp1253BtXyxS7yGq8ouuoyIyGQqFGbF6jxcT5HTTnzYJCvQYV//cz5MWqoc6sgK5X2Zil15mm4PXNeD9nMw55IU6SZTUlqfCpkosWLQLt0EMTuXwwAXkT+Aa1B/6EoVt+gagAyX4lXS4CcuukFfa0R0ZjY6MQppm+8+7ZjlGJwOgpSFR9jqar+u5tMVxHS/33iIlRo6GpFd3uGlpw9shZqBKnIPohe2QzEGHJ78DYugWTAim9AbrqfMxN/xvau5fqlU/2tMArFaNXwKysLGFDZq9UgAtlAh4hYIC+9o94Z9B8LPJx75z+4iK3TnLSIO88OqZOnSp8710+2AtQIzyiFYVln3YboRsufYgjDS9gyZLJQOFp1Igjc6qMvhVNDWokxj6FwP42UELPSVbhEyPakJk6A73u8cEE5EjAcO0UDp4fj9eSR8nSdu+IzEQ7P+3K9cwzzwhKn0y7tKLXJYcyBONnj4e2vgVtnQse7uDS2VNoSIyFRhOLRJSjrOZrU3EG6GpOo1D7GMKHm82stNXhZM5cqGn9hEIB9dydeL9ZNPKYm3T+B7qKdRgxeSu0OIqU8PvMzEF3oa0tROZEdcc6DEUsMvMq0KzvrJhLmmyZiaQVPlV26dKlmDFjBm++bCk5/uzzBAzaCux6ZyB+migqewP0jceQV3nd59vmTANEOz9561GsHgrfMHbsWGHg55y5ZyBCx0+FpvwUzl4y7TsgmGzqEBGuRkDgU4hNvBf1LddNdvy7aGu5BK1mKsaHDuxskrbgfdQ9thrNRiOM7VdRNOwUNK/moraHslYicFIWGs+shgozkdv0HVq3TUIg7uJayQpEvXQaQdtq0U753PoNwquWIWZZKa65UedLXuGLYRd2797dCZxPmIBvEzBA31yC1XMSsGLFZKgHmFbbKgbgwZHvAGqz0aRvN9Tp2rs6fIMyKASR5nZ8wWQzGrPHh0CJQYiOnYCGIx/ikqB09bjadA2a2c8i1ExTqjIysTZ+RMcbmXIYYubNhqbyv/G31rv2tVd3FrtTSxCRtRrLxqo61mIEjELynl1IPLkFu934g2/WDPvq6o1UNKvPYRe8QZ7LdAcBw7VSLIuZge2V2p7ZW4wmeybwzyvm4RsCAwMFTx/aQMlhO78wiofJjm8y2USII3glAoaHIkJ8AyDPqcLvERkypGuBXK/4rUwGW00rmonUPfPtZY7Bajb9vOgTCp/aRjP569atc9MMfj/p8WNMoB8ElMPikdtqhJFe5S3/lSUjzGe+lf1ovJOPkJ0/KSkJZOefPn06aFtGxw4axWsAwY7/NWrKqhBhNoJXhj6L2ZprgiePoa0F9dAgNtodEUBrsX3yUJP93vSGZ9ru0rH2OJbaZ7qW+Av/xhtvONZCTs0EmIDsCJCdn8w9tGbHscNsFN98BS31ISZzjikXYWJ3GArLPkKNMJk7BdGCe6VjpfSdusOm3+MH32g02fn7zqE/KXxG4VPjKOwCvcJx2IX+iJqfYQJMgAiIo/i/v/tfONLwKEKC7jED0zGxG1Ffjb/8/VrHZK7ZXedPxfUAF3Du/JfdF3npq5EzMdTK4i/nSxVz8CmFT7/qGzZsAIddEMXHf5kAE3CYgGkUvyN9ExpoQZXFCF6Y2G3YhPQdw7qP/h0uiB4wvVEIzxpwW38bhsDxWLp3Ak6mrseualNIbH0jSja9hnQswZZZYXbMGfSrMm7Lt3+1seMpMexCbm6uHak5CRNgAkzAksA9CAoJhQpR1hdUCRO7UYAq1GL0b5mPfZ+VobHIXH0TKeEP4IEZf8Qlwz0YFr8FlUUT0JYZhQHkz//gTBwf+ipqDi9x62pryYVHtgchxdyYMGECioqKeCNle4BxGibABJiA8L7hgxjItCOGXXBvwCUfhMNVZgJMgAn0QsCnbPjmbRDDLpw8edL8Mp8zASbABJhALwR80qQjtoWWWQ8ePBhXrlwBrcjlgwkwASbABHon4LMjfGoSLcIoLi7Gxo0be28h32ECTIAJMAGBgE8rfGoBhV2go7y8XPjL/zEBJsAEmIB1Aj5t0hGbRIuxEhIShHjaNOrngwkwASbABHoSkIXCp2YVFBTg/PnzQsydns3kK0yACTABJuDzJh1RhGLYBZdtliBmzH+ZABNgAjIhIJsRPslDNO1QJD3y1eeDCTABJsAEugjIZoRPTaKImrQl2s6dO7tayGdMgAkwASYgEJDVCJ9axGEXuGczASbABKwTkNUIn5pIphwa4dNuOBx2wbrQ+SoTYAL+SUB2Cp/ESBsjkHnn6NGj/ilVbjUTYAJMwAoB2Zl0xDZS2IWpU6cKO91z2AWRCv9lAkzAnwnIVuGTUEtKSkDB1Q4ePOjPMua2MwEmwAQEArI06YiyFcMukOLngwkwASbg7wRkPcIn4V69elWIt3Pq1Ckh2Jq/C5zbzwSYgP8SkPUIn8RK9vvU1FS88cYb/itlbjkTYAJMwFd3vHJUcmLYBY6o6Sg5Ts8EmICcCMjepCMKi8MuiCT4LxNgAv5KQPYmHVGwHHZBJMF/mQAT8FcCfjPCJwGLYRcOHDiAyMhIf5U5t5sJMAE/JeA3I3ySL4VdIGW/YcMGDrvgpx2em80E/JmAXyl8EjSN7Dnsgj93eW47E/BfAn5l0hHFLIZdKCoqEpS/eJ3/MgEmwATkTMAvFT4JlFw0Kbgah12Qc/fmtjEBJmBOwO9MOmLjNRqNcMphF0Qi/JcJMAG5E/DbET4JlsIuPPLII7hx4waHXZB7T+f2MQEmAL8d4ZPsKexCcXExh13gLwITYAJ+QcCvFT5JeNq0acLm5xx2wS/6OzeSCfg1Ab826YiS57ALIgn+ywSYgJwJ+P0In4RLfvlJSUnCXrhyFja3jQkwAf8mwCN8k/w57IJ/fxG49UzAHwjwCN8kZTHswsKFCznsgj/0fG4jE/BDAqzwzYROYRfi4uKEBVlml/mUCTABJiALAjJQ+AbomytRciwHc0PXoEJncEowixYtwt69ewXPHacy4oeZABNgAhIj4PMK39Ccj59ufBvFS9NRcNt5uoMGDUJWVhYyMjKcz4xzYAJMgAlIiIDPK3xlWDLe/f3v8OusmS7DSmEXyHOHwy64DClnxASYgAQI/EACdZBkFZYuXSqEXbhy5YqwIleSleRKMQEmwAQcIODzI3wH2upQUjHswu7dux16jhMzASbABKRKgBW+DcnEx8dz2AUbfPgWE2ACvkWAFX4f8tq+fTvWrVsH2jSFDybABJiALxNghd+H9GjyNjU1Ffv37+8jJd9mAkxASgQMzXmIVSig6PFPjYmZeaho1nVV19CIvNhQxOY1wjnH7q4spXjGCt8OqcycOROlpaWor6+3IzUnYQJMQDoEVNDkXkC70Qij+K+9FtuCqpAQswYl1+5Kp6oeqAkrfDsgc9gFOyBxEhcTcO2CQhdXzrezU6owNmUuElGC35V9LusRvaWgWOFbEunlsxh2ITc3t5cUfJkJuI6AqxcUuq5mcspJjciQIVZ2gTJAV7EGakU0MiuumzX4Oioyo6FQm6/ovwttbSEyJ6pNpqNYZOZVoFkvTcOQ7yt8XQUy1fchPOUooN2KyQ8Nd5sdbvny5SgoKOCwC2ZfAT51DwF3LCh0T019MFeDFtW5Jbievgu/jBniRAPu4lrJCkS9dBpB22o7zEa3foPwqmWIWVaKaxLU+b6v8AMnYVurmX3O2Iqy5BFWfrWdkKvpUTLtiGEXKJwyH0yACUidgBblKSMxwHzidoAaP1pRgs/bvsSt205oZd1Z7E4tQUTWaiwbq+rQOQGjkLxnFxJPbsHuSvO3A2lw8n2F72GOYtiFkydPerhkLo4JMAHHCViZtDXeRFPxPGB7Kl7dXwO945kCMEBXcxqFWitmoQA1wiNaUVj2Kcz8gPpViqsfYoXfD6KrVq3CjBkzcPXq1X48zY8wASbgKAFaB+O62FaBCItLQqIGqNxRimqnIuzWYvvkod1dPweMREq51tEmeiS9T8XSCQ4OdgrK5cuXnXpefJgiahYXF2Pjxo04ePCgeJn/MgEm4GICtN90eXk50tLSsGfPHtflrhyCkEg10OBsljOR21SA5LCBzmbkked9SuG7SmG7giyFXSCzDnVGMvPwwQSYgOsInDt3DsePH0dlZSXojfrGjRuggZbLDsN1tNS3QpU4BdGBSrLQmB1KBAwPRQTKza5ZnioRGD0FiaoTOHf+S8wLe7Rr3lBfjZyX5uD9xHdx0k3ziZa1sfczm3TsJWUlXXp6OoddsMKFLzGB/hAgRwgaQNGuc9nZ2Zg+fTqqqqpAgyuXKnvo0FxagMLy0VgxazQCrVRWGfosZmtaUXjoPTQKLpY6NJfswKbttV2pA8dj6d4JOJm6HruqtR2/GfpGlGx6DelYgi2zwrp+BLqe8uqZjyt8A/QX38fO+c+AzD3BwS9jbeEnaOv2a+0+vmLYhTfeeMN9hXDOTEDmBMg+T+7OEyZMwOnTp7FhwwZhZfu4ceNAnnHOHVa8dBQj8OrH/4pf1hzEiqiHrWevDMOsPW9jBf7/9s4FKKozy+N/yBYxkXQyiZmhW8ZQSkBnQoYCdBxjlo5mGk0lwSI+NkaQ8RXXR7LJCsTHGOMYjI+JY2BMxgSGEYpaR2Vx44yCwkI0riHAYEhGQZIlCI3JaDZpW6NW0Xfr3O6G7qYbmqYf994+t4qi+z6+x+/ce/q7/3u+c3diwl23ISRkDgqu6vDrvbbv3QjD6LRc1JQk41JOojkS6K45OHz/86gvXYnEcAm6V0HGS0/XUWH37kqh9WqPIAg3he66t4XF48cLKW/WCVf91K/r168Lqampwt/+9jc/1cjVBA+B74WWgjkC1GuFqu/oHFfW0tLSIuTl5QkAxP8XL15UVgcl2JsQapPznzipr72Bzz84izum/hya3h/SG7hQtBjTt43Dvo82QUvanB8WyrGzbNky8fZz+CMSPzSYq5A+AZpQOH46tvcGe1B4YbXkNGFPQDrq8zNnzvTCSN6TlgTfMTJ2+M6MZYKhZhN+/q/A2350+NSS3NxcsUHr1q1z1jBexwSCmgDp8ydPnsSePXtEDvT8iyQbXvxLQFZROu6iuT05ATF+1s8o7QJpkLNnzxbfh+tuW3k/JqBkAqTPHzlyBPn5+dBqtaD3S9CzL14CQ8A/moff+vYNGo/rsXCZ1kbm8U/lnHbBP5y5FnkQoPh5cvL33XcfDAYDjh07xs5eAqZT0AjfBGPjfpSMeh6/S3Dx5N3HwCken6IMDhw4gIyMDB/XxsUrhYBUJhR6gyfp80VFRTh79qwYP3/9+nXW570B1ktlKEfDN36Mt7a2ImXts4j1s5xjawu6hZ0xY4Y4DZxehM4LE1A6AdLnaRIihVbSwvq8dC2uDIdv6sD7+R8hZtEzAXX2VjNTzg+6ADjtgpUI/1ciARrclJaWio6e9PklS5awPi9xQ8tfwzfpUZN/DHfNTe1z9sbPUFx0OmCZ6mhmIC3eS/Yk8bOImxdUBEifp6g00udpYX1ePuaX9wjf2ILyrVl4odjxXbMjEb+5HOWZsQGb2kyZNMnx08Xg3Wnh8jm5uKXeIHALlxqrcbr2INbvqgTS9+Gj17VO0wF4o7aBynDU5zl+fiBa0twmX4dv6kD5irl44a96J2QfxeaqAmQ+GNgMdqRpfvrpp2J0gpNG8iomMDAB0yV8nL8BGW8Dy17PwJO6qXjQz8+nWJ8f2ERy2ypfhy8D0nSxPPvss/wQSwa2kl4Tv0XjrkV47jMd9m1ZhIkRYX5tIuvzfsXtt8rY4fsYNemdCxYs4LQLPuasrOIpxDgfs5/7AsurtmOWxn/Ons7XgwcPisnLKLR4/vz5LEkq6OSS/0NbiRuDZhVSutddu3ZJvKXcPMkQMF3AwU2FQNrDMO1fgQmUCXbCEuyquuDh6/gG7xnp80uXLkV2djbGjx8vDlBWrVrFzn5wdLLag0f4fjAXSTuUdqGkpITD1vzAW+5VmC4UYdb0MsS9/musfm4iIkINaCnKxqyNX2NZeSFe8tLEQlt9/v7770dmZibnt5H7yTNI+3mEPwggb2ymtAs0wqfRE11kvDAB1wRMMHZ9gQsj4/HL1EREiFeoCrEZLyEn/hx2bSrHhWG+74H0eUp7QIOQ8+fPi0EFNGeEk5m5topStrDD95Ml6WIieYfSLvDCBFwTuIWv2j/HNccdQh/AI2kJwIUv0CW+gclxh8G/W+PnaSY4LRQyTNldOZnZ4OyUsoeCculI3yT0bk662CZPnswXmfTNFaAWhuFHUeMw8trnaP/qFqByCC3TJ2CjAAAY40lEQVR+cCxGDzE0k/R5emUgLfQgll4byO9tCJB5A1wtj/D9aACagLVly5bei8+PVXNVsiEQClXCNKSN/BxnPvva5t3aRnRd+Arxab/AODeuWpIOaab3pEmTxGRmlN+mvLxcnAzIzl42J4PXG+rGqeP1OoO6QMqoSQunXQjq02DgzqumYOXWyahdm4t9LQYAt3Dp44MoaX4am2Y/OODscas+f+edd4r6PAUKeKzPG1tRXVaKnQtTkFN9eeA281ZZEGBJJwBmevXVV/HjH/8YV65c4bC3APCXfpVh0MzajP+6qwjbZj2EjddG4ifpG5C7dw4SXMg5pM+/9957qKmpEWWbYZ9bpvMo/JeNqL37HPaV3o7shdKnxi0cnACHZQ7OyCd70Aj/zJkznHbBJ3SDp1BHfd7b+W1MrYWYGbsH8VXHsG3aqOABq9CesqQTIMPShUmjssrKygC1gKuVKwHW5+VqucC3mx1+gGxAD87o/Z4bNmzg2PwA2UBu1XpVn5db57m9XiHADt8rGD0rhOKfKUyO0y54xi9YjqI7QZq0RyG9KpVKfPbD8fPBYn3v9pMf2nqX55BLW7x4sTjj8YknnkB8fPyQj+cDlEvAVp9fsWIFXnvtNY6fV665/dIzHuH7BbPrSkja2bt3LzZt2sTSjmtMQbPFVp8/fPiwmFqb4ucpnJfj54PmNPBZR9nh+wyt+wXTyJ7TLrjPS4l72urzer1eTLRHz3g4v40SrR24PrGkEzj2djVz2gU7HEHzxTZ+ntIRDzt+PmjIcUc9IcAO3xNqPjjGmnaBHs7RLTwvyiVAsk1jY2Nvig3W55Vra6n1jCdeScwi5PApuRq9AJ0XZREg2YZmwr7xxhvQarVITU2VsGRzGdU5MzB9e0OfEXQFaDm6CDEsBPcxkdkndvgSM1hnZ6eYduHixYuIjIyUWOu4OZ4QIJvSXdvq1auRl5cnPoDllMSekORjhkuAf6uHS9DLx5OTP3ToEN566y0vl8zF+ZtAU1OTGD9Pd2vW+HnS6dnZ+9sSXJ+VAI/wrSQk9p/eg0varjW7psSax81xQcCZPv/oo49ySKULXrzavwTY4fuXt9u1UfTGggULxLcS0QNdXqRNQF76fH+WY8aM6b/SzTUdHR1u7sm7BZoAO/xAW2CA+vft2wfSf2kaPS/SJMD6vDTtwq1yToDDMp1zkcTaOXPmcNoFSViifyNIny8tLRWjbjh+vj8fXiNNAjzCl6ZdeltFjmXZsmX8HtJeIoH74KjP02sDExISWJ8PnEm45iES4CidIQLz9+6UdoEe4B44cMDfVXN9FgKkz9MLa5KTk0H5bSjvEYVZUtoD5ea3odcqvoMlE8ZgzJgJmLG+HBeMJj4nZE6AHb4MDLh8+XLk5+eLL0yRQXMV00TS54n7fffdB8pvQ06f8tsoP6upCcamChzHLOw914H2ugI88dUbeHrrB6A37PIiXwLs8GVgO9u0CyQr8OJbArbx8xqNRsxvQzp90EyEM32JRkM8npsYIb4wPTRiMjIX6ICyajQaeJTv27PPt6Wzhu9bvl4tndMueBWnXWH0Q3ry5Ens2bNHXM/6vC2eG7hQtByr/28lDr40EeG2m/izrAiww5eRuUhLJnmB0y54z2jE9MiRI6J0Q/lt5s+fHwSSzRD4GS+g5mAxSr5IxpZN0xHBmsAQ4ElvVzaf9GziskUk7VDaBXrzkfuLCYbqddCEzEVh6w2HwyhBVhJCQjRIKTwP+5v1G2gtnIsQzTpUG67bfLbfy7ZAU2shUkKSkFN92bza2IrjO1/Hn/rVa3tUYD7TxDarPm8wGIJIn3eXtwmGmo2Y8JPpyNhYhMq6Wpz5nBV8d+lJdT92+FK1jIt2WbNoVlZWutjDcXUoVEmPI139BVo6jfYbTZfR3nQTWq0GzS162G01tePU/lNQpz+OJJV7p0lozCJUCPXYNm0UABMMdUVYmHUWPfa1BvQbvTaQpDGaxRyU+rzb9EOh0m7GuY421JVvRToO4IWnN6Jcf8vtEnhH6RFw70qWXruDukWvvvoqfvjDH7rPIFyD2Dg9iis+sYuyMLWdxv7mJ7By5XSg+ATqbR/IGfVoadYgPeVhqNyvSZJ7kj5PP5AU3rpjxw4xLXFtba2YgprTVgxmsjBEJDyH3+TlIP5aAz5u5VH+YMSkvJ0dvpSt46JtFC0ypNDA0ChMnTcV3U3tuNSryNxA26ljaE5PgU6XgnRUoqL+G0uNJhjqT6C4eyxiI20f0XWh8eibWKgJQUhICEI0C7HzeGvvnUGfpPM1DNUbMH76VnTjABbH3gFNTrXlx+YWuhuKkfOYxlxGSApyCqvR6oMYb9LnKT0Fxc+fOHEiSOLnXZw0w1wdOu4XSIu/fZil8OGBJsAOP9AW8Ev9IxA9dQZ0lcdwqs2i44uSTSPiYjUIVz2MlPTb0dR+2aLj38Kl9jZ062ZgavSIvhZ2H8dfGsdifWsPBOEm9CVj8Rfdy3in4du+fcRPoVBN24LzVWuhxhwUtHwP/bZpUOEWuspeRuJTJxCxrQE9ggDh6m8RW/sitC+Wo6v3x8ihuCF+ZX1+iMDc2d3YjQsdD2FijNzv99zprHL3YYevXNva9Sw0Igrxtjq+KNkkYN7UKITiXiSlJKN5/2m0iU7XiM6WLujmTUG07RmizsSv189CTDitDINaOw/puvM4fvaSwwNfu6r7vhhO4a1VZYjbshYvTlKLMd4IfwiL8nYj/Wgu3qqxPOztO2JIn1ifHxIulzub9OVYPuFprC/+2HxHaNKjZue7+OrfluJxTZjL4yS1wViHnXQX+dguNDi9e7QGM8RhcdmXg56/5rtXa+CDJaAhpRCtXhqk+Iud7eXsrzq5nkAQEEfxsOj4FskmzjqCD0V4ZDTirHcAhk9QUXwT8VGjzE55wPZe7//A1+n+VplI079cF88YnBbjsNKZPl9XV8f6vAOnoXwNVT2IyY98jeL1z2BS1BhMWLYf3876LfZm/lQ+MfjhSVi+Mwvamh1Y8059r+zYy8FYj72/KQIWvYbNsx5w4zzvPVLWHzhbpqzNN5TG0yheB2wjHf8bdFbUIm5eRu8IPjR6CubRO0s7jTChHU3QISfJF3n4G7B9+v3Y7qTp6ngnK12scoyfp5QH/CYpF7CGujr8p8h87wwyh3qcpPYPRXjir7BzRxWSsjbjncdKsCbxHksLv0XDO5uR1ZKGQyVPYnQQDXuDqKuSOhsD0BibUXzrRbQ3RVnkHEtTxAe7o1FccQb14sNc98Mxh9YZs6YvkH7v8GfW+QcuzVGfP3bsmJjfhp39wNyCc+s9SFy+ETu0jcha80eLtGOCseGPWJMF7Hh/C9JGmyUqU3cd/pSTYgkk0OCxnEJUDykiyYDW6kIXwQi2c1qsGpBVUrKZt0JGMlQjR+OwzovGY4fvRZhSL8o8iu/CZ0f+E/ubH0BUhK0ea36wG9dUhw8+6zI/zPVqh6zzAc7hw0+/stdMRb012snkL/sG5ObmIjY2VnwZOG2hl4LTzGMxYoiihvhP8Qzszwg3vjlKO8Z6vLPm98COjVhuGfGbusqwNHExqiM2Qt9DA5HzeDv2QyzQrkNZlzvzDgw4X/gStAtqe4MRevQbEVH7ImKf2o0GY5g5aKLbNhLuG9RXVKIbeptgCYvsCR1SfHJ3DdAoi5egIfC90FIwRyCzq7OrhO8c+/1dlZCthgDMEQpavrfZajlOvVao+q6nb33POaFAp+4tq6elQNAhUciu+oe4j/k7lXVNuHb1mtAj3BQ6D60U1OoM4c2P9IJY0tVzwqFsnQDtm0L9VZuy+2qx+3Tq1CkhNTVV/Dt06JBw/fp1u+38hQn0J2A576AVMjK0DufaP4Sq7EQBugKhxe70M6+3Xid95zJdF5brwXqMeN08JCw61G4+p60NENerBV3BOaFHvFbGmT/TdvF7ojA/Y5agtpYj2NdpLcab/3mE78YgQTm7hCEiKhpqJDqfUCU+2E0E1NEOo3/PCIRGpyBn7XdYHDsSI5/5D7SZwjA6LRc1Jcm4lJOI22hEftccHL7/edSXrkSiGP0zcF2Ug55y0ZNmf/78eTHGnlIkkKbPCxNwTiAMo2dlIX/RZezbNxI7dv6q71wTAxQaoI6PcsgTFI7I2LHodpyQ2K8CazDCBDzy0I/sH/6KwQgwBzWIkmkCKi2RcOZJjzpkrnwCcc1t6KRIIrEtcH5t9qvXwxXe/PXgspiAvwlcuXJFyMvLEyZOnChkZWUJLS0t/m4C1ycLApaRvONdau9dLd3ZOvmz7O96hG+9a3a8K7aO4tW9dw99ZVwTvqtaK4yju2xxpD9FvCsWtzu2z8tseYTv4Q8lHyYNApQagXLVU6qEyZMni3lyKIUCxeTzwgTcI6CGruCceSKgQyCBoM/FtAFzSVnvml3XZL17sD5Da+lsR31FLcbRpMfQUYiKB5raL6LVp8ES5vaxw3dtJ94iIwL0qkFKLEdyD+Wyp1cRTpo0ScyCyXKPjAzp76aKMqYGzR/+Hd3WABqxDd+iYeeTCBl0ctVAwQiUjwp9ARCic7+J4oJtKCgebT/psXgnthZ3+VbOAewlJ3+z5vqYgC8IkM5PGn9JSYn4akKK5CGdn15ZyAsTsCcwCtoX1mHm0VexbveHFqdvQGvZdnPoZm4aYgYbFquS8Kstk3B01Ubsrus2R6AZP0Xh6hexPTYLuXNjLI7WMhemdB9KYX1OZgmXrinFvpZk30XnWDo9WFfs2fA3JiAjAhSbT3LPlStXoFKpxDsASo1MrzDkhQlYCYSOnoXdNbuRfGkzNLdReO/d0B6+F6vr38XLvZO1rHs7+6/C+EW7HIIR/h0tybvR8v6LfQ+IYb0bgF3acbPUowbiohHpRuCCsxa4u47feOUuKd5P9gQoDUNjY6OYIpk6s2LFCjz66KMgOYgXJhAMBNjhB4OVuY/9CNAov7S0FDU1NeJdwJNPPgnOjd8PE69QGAGWdBRmUO6OewTofQKk85eVlYFecWjV+Sl1Ay9MQKkEeISvVMtyv4ZEgOSeo0eP4o033sDPfvYzZGZmgh7+8sIElESAHb6SrMl98QoBiuGnVyHSkpGRgZkzZ7LO7xWyXEigCbDDD7QFuH7JEiB55+DBg2JsPzn++fPns84vWWtxw9whwA7fHUq8T1AToIlb9ICX3o+r1WqxZMkSzr0f1GeEfDvPDl++tuOW+5mAVecnx08Lzehlnd/PRuDqhkWAHf6w8PHBwUqAdH5K30Bhna+88oo48uewzmA9G+TTb3b48rEVt1SCBEjnr6ysFF/GkpeXB0rcFhkZKcGWcpOYAMAOn88CJuAFAo7v2KUHvBTrzwsTkBIBdvhSsga3RfYEHNM3kM6fkJDAYZ2yt6wyOsAOXxl25F5IkACnb5CgUYK8SZxaIchPAO6+7whw+gbfseWSPSPAI3zPuPFRTGDIBKxhnZS+geL5U1NTOaxzyBT5gOEQYIc/HHp8LBPwkACnb/AQHB82LALs8IeFjw9mAsMjQGGd7733nhjPz+kbhseSjx6cADv8wRnxHkzA5wRs0zdQLP/s2bM5fYPPqQdfBezwg8/m3GMJE7Dq/Jy+QcJGknHT2OHL2HjcdGUTIJ2/qKgIZ8+eFdM3cJpmZdvbH73jsEx/UOY6mIAHBCgx27vvvouSkhLo9XrceeedyM/PR2dnpwel8SFMgFMr8DnABGRDgNM3yMZUkm0oSzqSNQ03jAk4J0A6/8mTJ7Fnzx5xB07f4JwTr+1PgB1+fya8hgnIhgCnb5CNqSTRUNbwJWEGbgQT8IwAp2/wjFuwHsUj/GC1PPdbkQRI56eXsnD6BkWad9idYoc/bIRcABOQJgFO3yBNuwSyVezwA0mf62YCfiDA6Rv8AFkmVbDDl4mhuJnBSsAEY+sHqPykHodf+QYLG7dgmsqzR2+cviFYz6G+frPD72PBn5iA5AiYWgvx9Gu1uPu/96EUa1F13nOHb+0cp2+wkgi+/+zwg8/m3GPZEbiB1sIMxG6I9orDt+0+p2+wpaH8z57dGyqfC/eQCQQFAU7fEBRm7u0kO/xeFPyBCQQvgZiYGKxatQpXrlyBSqVCWloasrOzQRO7eFEOAXb4yrEl94QJDJvAvffeC3oRS21tLR5//HFs2rQJlJ+fpB/S/nmRNwF2+PK2H7eeCfiEwB133AGdTofy8nLR6R8+fBjJycmgPP0U7cOLPAmww5en3bjVTMBvBDh9g99Q+7widvg+R8wVMAFlEIiMjOzV+TUaDRYsWCDq/CT38CIPAuzw5WEnbiUTkAwB0vnpoW5dXR1SU1OxY8cOUecvKytjnV8yVnLeEHb4zrnwWibABNwgQGGdpPNv374dZ86cEXV+eisX6/xuwAvALjzxKgDQuUom4DYBQzVyxk/H9m7rEWroCqpxdNF4SHG0xukbrHaS5n92+NK0C7eKCciaAKdvkKb52OFL0y7cKiagGAKcvkE6ppTiXaF06HBLmICfCYwZMwae/vm5qW5X59/0DZdRnZOEkJRCtJocmkjymCYEISFzUdh6w36j6TwKUzTQ5FTDYPvZfi+bb5TfaC5CNOtQbaCKKKtpBXauK+1fr81Rgf74T4FuANfPBJhAH4GOjo6+Lwr7ROkb6G/+/Pk4cuSIGOmj1WrF7xTr753lXiSl6KAubkOn0YQYm1TSpkvtaMIUaKd0oaXTCMSM6K3S1HYa+ys1SM95GCpc7l3v+sMIxCz6M4RF1j0uo65gPbKaVuBp6yoJ/ucRvgSNwk1iAkom4Nv0DaEIj4xGXHclKuptZwTfQNupY2hOX46VU2+iuOITGHohm2DsbEOzWoeUpHt71yrxAzt8JVqV+8QEZEDAV+kbQqOnYJ7uWzS1X0avqmNqx6n9XUhPSYEuRQcUn0C9KMUQqG9QX1GJ7rhoRIbbuMRLjTi6cyE0ISQDhUCzcBeOt1p/Jmwlna9RnTMD07c3AJWLEXtbEnKqLXcJpm40/CkHj1nKCHksB4XVrTAGyD42vQtQC7haJsAEXBAwwXjhOHYtmWzR9Z/G+uKPcanXi7k4TIarvZq+ITQKU+cloHL/abRZWJklm9GIjVRBlfQ40tGG9ku3zKRMl9He9C1086Yg2sYjdu87jsaxa9EqCBB6OlEy+hh0zxegwehogFGYtu0YqrITAV0BWnrqsW3aKMD0JcqW6vBU9Rhs09+EIPTg6ts/Qe2CZ/Bi2Zd9P0Z+tJdN9/xYK1fFBJjAoARM+koUHgWe/N1pdHS0oe7QE/gqNwOZuz8O2Ahx0EYPcwfvpG8IQ0RUNNTNZh1ffKBKko1uBqZGjwBUDyMlvQv7T7Wbna5Rj5bmBMybGmU3t0GdnYP1aeMRTn0KHQ1t5jzoav4HZ/WWH4oB+2qCoeYPWFU4AVvWL8YkdRgVgvDx6cgreQpHV/0BNb13GAMW5NWN7PC9ipMLYwLeInAD/9v2A8xe9Us8KMoMYYiYmImcnAT8fe/7NnKEt+qTVjmu0je418pQyyjequOTZFOLuN4RfDgiY0db7gBMMNSfQDGiERVBTnmw5QvzA9/BdrPKRGrHcl09Yxi0QK/swFE6XsHIhTABbxMYgXH//HOHQsPwo6hxGOmwVulfKayT/jo7O93vqjiKvx3bSMc36FFRPBrzaqwj+BGInjoDug10B3ADaG8D0pcjySaix/2KBtmzeyum373VyU6J8FZckpPCXa7iEb5LNLyBCUiTwO3JCYixfbgozWZ6vVUk97i/9I3iW/XtaIqzyDmWAsQHu3G1qPiozvIwl8IxfbCImr4AgZ4D2P1ZdH4fVDlQkezwB6LD25iApAh8g8bjeixcpoWGr9xBLGMZxTc34kj5+2iOj0KELTPxwW4UmhrO4LNmepgrKvWDlDmUzZb5AM2N+LTbVvM3wdiwC485m/w1lOI93NcWgYdF8GFMgAn4noAJxsb9KBn1PJYm3OP76hRQg3kUX4astReRnuI4gqcHuw+gee1avOkw+ves63RHMdZy6A0YjSaotM8jf2YtVq17F3Wi06fZuOX4zZrfAzvWYK7NxC/P6hz6Uezwh86Mj2AC/idgbEDhoXuQtTTRHDXi/xbIr8bQUYiK1wBOJ1RZHuyqAbXj6N+jno5A9MylWHtrA2JvG4tn/twGU+gDSNt9CCXJHcjR3I6QkNtwl/Yw7l+9H6UvTwqIHTl5mkfG5YOYgB8JmDrwfv5HiFn0DGKDULv3I2nFV8UjfMWbmDsoawImPWryj+Guual9zt74GYqLTtukBpB1D7nxfiTAI3w/wuaqmMCQCBhbUL41Cy8UNzkcNhLxm8tRnhlrN1HIYSf+ygT6EeA4/H5IeAUTkAABUwfK1yzEC3/VO2lMAtIeeYCdvRMyvGpgAjzCH5gPb2UCTIAJKIYAa/iKMSV3hAkwASYwMAF2+APz4a1MgAkwAcUQYIevGFNyR5gAE2ACAxNghz8wH97KBJgAE1AMAXb4ijEld4QJMAEmMDABdvgD8+GtTIAJMAHFEPh/0LvByLDK+hYAAAAASUVORK5CYII="></p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that both spins are yellow.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that at least one of the spins is yellow.</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>Write down the probability that the second spin is yellow, given that the first spin is blue.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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 style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3} \times \frac{1}{3}">
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
  <mo>×</mo>
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
</math></span>  <strong>OR </strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{1}{3}} \right)^2}">
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mfrac>
            <mn>1</mn>
            <mn>3</mn>
          </mfrac>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span>  <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for multiplying correct probabilities.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{9}">
  <mfrac>
    <mn>1</mn>
    <mn>9</mn>
  </mfrac>
</math></span> (0.111, 0.111111…, 11.1%)      <em><strong>(A1)   (C2)</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 style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{2} \times \frac{1}{3}} \right) + \left( {\frac{1}{6} \times \frac{1}{3}} \right) + \frac{1}{3}">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>1</mn>
        <mn>2</mn>
      </mfrac>
      <mo>×</mo>
      <mfrac>
        <mn>1</mn>
        <mn>3</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>+</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>1</mn>
        <mn>6</mn>
      </mfrac>
      <mo>×</mo>
      <mfrac>
        <mn>1</mn>
        <mn>3</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>+</mo>
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
</math></span>       <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{2} \times \frac{1}{3}} \right)">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>1</mn>
        <mn>2</mn>
      </mfrac>
      <mo>×</mo>
      <mfrac>
        <mn>1</mn>
        <mn>3</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{6} \times \frac{1}{3}} \right)">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>1</mn>
        <mn>6</mn>
      </mfrac>
      <mo>×</mo>
      <mfrac>
        <mn>1</mn>
        <mn>3</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span> or equivalent, and <em><strong>(M1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3}">
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
</math></span> <strong>and </strong>adding only the three correct probabilities.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - {\left( {\frac{2}{3}} \right)^2}">
  <mn>1</mn>
  <mo>−</mo>
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mfrac>
            <mn>2</mn>
            <mn>3</mn>
          </mfrac>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span>       <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{2}{3}}">
  <mrow>
    <mfrac>
      <mn>2</mn>
      <mn>3</mn>
    </mfrac>
  </mrow>
</math></span> seen and <em><strong>(M1)</strong></em> for subtracting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{2}{3}} \right)^2}">
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mfrac>
            <mn>2</mn>
            <mn>3</mn>
          </mfrac>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span> from 1. This may be shown in a tree diagram with “yellow” and “not yellow” branches.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{9}">
  <mfrac>
    <mn>5</mn>
    <mn>9</mn>
  </mfrac>
</math></span> (0.556, 0.555555…, 55.6%)      <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>   (C3)</strong></em></p>
<p><strong>Note: </strong>Follow through marks may be awarded if their answer to part (a) is used in a correct calculation.</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 style="text-align: left;"> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3}">
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
</math></span>  (0.333, 0.333333…, 33.3%)      <em><strong>(A1)</strong></em><em><strong>   (C1)</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> moves along the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. The velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>&#8202;</mo><mi mathvariant="normal">m</mi><mo>&#8202;</mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds,&nbsp;where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>4</mn><mo>+</mo><mn>4</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&#8804;</mo><mi>t</mi><mo>&#8804;</mo><mn>3</mn></math>. When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>&#160;</mo><mi>P</mi></math> is at the origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">O</mi></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> reaches its maximum velocity.</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>Show that the distance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">O</mi></math> at this time is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>88</mn><mn>27</mn></mfrac></math> metres.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, clearly showing any points of intersection with the axes.</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>Find the total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find turning point (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>'</mo><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math>, average of roots)                 <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>0</mn></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>4</mn><mrow><mn>2</mn><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo>+</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> (s)                 <em><strong>A1</strong></em></p>
<p>  </p>
<p><em><strong>[2</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math>                 <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>v</mi><mo> </mo><mo>d</mo><mi>t</mi><mo>=</mo><mo>∫</mo><mfenced><mrow><mn>4</mn><mo>+</mo><mn>4</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo> </mo><mo>d</mo><mi>t</mi><mo>=</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><msup><mi>t</mi><mn>3</mn></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math>                 <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>t</mi><mo>+</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></math>, <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>t</mi><mn>3</mn></msup></math>.</p>
<p><br>attempt to substitute their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> into their solution for the integral                 <em><strong>(M1)</strong></em></p>
<p>distance<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo>=</mo><mn>4</mn><mfenced><mfrac><mn>2</mn><mn>3</mn></mfrac></mfenced><mo>+</mo><mn>2</mn><msup><mfenced><mfrac><mn>2</mn><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mfrac><mn>2</mn><mn>3</mn></mfrac></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>8</mn><mn>3</mn></mfrac><mo>+</mo><mfrac><mn>8</mn><mn>9</mn></mfrac><mo>-</mo><mfrac><mn>8</mn><mn>27</mn></mfrac></math> (or equivalent)                           <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>88</mn><mn>27</mn></mfrac></math> (m)                   <em><strong>AG</strong></em></p>
<p>  </p>
<p><em><strong>[5</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p>valid approach to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mn>4</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math>   (may be seen in part (a))                 <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>-</mo><mi>t</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>t</mi></mrow></mfenced></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>4</mn><mo>±</mo><msqrt><mn>16</mn><mo>+</mo><mn>48</mn></msqrt></mrow><mrow><mo>-</mo><mn>6</mn></mrow></mfrac></math></p>
<p>correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>- intercept on the graph at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn></math>                 <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The following two <em><strong>A</strong></em> marks may only be awarded if the shape is a concave down parabola. These two marks are independent of each other and the <em><strong>(M1)</strong></em>.</p>
<p><br>correct domain from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> starting at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo>)</mo></math>                 <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> must be clearly indicated.</p>
<p><br>vertex in approximately correct place for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>&gt;</mo><mn>4</mn></math>                 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognising to integrate between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>3</mn></munderover><mfenced open="|" close="|"><mrow><mn>4</mn><mo>+</mo><mn>4</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo> </mo><mo>d</mo><mi>t</mi></math>                <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>2</mn></munderover><mfenced><mrow><mn>4</mn><mo>+</mo><mn>4</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn></math>                 <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>2</mn><mn>3</mn></munderover><mfenced><mrow><mn>4</mn><mo>+</mo><mn>4</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>5</mn></math>                 <em><strong>A1</strong></em></p>
<p>valid approach to sum the two areas (seen anywhere)                <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>2</mn></munderover><mi>v</mi><mo> </mo><mo>d</mo><mi>t</mi><mo>-</mo><munderover><mo>∫</mo><mn>2</mn><mn>3</mn></munderover><mi>v</mi><mo> </mo><mo>d</mo><mi>t</mi></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>2</mn></munderover><mi>v</mi><mo> </mo><mo>d</mo><mi>t</mi><mo>+</mo><mfenced open="|" close="|"><mrow><munderover><mo>∫</mo><mn>2</mn><mn>3</mn></munderover><mi>v</mi><mo> </mo><mo>d</mo><mi>t</mi></mrow></mfenced></math></p>
<p>total distance travelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>13</mn></math> (m)                 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<br><hr><br><div class="specification">
<p>Rosewood College has 120 students. The students can join the sports club (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
  <mi>S</mi>
</math></span>) and the music club (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M">
  <mi>M</mi>
</math></span>).</p>
<p>For a student chosen at random from these 120, the probability that they joined both clubs is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{4}">
  <mfrac>
    <mn>1</mn>
    <mn>4</mn>
  </mfrac>
</math></span> and the probability that they joined the music club is<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3}">
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
</math></span>.</p>
<p>There are 20 students that did not join either club.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the Venn diagram for these students.</p>
<p><img src="images/Schermafbeelding_2018-02-13_om_08.15.35.png" alt="N17/5/MATSD/SP1/ENG/TZ0/07.a"></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>One of the students who joined the sports club is chosen at random. Find the probability that this student joined both clubs.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether the events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
  <mi>S</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M">
  <mi>M</mi>
</math></span> are independent.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><img src="images/Schermafbeelding_2018-02-13_om_08.19.04.png" alt="N17/5/MATSD/SP1/ENG/TZ0/07.a/M">     <strong><em>(A1)(A1)     (C2)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(A1) </em></strong>for 30 in correct area, <strong><em>(A1) </em></strong>for 60 and 10 in the correct areas.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{30}}{{90}}{\text{ }}\left( {\frac{1}{3},{\text{ }}0.333333 \ldots ,{\text{ }}33.3333 \ldots \% } \right)">
  <mfrac>
    <mrow>
      <mn>30</mn>
    </mrow>
    <mrow>
      <mn>90</mn>
    </mrow>
  </mfrac>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>1</mn>
        <mn>3</mn>
      </mfrac>
      <mo>,</mo>
      <mrow>
        <mtext> </mtext>
      </mrow>
      <mn>0.333333</mn>
      <mo>…</mo>
      <mo>,</mo>
      <mrow>
        <mtext> </mtext>
      </mrow>
      <mn>33.3333</mn>
      <mo>…</mo>
      <mi mathvariant="normal">%</mi>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>     <strong><em>(A1)</em>(ft)<em>(A1)</em>(ft)     <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(A1)</em>(ft) </strong>for correct numerator of 30, <strong><em>(A1)</em>(ft) </strong>for correct denominator of 90. Follow through from their Venn diagram.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(S) \times {\text{P}}(M) = \frac{3}{4} \times \frac{1}{3} = \frac{1}{4}">
  <mrow>
    <mtext>P</mtext>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>S</mi>
  <mo stretchy="false">)</mo>
  <mo>×</mo>
  <mrow>
    <mtext>P</mtext>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>M</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mfrac>
    <mn>3</mn>
    <mn>4</mn>
  </mfrac>
  <mo>×</mo>
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>4</mn>
  </mfrac>
</math></span>     <strong><em>(R1)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(R1) </em></strong>for multiplying their by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3}">
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
</math></span>.</p>
<p> </p>
<p>therefore the events are independent <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{as P}}(S \cap M) = \frac{1}{4}} \right)">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mrow>
        <mtext>as P</mtext>
      </mrow>
      <mo stretchy="false">(</mo>
      <mi>S</mi>
      <mo>∩</mo>
      <mi>M</mi>
      <mo stretchy="false">)</mo>
      <mo>=</mo>
      <mfrac>
        <mn>1</mn>
        <mn>4</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>     <strong><em>(A1)</em>(ft)     <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(R1)(A1)</em>(ft) </strong>for an answer which is consistent with their Venn diagram.</p>
<p>Do not award <strong><em>(R0)(A1)</em>(ft)</strong>.</p>
<p>Do not award final <strong><em>(A1) </em></strong>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(S) \times {\text{P}}(M)">
  <mrow>
    <mtext>P</mtext>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>S</mi>
  <mo stretchy="false">)</mo>
  <mo>×</mo>
  <mrow>
    <mtext>P</mtext>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>M</mi>
  <mo stretchy="false">)</mo>
</math></span> is not calculated. Follow through from part (a).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mo>&#8202;</mo><mi>cos</mi><mo>&#8202;</mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mn>3</mn><mo>&#8202;</mo><mi>cos</mi><mo>&#8202;</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>&#8202;</mo><msup><mi>cos</mi><mn>2</mn></msup><mo>&#8202;</mo><mn>2</mn><mi>x</mi><mo>-</mo><msup><mi>cos</mi><mn>3</mn></msup><mo>&#8202;</mo><mn>2</mn><mi>x</mi></mrow></mfenced></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand and simplify <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using a suitable substitution for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi>m</mi></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is a positive real constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[5]</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="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;">attempt to use binomial expansion           <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><mprescripts></mprescripts><mn>3</mn></mmultiscripts><mo>×</mo><mn>1</mn><mo>×</mo><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>3</mn></mmultiscripts><mo>×</mo><mn>1</mn><mo>×</mo><msup><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mn>3</mn></msup></math></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>a</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mfenced></math>           <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mn>3</mn><mi>a</mi><mo>+</mo><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>3</mn></msup></math>          <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math>                   <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math>             <em><strong>A1</strong></em></p>
<p style="text-align:left;">attempt to substitute any double angle rule for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math>                   <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math>             <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></math>             <em><strong>AG</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> Allow working RHS to LHS.</p>
<p style="text-align:left;"> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">recognizing to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>×</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>d</mo><mi>x</mi></math>                   <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>EITHER</strong></p>
<p style="text-align:left;">applies integration by inspection                   <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn><mo>∫</mo><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>×</mo><msup><mfenced><mrow><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced><mn>6</mn></msup></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math>             <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mi>m</mi></msubsup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mn>0</mn></mrow></mfenced></math>             <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>cos</mi><mo> </mo><mi>x</mi></math>                   <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mn>32</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo> </mo><mfenced><mrow><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mo>∫</mo><mn>32</mn><msup><mi>u</mi><mn>6</mn></msup><mo> </mo><mo>d</mo><mi>u</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>u</mi><mn>7</mn></msup><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math>             <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mi>m</mi></msubsup></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>u</mi><mn>7</mn></msup></mrow></mfenced><mn>0</mn><mrow><mi>sin</mi><mo> </mo><mi>m</mi></mrow></msubsup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mn>0</mn></mrow></mfenced></math>             <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math>             <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mfenced><mrow><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup></mrow></mfenced><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math>                   <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></mrow></mfenced><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math>                   <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>m</mi></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math>                   <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>+</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math>                   <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>128</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><msup><mn>2</mn><mn>7</mn></msup></mfrac></mrow></mfenced></math>                   <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>                   <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math>             <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[5 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;">
<p>Many candidates successfully expanded the binomial, with the most common error being to omit the negative sign with a. The connection between (a)(i) and (ii) was often noted but not fully utilised with candidates embarking on unnecessary complex algebraic expansions of expressions involving double angle rules. Candidates often struggled to apply inspection or substitution when integrating. As a 'show that' question, b(i) provided a useful result to be utilised in (ii). So even without successfully completing (i) candidates could apply it in part (ii). Not many managed to do so.</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.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>In an international competition, participants can answer questions in <strong>only one</strong> of the three following languages: Portuguese, Mandarin or Hindi. 80 participants took part in the competition. The number of participants answering in Portuguese, Mandarin or Hindi is shown in the table.</p>
<p style="text-align: center;"><img 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"></p>
</div>

<div class="specification">
<p>A boy is chosen at random.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the number of boys who answered questions in Portuguese.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the boy answered questions in Hindi.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Two girls are selected at random.</p>
<p>Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>20     <em><strong>(A1) (C1)</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{{43}}\,\,\,\left( {0.11627 \ldots ,\,\,11.6279 \ldots {\text{% }}} \right)">
  <mfrac>
    <mn>5</mn>
    <mrow>
      <mn>43</mn>
    </mrow>
  </mfrac>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>0.11627</mn>
      <mo>…</mo>
      <mo>,</mo>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>11.6279</mn>
      <mo>…</mo>
      <mrow>
        <mtext>% </mtext>
      </mrow>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>     <em><strong>(A1)(A1) (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for correct numerator, <em><strong>(A1)</strong></em> for correct denominator.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{7}{{37}} \times \frac{{12}}{{36}} + \frac{{12}}{{37}} \times \frac{7}{{36}}">
  <mfrac>
    <mn>7</mn>
    <mrow>
      <mn>37</mn>
    </mrow>
  </mfrac>
  <mo>×</mo>
  <mfrac>
    <mrow>
      <mn>12</mn>
    </mrow>
    <mrow>
      <mn>36</mn>
    </mrow>
  </mfrac>
  <mo>+</mo>
  <mfrac>
    <mrow>
      <mn>12</mn>
    </mrow>
    <mrow>
      <mn>37</mn>
    </mrow>
  </mfrac>
  <mo>×</mo>
  <mfrac>
    <mn>7</mn>
    <mrow>
      <mn>36</mn>
    </mrow>
  </mfrac>
</math></span>     <em><strong>(A1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for first or second correct product seen, <em><strong>(M1)</strong></em> for adding their two products or for multiplying their product by two.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{14}}{{111}}\,\,\left( {\,0.12612 \ldots ,\,\,12.6126\,{\text{% }}} \right)">
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>14</mn>
    </mrow>
    <mrow>
      <mn>111</mn>
    </mrow>
  </mfrac>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>0.12612</mn>
      <mo>…</mo>
      <mo>,</mo>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>12.6126</mn>
      <mspace width="thinmathspace"></mspace>
      <mrow>
        <mtext>% </mtext>
      </mrow>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>     <em><strong>(A1) (C3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A small cuboid box has a rectangular base of length <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x">
  <mn>3</mn>
  <mi>x</mi>
</math></span> cm and width <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span> cm, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
  <mi>x</mi>
  <mo>&gt;</mo>
  <mn>0</mn>
</math></span>.&nbsp;The height is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
  <mi>y</mi>
</math></span> cm, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y > 0">
  <mi>y</mi>
  <mo>&gt;</mo>
  <mn>0</mn>
</math></span>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>The&nbsp;sum of the length, width and height is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12">
  <mn>12</mn>
</math></span> cm.</p>
</div>

<div class="specification">
<p>The volume of the box is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V">
  <mi>V</mi>
</math></span> cm<sup>3</sup>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <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 an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V"> <mi>V</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>V</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V"> <mi>V</mi> </math></span> is a maximum.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Justify your answer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum volume.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 12 - 4x"> <mi>y</mi> <mo>=</mo> <mn>12</mn> <mo>−</mo> <mn>4</mn> <mi>x</mi> </math></span>        <em><strong>A1</strong></em><em><strong>   N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into volume formula        <em><strong>(A1)</strong></em></p>
<p><em>eg </em>    <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x \times x \times y{\text{,   }}x \times 3x \times \left( {12 - x - 3x} \right){\text{,  }}\left( {12 - 4x} \right)\left( x \right)\left( {3x} \right)"> <mn>3</mn> <mi>x</mi> <mo>×</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> <mrow> <mtext>, </mtext> </mrow> <mi>x</mi> <mo>×</mo> <mn>3</mn> <mi>x</mi> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mn>12</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>12</mn> <mo>−</mo> <mn>4</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = 3{x^2}\left( {12 - 4x} \right)\,\,\left( { = 36{x^2} - 12{x^3}} \right)"> <mi>V</mi> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mn>12</mn> <mo>−</mo> <mn>4</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>36</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>12</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>       <em><strong>A1</strong></em><em><strong>   N2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for unfinished answers such as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{x^2}\left( {12 - x - 3x} \right)"> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mn>12</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}x}} = 72x - 36{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> <mn>72</mn> <mi>x</mi> <mo>−</mo> <mn>36</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span>           <em><strong>A1A1</strong></em><em><strong>   N2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="72x"> <mn>72</mn> <mi>x</mi> </math></span> and <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 36{x^2}"> <mo>−</mo> <mn>36</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span>.</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>valid approach to find maximum           <em><strong>(M1)</strong></em></p>
<p><em>eg</em>      <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V' = 0{\text{,   }}72x - 36{x^2} = 0"> <msup> <mi>V</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> <mrow> <mtext>, </mtext> </mrow> <mn>72</mn> <mi>x</mi> <mo>−</mo> <mn>36</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct working           <em><strong>(A1)</strong></em></p>
<p><em>eg</em>      <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\left( {72 - 36x} \right){\text{,   }}\frac{{ - 72 \pm \sqrt {{{72}^2} - 4 \cdot \left( { - 36} \right) \cdot 0} }}{{2\left( { - 36} \right)}}{\text{,   }}36x = 72{\text{,   }}36x\left( {2 - x} \right) = 0"> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>72</mn> <mo>−</mo> <mn>36</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mo>−</mo> <mn>72</mn> <mo>±</mo> <msqrt> <mrow> <msup> <mrow> <mn>72</mn> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>36</mn> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mn>0</mn> </msqrt> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>36</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mn>36</mn> <mi>x</mi> <mo>=</mo> <mn>72</mn> <mrow> <mtext>, </mtext> </mrow> <mn>36</mn> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </math></span>           <em><strong>A2</strong></em><em><strong>   N2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </math></span> and <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><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to explain that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V"> <mi>V</mi> </math></span> is maximum when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </math></span>         <em><strong>(M1)</strong></em></p>
<p><em>eg</em>      attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{V''}"> <mrow> <msup> <mi>V</mi> <mo>″</mo> </msup> </mrow> </math></span>, sign chart (must be labelled <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{V'}"> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> </math></span>)</p>
<p>correct value/s         <em><strong>A1</strong></em></p>
<p><em>eg</em>      <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V''\left( 2 \right) = 72 - 72 \times 2"> <msup> <mi>V</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>72</mn> <mo>−</mo> <mn>72</mn> <mo>×</mo> <mn>2</mn> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V'\left( a \right)"> <msup> <mi>V</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </math></span>  where  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a &lt; 2"> <mi>a</mi> <mo>&lt;</mo> <mn>2</mn> </math></span>  <strong>and  </strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V'\left( b \right)"> <msup> <mi>V</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </math></span> where  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b &gt; 2"> <mi>b</mi> <mo>&gt;</mo> <mn>2</mn> </math></span></p>
<p>correct reasoning         <em><strong>R1</strong></em></p>
<p><em>eg</em>      <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V''\left( 2 \right) &lt; 0"> <msup> <mi>V</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>&lt;</mo> <mn>0</mn> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{V'}"> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> </math></span>  is positive for  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x &lt; 2"> <mi>x</mi> <mo>&lt;</mo> <mn>2</mn> </math></span>  <strong>and</strong> negative for  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x &gt; 2"> <mi>x</mi> <mo>&gt;</mo> <mn>2</mn> </math></span></p>
<p><strong>Note:</strong> Do not award <em><strong>R1</strong></em> unless <em><strong>A1</strong></em> has been awarded.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V"> <mi>V</mi> </math></span> is maximum when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </math></span>           <em><strong>AG</strong></em><em><strong>   N0</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into <strong>their</strong> expression for volume        <em><strong>A1</strong></em></p>
<p><em>eg</em>      <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 \times {2^2}\left( {12 - 4 \times 2} \right)"> <mn>3</mn> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mn>12</mn> <mo>−</mo> <mn>4</mn> <mo>×</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="36\left( {{2^2}} \right) - 12\left( {{2^3}} \right)"> <mn>36</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>12</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mn>3</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = 48"> <mi>V</mi> <mo>=</mo> <mn>48</mn> </math></span> (cm<sup>3</sup>)           <em><strong>A1</strong></em><em><strong>   N1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{1}{3}{x^3} + {x^2} - 15x + 17">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−<!-- − --></mo>
  <mn>15</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>17</mn>
</math></span>.</p>
</div>

<div class="specification">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> has horizontal tangents at the points where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
  <mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
  <mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
  <mi>a</mi>
</math></span> &lt; <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
  <mi>b</mi>
</math></span>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
  <mi>a</mi>
</math></span> and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
  <mi>b</mi>
</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)">
  <mi>y</mi>
  <mo>=</mo>
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </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>Hence explain why the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> has a local maximum point at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = a">
  <mi>x</mi>
  <mo>=</mo>
  <mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( b \right)">
  <msup>
    <mi>f</mi>
    <mo>″</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>b</mi>
    <mo>)</mo>
  </mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, use your answer to part (d)(i) to show that the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> has a local minimum point at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = b">
  <mi>x</mi>
  <mo>=</mo>
  <mi>b</mi>
</math></span>.</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>The normal to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = a">
  <mi>x</mi>
  <mo>=</mo>
  <mi>a</mi>
</math></span> and the tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = b">
  <mi>x</mi>
  <mo>=</mo>
  <mi>b</mi>
</math></span> intersect at the point (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
  <mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
  <mi>q</mi>
</math></span>) .</p>
<p> </p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
  <mi>p</mi>
</math></span> and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
  <mi>q</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = {x^2} + 2x - 15">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mn>15</mn>
</math></span>     <em><strong>(M1)A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct reasoning that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>0</mn>
</math></span> (seen anywhere)    <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 2x - 15 = 0">
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mn>15</mn>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p>valid approach to solve quadratic        <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 3} \right)\left( {x + 5} \right)">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mi>x</mi>
      <mo>−</mo>
      <mn>3</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mi>x</mi>
      <mo>+</mo>
      <mn>5</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>, quadratic formula</p>
<p>correct values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span></p>
<p>3, −5</p>
<p>correct values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
  <mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
  <mi>b</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
  <mi>a</mi>
</math></span> = −5 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
  <mi>b</mi>
</math></span> = 3        <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><img 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gakZ6QjQ0+f9keCnxxliIwERN1syoLH60U4mPyekdKHaexAzSxj77AoApqaRImAAJGRgKAas7MRDofg9fJrXAQUrThRSh+msfjXOXm0SVZxHWcQhYiMBgEn3lfmzs7vTIHDHJXuGTW5W5CRwYc/ideOlF8eBIiMBMTdZObDvDpdTgGlKlOU0j0jr88Dq4UCqymz98TWisgoNi4jSs2z8sMCZ139iMpTIWEQcNXx4X+NObRJVhhEpZFCZCQgzraczv1pnuGdyipg1ZKLUvIwrWtLTo6RyEjyjpFAhURGCYDXt6jJwpORy0nDtL7YSPnc3MCf0mKxdAeTk7J+qmtkCBAZjQy3AUsVFhSivqFxwPfJ8kLRnlFLCwezxUxzRmrqb0RGAlvLbDOjqUUZMY0EblovcUqewG5u5P8YmO20Y7+X0RT+QGQksIGKrEVwOZKfjASGTVBxbhcfPqQgzy6oXBImLgJERgLjW2S1Ioww2lr5oYLA4hUjTsnDtEAwwB03npY+SjF4kSJDI0BkNDRGceUw5OTB7w+iK+xpXIVVlFnJw7TWllYY0gxDnGyiIrBTRFUiI4ENbTHnQsO2hITaBJasLHFK9ow8Pi8ysnMRSYFQLsrqFYlpQ2SUGH79Sufk5CAQDuFYW6jfu2RKULJn5GtpRVaeAaN1tGNfTX2OyEhga+Xm8ltCPJ1fdAQWT+KGQIAdQNka8MJkoC9pQ0CluNdERgKbJCuLP1nWqbCTZQVuJndAotAyhZLn9/iRRptkhYJTMjlERgJDnW3KRZpWh3afX2DJyhKn1GFa5OsOHOvoQE4WeUbK6jFDa0NkNDRGceXI0usRMWjRnOQLH5U6gR2OAJ7mRmRm88PluIxHmWVFgMhIYPiNBhPSoINTIcdcC9y8qDjFekbhCIKhEPLyKMpj1FgquSEyEthQozL4wxzrHHwYC4HFk7ghEKh38rizT/t0qQsBIiMR7GWxWXHk6BERJCtHpFKHac7OHfv5+RRyVjm9ZXiaEBkND6e4clmsVriS/GuaUodp7s4om7nZmXHZjDLLjwCRkQg2sFut8Af8aGtL3lXYSvWMXJ2bZPM6l1iIYF4SKRICREYiAJtfUMBJ7QplIUIVsotUqmfk83k4bMxmk+wYkQLxIUBkFB9ew8ptMXVGfEyBU0KGBYiEmRqdfPgWi80mYa1UlRAIEBkJgWIfGWYbT0ZNSUxGSh2mOT0e5NN5aX16pDoeiYxEsFNBYSEntbExeU8JUeowraWpCbk0RBOhV4svkshIBIxNmdnQanVoIc9IBHQHF+nxNCMvn98fOHhOeqs0BIiMRLCIXqdFWo4ejYHkPeZaiZ4R27Hf3toOk4lOBRGhW4sukshIBIi1GRkwarPQ4kru0LMiQJeQyOCxYwiHw7CZ6EtaQkDKVJjISATgM/R6ZOamo9GdvIH5lTiBHfAHEIj4YaTDG0Xo1eKLJDISCeM8Yx5am2mYJhK8McX6Q0FEfGEYOgPcxcxEiYpFgMhIJNOwuEZ1Lcl7sqwSPaNjbW1cyF9TDsUyEqlbiyqWyEgkeLOysuDt3LQpUhWyilXiBDYLxM8ug5HCh8jaOUZYOZHRCIEbqpjVYuXi6rR0HrU8VH56nzgCbU1NnBB2Qgtd6kOAyEgkm+Xl8Z+XHTUOkWqQV6wSh2l+X5ADxWAgz0je3jGy2omMRobbkKVsxWO4PPXO5Jw3UuIwzR1wQafRIDs7e0j7UAblIUBkJJJNTrWXcJI/r64SqQZ5xSrSM2oKQKNLg8FgkBccqn1ECBAZjQi2oQuNKeJ3jTc0NAydWYU5lOgZ1dbWAgjDkEmB1VTYpUBkJJLVMjMzoTVo0draKlINJLYvAo31DTDmGDFKq+37ip5VgACRkYhGGpszFvUNjSLWIJ9oJQ7TamvrYLcWywcK1ZwQAkRGCcE3eGGz1YS2Fv5z8+A51fdWicM0Z5MbJhN91ldfb+I1JjIS0XJmswVerx9sN3myXUrzjI4f74DX0wxrZyypZMM7FdpDZCSilbNyctDW2opQR7uItcgjWmmekdfDr74usPJRNuVBhWpNBAEio0TQG6KsJdcEf8SHYCD5PKMhmi75a09zM1enicKHSI69UBUSGQmFZAw5OWYL/J4AAsFAjLfqTlLaMK2phf9qmZdHUR7V2rOIjES0nKUz/Kmz8ywvEauSXLTShmktrXwgu2wKHyJ5XxCqQiIjoZCMISe388uOu/P89xhZVJukNM+owcsP0/LoZBDV9ikiIxFNV9B5mGNDEq41UppnFPR4uUMQzPRpX8QeLa5oIiMR8TV3BoZ3N7lFrIVEMwSa3C1IS9NBQ5sKVNshiIxENF1O5/ldDm7PlIgVySBaacO0prY26HVpSDOkyYAGVSkEAkRGQqA4gAy2R2pM4VgcPZJ8hzkqbZjm93mRkZkB3ehRA1iDkpWOAJGRyBayFdlQW1sjci3Si1eaZ9TocsOQlobROr30YFCNgiBAZCQIjAMLGTeuBE6nG5FIci18VJpn1OL1II1iXw/cEVXwhshIZCPZbFauhpYmCiUiJtRNTW6YafW1mBCLLpvISGSI2WbZcDgAj9cjck3SilfcMK3Rg5w8OqJI2l4gbG1ERsLi2U9aTlYewuEIku2UECUN0xi24XAIpmwKH9KvA6oogchIZGNZLPwu8ubOFcIiVyeZeCV5Rk43v44rh/alSWZ/MSoiMhID1R4yc8x8cHi3K7mCrCnJM2qo55dOWDuJvwf8dKsiBIiMRDZWpiEXer0BLS6XyDWlrvg6Rx3XeKu1KHVBSIKWExmJbERDmh4Ggx4NjckVC1tZwzT+bLriErvI1iTxYiJAZCQmuuzc9+xcbr+UsyW5PCMlDdM8nURvs/HHQ4lsUhIvEgJERiIB2yWWbQnJtuairoYfSnSlq/2nkjwjdhyU0ZildkhTXn8iIwm6QJHFjrra5CIjRXlGXh/MJlpjJEFXFrUKIiNR4eWFjx1TiPqG5NssKwF0w6qipcWNnCxaYzQssBScichIAuOYzGauFv74ZQkqjFEF2xvXNMQXPafTiaqqqmEdraSUYRo7Bsrn9yGbgqrFsLq6koiMJLCXzcLvT3O3SB9kjf2yfnb4U1y8bBmu/X8/i9laluf2227H7353Dza/9RbOO3cRnK7Bh5VKGaaFjnego+04srJoziimcVWUSIeSS2Cs4pISrpaaSgdmTZslQY3dVdQ7nThUVY6d23dhxqzYdd//wH0oLy/HW5v/A41Gi0g4gmUXXoL3d+0c8Nx6pXhGTFeP1wurlb6kdVtdnXfkGUlgt2I7vxjP1dwgQW29q7AXFeGKZZdh3Dh2Bn3/MCaOWgcee+gR/PDySzkiYqVvvOlG7DtwAM8//VRvYT2eFOMZBQLwBwMwGvmV7j1UpFuVIUBkJIHB8nMKAI0GzTJuCdFCB62mv7m3bt2KQCCAiePGRZEYnZ6O8eNL8do//x1NU+qNPxBEOBREbm62UlUkvYaJQP/eOcyClC0OBPRAfk4eXG5+pXAcJQXLyngoHOlv7kMHDnGnapj67OsyW6yoqqxCezAYUwelDNOcnZPyBgN92o9pKBUl0pyRBMZKHz0amaZseBW4c/+LL/7LkRHbttLz0uq03FeqSMcxQK/H4coKbHrtP4hoQtDodKisrEFrSwsefPhBbo6JlV22bBlKOufHesoS897p4gne1odMxayTZIuDAJGROLj2ksomhXOMWXDJOEzjFeo/ZxQKBAF9BNpIHzJCBJFwGNDwXaT68Oe46+67erWLPdx1R3fahCkTJCej5np+Hi7PSl/T+hlHZQlERhIZzJRrQnV1DTpCQVmCxocRQXqMOSOTzQIciiCsYcOxzCgare3tMGZlQaPlh3bnX7wEXk8z5wWxtB9efhU+/ewwPt1/MFomLVv6eZumzmOtLTl83KioMnSjOgT6TyKorgnqUDh/TCFaWuWLaaSFJuac0eypUxEMhdDs8fYC0tfsg81qAxtisovtscvMzkW2ycT9ZMO4Uf8zmntmaewfyyP11eDkNyAbKf611NALXh+RkeCQxhZoycxGc3MzQh3HY2eQKXX+wvnQarRgn/i7LrZau7qmCgvPWRD93N/1ruunUiawHUePIsuYIwsRdmFBP4VBgMhIGByHlGIp4hflORzdv/RDFhIwAxumxVpnNGfaPJw+dy42bXwjWtt/3tgEY5YRP7355mha3xt+nVH/Oai++cR+drucKCqhBY9i4yyFfCIjKVAGUFhYyNVUK/HufbbVg+05q607iv9+7uDuWVrPa+36V7Bt127861//wiefHMTv7/kt1r+4Fl176nrm7bpXimfEjigqsrEFnXSpHQHpB/lqR2yE+tvtfBTC6qoqABeMUEr8xULHjqHO6cCLzzzLFa6proHRYMCozO7JahY7+qMPdmDjpk0oK6/A5s1vw9y5uXegGnnPSN7u09bWBr8/iAIrTV4PZCc1pcvbm9SEVIK6dkUhbGyUNpRIRmYm5sw6Y0jtWb4rL7tsyHy9M/T2sHq/E/8pGAggFArCWpAvfmVUg+gI0DBNdIj5CnJzc5GWloa6zq8/ElUrWjVKGKa5W1uhCUeQnUmxjEQztISCiYwkBLuoqAgup/RhRMRoohI2ygb8HgTCIeTl01YQMWwstUwiIwkRLywcgyONX0lYY3JX5fX4uQbm5pqSu6Ep0joiIwkNbbNZ0dHSATbxqvZLEcO0zpNk87PIM1J7f2L6ExlJaEWz1YRAKIiAPyBhreJUpYRhWovXy23yzcjMEKeRJFVSBIiMJIQ732RDIODjdsNLWG3SVtXsckGv08FgoMBqyWBkIiMJrWjLz0c4HEFLKw3ThIC9yePhPCNjDu3YFwJPuWUQGUloAZOZX5zX1BmDR8KqBa9KCcM0tkk2zZCG0aP4zbyCN5IESooAkZGEcOdY+L/g9Z0xeCSsOimrqnPWgc3D0ZUcCBAZSWhHS5aJG1a4W+QLJSJUc5XwNa3WUYciK3/YgVDtIjnyIUBkJCH2uWaejKoqKyWsVZyq5B6mdRw7Bq/PA2s+bQURx8LSSyUykhBzFn7WZrdCrjAiEjZV9KoaO4PB5VHsa9GxlqoCIiOpkO6sZ5y9BBXlFRLXKnx1cg/T3J2nglhMtGNfeOvKI5HISGLcS0rs8Af8YMMMNV9yD9OcTv747aJimjNScz/qqTuRUU80JLgvKBzD1VJfL20oEQmaJmkVjQ38qSCFNpozkhR4ESsjMhIR3FiiWSAzdjllP7YolnbDT5N7mNbS2sopa8mjkLPDt5qycxIZSWwfa+en6JYm/lQLiasXrDq5h2meJg+MxizodNSFBTOqzILIkhIbwGI1czXW0jAtIeSdjS5uT5pmdHpCcqiwchAgMpLYFkaDEYY0Azyt6vaM5B6msTVGJlMu0nUUOVniLixadURGokEbW7Ahy8CFn3XWtcTOoJJUuYdprU2tSEvTD3ium0pgJDV7IEBk1AMMKW4z0jOQacpGc/1RKapLyjrYIZPOOhdMdIpsUtmXyEgGc+ZlZaFapsMchWqu3MM0n98HFsaXruRBgMhIBlsWFdnhVPkpIXIO0453HIfP50V2Nu3Yl6H7ilYlkZFo0A4smA0v2C9TexLEwh64leK9OXKUH+JmGelLmngoSy+ZyEh6zDF2zFiu1hoVD9XkHKY5av7L4VdU/C0ZrEdVioUAkZFYyA4it7i4hHtbVV09SC5lv5JzmOZwHOHAsdloX5qye0l82hEZxYeXILknTTqNk+Oo4X+pBBGaQkLqOsP22q20Yz+ZzE5kJIM1x9jtXK3Nri9lqF2YKuUcpjV1npdmsdG+NGGsqQwpREYy2GGUVov8vHzUehtlqF2YKuUcpjXUN6CwoFCYhpAUxSBAZCSTKSxWC5oc6l6FLRN0aGltgtnG7/GTSweqV3gEiIyEx3RYEs0WKzzNzTgeDg8rv9IyyTVMY3i5nU2wmqxKg4T0SRABIqMEARxp8cL8PLS0tiCo0oiPcg3TGF7BcAh5+fSTPb0AACAASURBVHkjhZ7KKRQBIiOZDGMxW9Do9SISCsmkgTqrDQYC8Pv96ApSp85WkNaxECAyioWKBGlWqwXhUBDNfq8EtQlfhVzDNH+4DYFAgDbJCm9S2SUSGclkgsKxY4FIBDVVNTJpkFi1cg3TWuq8CIdDyKHz0hIzoAJLExnJZBSblV8jU+uolUkDdVbraeG/QJrN9DVNnRYcWGsio4GxEfWN3c6TkcOhTs9IrmGau7mZs0uOMUtU+5Bw6REgMpIec67GXJMZWq0OtXX8+V8yqTHiauUaptU5nZzO5s5Y4iNuABVUHAJERjKaZFxpKR11HSf+zjp+WJufR5/244RO8dmJjGQ0kd1mg6PGIaMGI69armHa0SNHkJOTh1GjRo9ceSqpSASIjGQ0y7jx4xAMhdDa1CSjFiOrWq5hWlV1DUo6Q7CMTHMqpVQEiIxktEyB1YZA0I9Gj0dGLdRVtaO2FsXjaLe+uqw2PG2JjIaHkyi5rHk5CIfCaGltE0W+mELlGKa1tbYgHAmjyMwHpxOzfSRbegSIjKTHPFqjxcJHKnQ51bfWSI5hWkNjM7dQNL+YJq+jnSiJboiMZDSmyWbkPu/XN6g3rpGU8DU18QsebTk0TJMSd6nqIjKSCukY9eQbzPzpsi53jLfKTpJjmNbUxOOUS4c3KrtzjFA7IqMRAidEMaPJBJ1OC5cKV2HLMUw72nkKr4UWPArR/RQng8hIRpOw8LNWqxVlFRUyaqGeqttb2mBIM0Cv16tHadJ02AgQGQ0bKnEy2m1FqKlW3/40OYZpLp+LG9YaaF+aOJ1RZqlERjIbYFzpOPgDfrSp7HRZOYZp9Y5GjMocDUOaTmarUfViIEBkJAaqccgsKOC/DFVWqfdAxziam1BWti+twJhPW0ESQlG5hYmMZLbNqafxC/jcLpfMmsRXvRzDNIejFmZzbnyKUm7VIEBkJLOp7CedxGlQ89VXMmsSX/VyDNMaPc0oGDMmPkUpt2oQIDKS2VT5ubnQGnXwdh7ZLLM6iq2+traWW31togiPirVRoooRGSWKYILltRkZyE/LU12QNamHaTW1fKgVexF/NHiCsFNxBSJAZCSzUdJ1WpjMJjjd6gojIvUwzVnDR8Sk8CEyd1gRqycyEhHc4YjWaLTIy8tHQ2P9cLKnbJ46J09G9mJ+c3HKApHEDScyUoBxrRaz6iI+Sj1Mc7kboTVoYbPRsdYK6LKiqEBkJAqs8Qktstu5gwlbWlvjKyhjbqmHaV5PC/LS8vB1RMZGU9WiIkBkJCq8wxNebC/mMpYfKh9egRTM5XI7kWs2g+3noys5ESAyUoBdi0v4L0QVleohIymHacfDYTQ3e2AtyFeAtUgFsRAgMhIL2TjklpaO53LXqiiUiJTDtMjXHWCB1bJMdIpsHN1KdVmJjBRgstzcXC40Rk0df0ChAlRSlArhAOD1elCUn6MovUgZYREgMhIWzxFLKy4phqvztNQRC5GwoJTDtGZ/AzfBn2+hL2kSmljyqoiMJIc8doVF9mJ4Gj3oCAVjZ1BYqpTDNEct7zFarERGCusGgqpDZCQonCMXxo5rbmltQrCtfeRCkrRk1+prC60xSlIL880iMlKIeYuLbPB6fQgE1eEZSTlMq6nlI2GW2Gj1tUK6qyhqEBmJAmv8Qm0lxQiHQ6h1qiOukZTDNGfnXFq+jY4oir9nqacEkZFCbGXKNXGa1HXuTleIWopQgwVVKywcQwseFWEN8ZQgMhIP27gklxTxQxAubk9cJeXJLOUw7ciRr1BcTKFD5LG0dLUSGUmH9aA1FRQWcu+rq9URC1uqYVrHsWNoa22DvZiGaIN2oCR4SWSkECOOTk/HSWNPQnkZnaHW0yR+nw9BTRA2K3lGPXGR4p79IZDyIjKSEu0h6jptxmmorlGHZyTVMM3j8yHgCcKYmTkEevRaSARYZM25c0/HfQ/cJ6TYQWUlPRk5ah244uqrsX3nzkGBUMLLyfap8Hiacdlll6GlpUUJKg2og1TDtNq6Ou4rY8nJJw+oSzK8YOfmXXfd9XjhlVcU0Ry9To9QJILf/O9vcOftvwLbrCz2lfRk1NTSipf/8Q9VBC+z2fkVxuvWrUMoEBDb9qqQ39rSxumZmZ7knlEkjFdffRXlZWWKsIvVYsGO997F6bNn46FH/oBbfnrTsAiJ7SBwutzYsumtaH5GZK+98hqGiteV9GQk1XBCiB5kt4+NilF6DDGpcG32NHCYaJK+p3aaPiK+BxLtZEPcsJNYNr/9Nhadez7++te/Yfmyy3D8eMegpWpqHHh6zZO4YMlF2LB+Pdi808XLLsTlVy3HU2vWDFo26SNVdQ0n1NCZCwq7914p/XevC9dBe5cAL5tUsghUgKbyIjTK+pXMyMzE6xtfxaXLL8eGDRsQWBrEunUvgaXHuiaUlmLCPffhwKFD2Pj6RgSDftx39/247+7fwzrERudht/zJ1au5ug8fPMj9ZH+5IxGl//0GfB4vp+8zTz+NnTt3KlZnjUbD7UwHY81IBLfcfjuMaWkIKwxjZvHRGg0+PfgJPB4Pbli5Ev/H+gIARqBC6/vhBx9y9ntk1WNY+8paweVzwmX+H/dbFI4gEPHjnS1bsMLtllkjvnrWJ7v+KGYZspCWloZNW97E2d87D++9/daAhMRKLz7/Itx+5+1YfP5iTJkyaXjtOTHMa9HixSfYPwD0jzCgPpDCfaCgsODEoUNlgzLHV/X/PaHV6k588MGOQfP1fImeD8l4//GBA9wvzvPPv6SK5pWOH8/pW3/kiKL1ZX+Y7PYS0XU0pBk4PN5/9z3R65KzAq/Xe8JozDpx6x13yKnGgHX//e/PcuQydszYEx/vOzBgvq4X615edyIrK+fEH+7/Q1fSkD+7vLDhuVEqzNU90aqcicHBYCxWyWbQblwHa01i79jyBn/AzwlRw5xfQq2NhBGR4PP5SHR84MGHce21P4LNZsP27Vsxa8a0QcU46urg8/pw1sJz8OGHH3B5a2r4yAuDFUx6MmKHJOr1euhUcqqEvaRkMHsp5h0/gS0uwTs75060Wh0iGPb0pmIwiksRjRYGoxF6nXJ+Jdkn+V/96k78/q47MHHyZHz04Q7Yi2P3T5Z3zeo1qDj8KdY+9xyWX3U5Tp89C9t378auPbtweP8nQ8KR5BYGSidMwGeff468rKwhwVBChrFj+M/7rpZWWGTykth6kNtuuQMWWw7qHG7cd99vwc5263nxnpG43ae+xsFV+Ze//gVzT5/Zs/qku8/MzMTBvXugMxgU0Tb2Cf/nt/4Sa1avxne+/R2sX/8K2Kf+Aa//+xr/XP8atu/chj8+8Sew7U1Xr/gRXn75FezZtRe/+MXPBiwafTHkQI4ySIrA2+++y82RvPQPeea4mj2eE6XjJ554fd2rXLsPHdh3YszYohNH6+t74cDPGdl7pQn98Jdn/8ZhUVZRIbRokjcEAvv2fXwiLS3txNLFi0+wPiHFpRyfMEqPqX0zcSJ/bFGNe+gxthhIrVr1JNrbfViy7EJO/JRpM2AvPgm/uv3OXtVJMWfUWucGG6JZsrN71U0P4iMwY8YsvL/jA7z88svIlQj/pCYjtvrzxbUvDmg5Ns791z//hQcfeACHK5WxWz4rOxtGYxYaKur66d3e1oZt72zr9W+oFbH9hAySwOQ//thjWHTB+WBzbV3XrElT8Prrr3OrabvSpFj0WOdyIivLiDfefBNr166FWyHrb7owSOSnq6kJ/9q4cUARzBZsDob9cXC75DnCik1Us+GWVFdSkhEjoVVPrML06TPxi1t+ERNLNi/yg8UX4khDDRYtXYo//PYePPXUUzHzSpnIjm+2mC2oaajvV+39Dz2Es793dvTfb+/5LUaNGt0v30gTKisq4ff7cNrY7m0pTBabVGdftd7e8X5UtBSe0dEjR+ENesEW3F5zzY9w8rdOxqZ//zuqgxpvnC4XfvGL2zDh1FPw8gvPxWzC4YpKnH3ed1FksWPmzOm4+OIrsGv37ph5kykxKcmIhZ245urlGNvnl6qn4e69/160+Vpw8023YdJpp+HXv/kNbrnlNnx2+NOe2SS/Zx5J0Ulj4HbWRTcaMiXYX8f9+w/i9Q0bsGHdBu7nM889I6h+DifvjfU9udVoNHL1fPHpZ9H6xPaMIpEw2OfgmROn45NPPsGnnx3mVgD/4o67wN6p9QoFg7j77t8gEtEg0sP77GoPa9uPr1uJBWcswKKLzsW8efNw3Y3X4arll6OtVdmRHLraMNKfSUlGbMdxZnYuJkybEhMXl7MWq1etxoUXL4u+L51wKgry87Bq1Z+iaXLd5JstcLncCPVYd/LSuldx52234cKLL8bFl1zM/RxXMk5QFf1Bfk2PPi2tl1ytRsc9B3pEEhDbM/o6ArjdLlg6TwRhbV125Q84ImLv1HrZ7XawL2f2ktgnnfzrX//Gxx99hOU/vDzaxPlzZqGuzoU3Nm2KpiXjTVKSUdRQA/wFfX3jm1yMnMnTp0azMo/klNNOw7oN66Npct3YbVY0Njci1M6Hz2DzBw88cB9WrlyBZRcvw57de0RRLcfAL38Io7fnEQj6uPqys3Oj9YruGR0/Do/XC0uPJRnHWoO49eafJ0Vg/kgoNqOyzah6nQ7jJk2IYl1UbOcm8l9/9fVoWjLeJDcZhbonYXsa7/PP+S9V9j67iO2FBfD5fKip+7xndsnvS0pKuTorq3g96+vqsOLq63DKqadg05ZNOPPb38Y999zDbU4VUjmbxcKJczsbe4l11fHHJ02fOjGaLrZnVN25xuikzkV227ZugwYRXH/TjVEd1HyjGWBxo8Ph4D5gsLnDrov9obRazdi+XfkBArt0HsnP5CYjXe+/8F0ANTbwcyN5hQVdSb1+trbIe5BiaSm/ynV/+V5Or3ETJuDRRx/Eu+++h08Pf4rpM6fj7rvvxj8H+RrTq0HDfJg4YwaysnLw5X+repVw1NUiJy8PZ8xfEE0X2zOqruJ1yDWZ8MDDD+C8C87Ds88/jzt//etec2lRhVR2E8szag8G4fN6YTb1X1yo0Wnh9XlU1sr41O2m3/jKSZqbDVMcR2sQPh4BC2sw0DVlyuB7ZrrKabWdMhjn9Pxy2TmhKAUo7GDCpqYmLvgGC8XCJjN5b0MDfyjEqVrxSWWXytGf9mI73t28GXNPPx2rHn4Uy5Yujb5L9Ib9Nb7rl7/E46tX4bE//YkbDrHlDxXllbjy6it7iRfbM6rsJKMFZ8xDns2Gc844C9fe8mM8fv/9WHTWOZh/Vjcx9lJMJQ+xPKOuLUtB8Pbv2RRNeOB+3zOfmu+l+L1LGJ+GZg+eXfMsNBEdoGMbCiPQavQIR4K90p748/DIyFrITx42BhuRgYyofu5mfnhS3GfrQzSDgDdVlVV4882NA7bFkGbAxx9+FLNGFtiKxSZefs3ymO8TSbzxpz/Fhtc34Pmnn8P1N16HDetfA/urzOrreYntGdXWOrh5kkK7nSPFWfPm4J/P/RunTDoZO3fvVj0ZxfKM2B+D/PzCmIcyuNwuTCzlF8T2tEMy3auCjEqK7Xjiz38WDPfZc2ZzshzlDhRbi6Ny2YkIE8dPHjRoVDRzgjdnnXMW2L+Brk8+/xz1R44M9BrF44tRYo+9aXHAQsN4kZGRjvfeew/3P3Y/7rz9NqQZs/D2W//ph4nYnlFDfQMKCvJ6TVaXjLdj9tTp0OlU0W0HRTuWZ8QKTJp0Gj786EMuvGvXGjIWvSAU8uOcRYsGlan2l8nv+8Ww0PmLF3FzINt3bo++ZUPBupo6XHPNVdE0OW9K7Hbu8EJ2akSsq7KsivNcYr1LNI15Xg/e8zAefPhR/PbXv+5HREy+2J6R2+WGraj35lw2ZPQFAjhjwfxEmyh7+VieEVPqR9f+COFIGO+9z4feYGmOGgeCoQguEnBILjsAMRRISTJif3EeffBhrNuwLrrFYdv7O2CymHD1yhUxYJI+yWYtgs8f4D5vr3nySaxYsRKMMNkH4YN79+P1Devw42t/LL1inTWK6Rl1tAXhbHGhubkZhysOR9v4j+eew9yZszFnxoxoWrLdsHnPa6/6EV54+ulo017dsAFLF5+PmXNmRdOS8Ub9/m4Mq7S3H8OWN97k9nDptWl4bPXjWHzuUrDhXtd19YoruVXN11xzLYrtRXC4avH2W5sl2xTYpcdAP8eNt3PBzFtcLmgNOrz00kvYuHEj5s6eifnnnIV//OMlSfcN9dVTTM8oED4Gf9CH+iNHMXvaTMw8fSaKrCWYNrkUf3nmqV775vrqpfTn2to6bHlnE1rcrQiHqvHii2ux6PxFvfrdH//0BH52y89w000/AVsql56hxz+e/0evIavS2zkS/b7BQgOMpGCylGEbTYOBALdiW0lt2r5tJ75z9rex7tVXccmyZdznbI+LrUhWxpnzS5ZcgPLyCnz55ZeCw1ZbW4tTTjoJdz/4MFZcfinSDGmKs4/gjY4hkHnCWq1W1j86MdQSLSkpPaN40GJDtq6JwnjKiZ3XXlwErU6LWgcfYIzbQKsQImJtF9MzYh8S2Mdt+5ixiiFfse0dS/5AxwHFypsMaSk5Z6QGwxVYrdBqtKiqUEZok76YiTlnVFdTy1VXUhx7/1ZfXeg5ORAgMlKoHZknNK6kFB8fUsZxx31hEtcz4slIKUPSvm2nZ3EQIDISB1dBpJaOH4eKcmWSkZieUXVVJbfg0WrtPmFXEEBJiKIRIDJSsHkmT+Q3pjqqqxWnpaieUVUNSlVySoriDKNihYiMFGy8rmOLqmv5YYuSVBXLM2ILG2sc1SjpjAWupDaTLuIiQGQkLr4JSR9Xwq+LYp+6lXaJ5Rn5mprgCwQxPsn3YSnNnkrQh8hICVYYQAezyQp9VhocVcobponlGXEHN0ZCsFr6h9EYACZKThIEiIwUbMgsQzbMmblg8YSUdonlGTXUOxEOR1BkG6O0JpM+IiNAZCQywImIH52pR35OARzDOKc8kXpGUlYsz+jLWv5UlMKTurfujEQ/KqM+BIiMFG4zW4kd5ZX9g6zJrbZYnlGjuw4sllP2aOGOYJIbK6p/eAgQGQ0PJ9lylRQVIeD3gx36p6RLLM/IUVvDHUk0UEhgJWFAugiLAJGRsHgKLm38ZH6t0cd7dgkuOxGBYnlGVRUO5JpNGC3g4ZSJtJPKSocAkZF0WI+oplnTZ3LlvihX1lBNLM+ooqIcpeOTO7zqiDpCChQiMlK4kcedeiq3NaJSYZPYYnhGLLwqO0bbTttAFN4rxVGPyEgcXAWVyoLC1Tj4M9QEFZyAMDE8o66J+uLOs9ISUI+KqhABIiMVGK24tASNRxsVdV6YGJ5RXTUfu2ny9O7DIlVgHlJRIASIjAQCUkwx9kI72FE1gdZWMauJS7YYnlGNiz+4cZyNP1E3LoUos+oRIDJSgQlPOqUEPp8XzT7lfN4XxzNyIqcwB7osnQqsQioKjQCRkdCIiiBv3Enf4qR+8knvY6dFqGrYIsXwjGrrnLBmWpFOCx6HbYdkykhkpAJrnnbaKZyWVQpaiS20ZxSJhFHfUI+ConxVn/6hgu6kWBWJjBRrmm7FWFwjrVaHQwqK+ii0Z/R1BHA5HLAVUNzrbsun1h2RkUrsPX3yZFQcOqQYbYX2jJrdbngDARQXdx83rpjGkiKSIEBkJAnMiVcyffZ0OJ0uDHTcdeI1xCdBaM/owAE+1neRnTyj+CyRPLmJjFRiy6JTixEMBdDQ0KAIjYX2jD7//FOuXROnTlVE+0gJ6REgMpIe8xHVOKlkAhd0zNF5qOOIhAhYSGjPqLJzcn5cjyPIBVSXRKkAASIjFRiJqVhQUAC2+ubzzlXKcqsttGfETpEtKSlR5Om+cmOdKvUTGanE0laLFXpjFo78lx/OyK22kJ4ROxHEWVOHEjtNXsttVznrJzKSE/046jaZTcjLysH+cmUcdy2kZ+Rv5Xfrl5xcEgcilDXZECAyUpFFT51xKg59fEARGgvpGXm9Pni8XhSeQnGvFWFcmZQgMpIJ+JFUO7l4OhfvRwknzArpGdW6ahEOh1A8hk4EGUm/SJYyREYqsuTU2fxn7yoFTGIL6RnVVPFHMdmLx6nIGqSq0AgQGQmNqIjyZkyawEn/7LNPRKxleKKF9IxqqvkNwEU26/Aqp1xJiQCRkYrMmpdTgKz8LFQ45D9hVkjP6FBZGQry8pCbm6sia5CqQiNAZCQ0oiLKG509GkWWItRW8BERRaxqSNFCeka7d+3GpFkzhqyTMiQ3AkRGKrKvVqOFzWpDZeewRk7VhfKM2HlwLAj/xHEU3VFOeyqhbiIjJVghDh1KTy1Fff1RtLW2xFFK+KxCeUbOzlNPJk6cLLySJFFVCBAZqcpcwPgJ/JliH30k73ojoTyjys69dsUltOBRZV1RcHWJjASHVFyBU2fzhzru3rtb3IqGkC6YZ/TVV1zgOBt9SRsC8eR/TWSkMhtPOu007iz6qip542EL5RlVffU5crKykJWVpTJLkLpCI0BkJDSiEsibOH4yKivl3aMmlGdUXVaDbGsuMjMzJUCOqlAyAkRGSrbOALqVTixFk7NJ1klsoTyjiqpKlNpovmgAU6dUMpGRCs09/rRT0eBpRKPXJ5v2QnhGTqcTXq8H40ppG4hshlRQxURGCjLGcFUpPfk0IBLBF4c/G24RwfMJ4Rkd6jxgYOJk+qwvuIFUKJDISIVGO2UCH2qjokq+eSMhPKPDnxzm0J8zZ44KrUAqC40AkZHQiEogb2zRKdwXtQ8/+liC2mJXIYRnVFNdzbWDjieKjXGqpRIZqdDio7RazJs3B7t27pRNeyE8o+o6B+zFJWCnydJFCBAZqbQPzJ45Fx5PM9gksBxXop5Rx7FjfNzrYjsdZy2HARVYJ5GRAo0yHJUmTuUnfatqKoeTXfA8iXpGgWAQDc0NKJ1In/UFN45KBRIZqdRw48bxv8SfyxROJFHPqKqmGgF2nLWNduurtAsKrjaRkeCQSiPQajIjLy8PX3xaLk2FfWpJ1DM6tI8/znrieCKjPtCm7CORkUpNb7JYYDKbsb9MnhC0iXpGhyt5Ei0lMlJpDxRebSIj4TGVTOLkiRPxcdlHktXXs6JEPaPygwdROq4UGbQnrSesKX1PZKRi84+fOBFhfxgHDx6UvBWJeEbsS1p1dTUmT+VjM0muPFWoSASIjBRpluEptWD+GVzGffv2Da+AgLkS8Ywa6uvh9fkxfjxtAxHQJKoXRWSkYhNOnjGLW8H8ySfSzxsl4hnVOGsRDgVRMo42yKq4+wmuOpGR4JBKJzBdp0WR3Y6ycum/qCXiGVVX1kAHoHQ8DdOk6y3Kr4nISPk2GlBDjUaL8aXjUeuoRXswOGA+9oJtuait5U9uHShj1eefY/eeXWBzOkNdiXhGLDCc1mCA1Wweqhp6n0IIEBmp3NjjS8dxp4X4mtwxW3I8HMauXbux8KyF+PENP4mZh5HPyhtuwIN/eAi7PjqA875/Htgm1sGuRDyjXbv2wGIx06GNgwGcgu+0KdjmpGryzOmzufZs37kLV1xR1K9tR2rrEAj4cKiqAjOmxD4o8a7//Q28LU3452v/5MpnpulxwYUX49AnB8E25ca6RuoZMeIrLz+Ay394aSyxlJbCCJBnpHLjzzpjFteCnR/FXm9UUmzHwoXnYrydTRb33x1fU1OFp59+CksvWhpF4pprrkKtowZrnngymtb3ZqSeUUVlFcLhCCZOnd5XJD2nOAJERirvAKZcE4pLinFgz54hW6LV9Df35i3buT1ipcXd2zJGjRqNkpISvL7p3wPKHKlnVHaI3wYyb/68AWXTi9REoH/vTE0cVN3qb889HXW1dWhraxuwHYyHwpH+5i4rOwCdRgOT2dSrrNliRU1lNdqPtfdK73oYqWdUXV3FLUcosfUfUnbJpp+piUDsCYHUxEK1rZ44Yw6efeklOBwOTJkyJa52OL78LzS6NGQZDb3KaXVa+AMBREIhIB3Yu3sPHnvsET6PRouD+/fC5/XhsssuQyQcQUQXwh0/vwszZvDDxl7COh/YF72KykpYLQUwmnqTX6z8lJZaCBAZJYG9p0wczwXor6ioGIKM+s8ZhQJhQB8BQr27Qt9hmKupCdt27oSm87/mYDM0gTC2bt3KpTAYr1sx+GklbClCZeVhFBefMuDEeBKYg5owQgR698ARCqFi0iHwzDPP4KHHHoEeOgQRgjYChDXgFhHe8OMb8ND9D8FcZMUb//4XRuv0UcXCiCA9xpyRtciKfYf2IRQJAOg+SLG1tQ0GgwGa0emcjCVLLsCSJU1ReRdeeCHKyirw5ZdfRNOGumlpbUVNzVf43vkXDpWV3qcgAkRGKjP6ypUrsfzqq/E/COP/wJuPfX4/5ZRToNel4eODezivg3khPS8tNDHnjGZOn4V1GzagubUFLCxJ19Xs8cButyFD301oXe/Yz1Ak0vNxWPfbt27n8s2fzS9HGFYhypQyCPSf0UyZpqu3oYx8tFo9Rzpd64DY5/vyijKEjh2LK6b0gnPOBhuS/beqphcgtQ4HzllwTq+0/g/9h33983Sn7Nv3Ifdw1lkLuhPpjhDoRIDIKEm6wuy5M7mWfHzgUMwWsWFarHVGUyadhsXnX4SNW96Ilntl/XqY8/Jw6y9vjab1veHnlHp7X33z9H0+VFaO/Px8Lihc33f0TAgQGSVJH5g1nSejg3v392pRe/sxHPzkMLcv7bPPqrB/7/5ee8/YcO7Z5/+OA/sO4Zmnn8PWd7biyVWr8PJrrw0a+Iz/tD98z+hYexscNQ7MOCP2KvBeStNDSiIQ35+2lIRIHY0eY7VgTOFY7Cvr7RlpNUBudia2bt/GFhpBF2MOKDMzEx998AEOHDoAvy+ItzdvHpSIGCJ9v7YNhZLfH4CjthaXLr18qKz0PkURIDJKEsOnZ2SiuLQEB/fs7dWi0enpKErnv4j1etHngeU7wFj5VQAAD9xJREFU44z5fVIHfox30WNFVSXC4RBKT+9e6T2wdHqTigjQMC2JrD5j0iR8deQrOF3iH+wYr2e0e9cuDulppRTdMYm6nKBNITISFE55hZ1zzrmcAhtff1N0ReL1jHbs2AFDmgHjTj1VdN2oAnUiQGSkTrvF1Po7C8/i9n3t3b0z5nshE+PxjFjYkL17D+Lcc3myFFIPkpU8CBAZJY8tuXVHU6dOxr6yMoQjw//SNRII4vGMKqtr4PP7cMYCWl80EqxTpQyRUZJZmm1UZQsWW93dWzfEaGI8nhELM4tIBNOnThRDFZKZJAgQGSWJIbuaMX32bC4+Ud9P/F3vhfoZj2dUfugAjEYD7PZioaonOUmIAJFRkhl1wZw5XIt2bhd33igez2jbzt0wmy1c3Oskg5uaIyACREYCgqkEUTa7nYv8+OZGcb+oDdczam9rw6EDB/DtBQvi2jOnBCxJB2kRIDKSFm9Jalt8/lLUuB1wOsVbbzRcz+j9HR9yix3nzJ0rSdupEvUiQGSkXtsNqPmCM89E0BvA/gN8vOkBMybwYrie0d69/GLHc89dlEBtVDQVECAySkIrT542BXqdDvv2iDdvNFzPaM+e/Vxw/zxTbhIiTU0SEgEiIyHRVIisoqIi2IqKsGnLNtE0Go5n1BEKoqysHBMnT6Yws6JZInkEExkljy17tWThOefg0KF9cNYNfqR1r0JxPAzHM6o4fBjNzQ2YPX3gIP1xVElZkxwBIqMkNfCixfyhjFu2ieMdDccz2vTmJg7dcxd+N0lRpmYJiQCRkZBoKkjW6XNncxtTd28TZ95oOJ7RO1u3IT+/AJOmxXd8koJgJFUkRIDISEKwpayKBUybOns6dnz0Adh5ZUJfQ3lGba0tqCivwKJFC4WumuQlKQJERklqWNass+bNR011Dar7BNsXoslDeUYV1dXwB7yYN4+OsRYC71SQQWSUxFY+f+n5XOvWr39F8FYO5Rnt3PEBwuEI5s0+XfC6SWByIkBklJx25VrFdvCPGVuETZveEXyoNpRntHH9aygsGIOTT6NgakncxQRtGpGRoHAqT9jFFy5FeWU5XC63oMoN5hm5XU58fGAfLr5sGa0vEhT15BZGZJTc9sWZp5+JgN+PPbv3CNrSwTyjbds7t4AspMiOgoKe5MKIjJLcwAsXngNoNNiwYYOgLR3MM9rx3nvcsoLZ0yn4vqCgJ7kwIqMkN3BGZiZ+uPQibNi0Aexzu1DXQJ4RW0bw1tubMXXmZOSazEJVR3JSAAEioxQw8g9+8EOE/WFs3bpdsNYO5Bnt3bMfR48excIFtL5IMLBTRBCRUQoYet78eTAYjNj0pnAB1wbyjJ5b/wyH6Irrr0sBZKmJQiJAZCQkmgqVZbVaMWfODGzY+DqOh4VZjR3LM2KyP3r3I8ycPRdWi0WhaJBaSkWAyEiplhFYr8svXQ6fz4dXX1kviORYntFX1TWorK4GixhAFyEQLwJERvEiptL8l155GYxGIzZseFWQFsTyjNaufZELMbu0c+W3IBWRkJRBgMgoRUw9WqfHuecvwrat29DSlPgCyL6eERuivfzqOkyeOhVs5TddhEC8CBAZxYuYivNfccll8Af8ePnVxL2jvp7Rvr17UVddg2uWL1cxQqS6nAgQGcmJvsR1L168GAVjCvHUk08lvFetr2f05htvIKLV4YIlSyRuFVWXLAgQGSWLJYfRDrYg8eabfoaK6kqw9UB9L/Y+HAlzRMXu2b+BLt4z0nKvO44dw+onV2Pu6aejuJhOjR0IM0ofHAEio8HxSYq3TW43br/tTqxceT0uueyH0Gq0ePrFp3q1rbrGgVnz5mHOvHmYN+8M7t7tbO2Vp/8DT1Yvr1vHDf9u/ulN/bNQCiEwTASIjIYJlFqzVTo+xeon12DNmtWob2iAvciOpUsX49m/Pg9HXR3XLOYBPfrIYzBo9cjU6aHX6jBvxkxYbKYBm80P03jP6IUXXoDdXoRF59LG2AEBoxdDIsD3piGzUQa1IlBqn4Df3TcBW7ZugVYT4Zpxxx2/xMaNb+CZNWtw3wMPoKW1FZGvj2P7ruHHy+aHaWHs3X8QH3zwAW699VaMTk9XK0yktwIQIM9IAUaQQgWdRhetZtq0aThz7lysXvMU2tva8If7H0KztzGuMCNdE9j3/v43SEtLw89vvTUqn24IgZEgQGQ0EtRUWEbTw9IajRa3/ep2eL0e/OSWn3ELITe+/gbmnj4XkyZNwvbtQ2+oZZ5RXZ0T27Ztx/Lly2n7hwr7hNJUpmGa0iwikT6Lzl2E75x9Nt54fSM+/fQwQqEwnlqzBk+98AyXvuP9HZg//4yoNlXVVdi0cSP+T6PFNzWAo8bBrbYOh0OA5n+w6vHHcP75i1FcUhItQzeEQFwInKArJRD49ulnnli06Nxebd2xY8cJACcu/eGl0fTKz784UVBYcGL6zJnRNHazYd2GE3q94YRer+f+sXLsn1ar4561Ou2JzZs39ypDD4RAPAh8g2WOi70os6wIrF27Hk+tXj2oDjabDX9/8RmwLSBd1xlnnIHMzAy89dbmriTu53Urr8ffn/0bl75oEf817MUX1+Kqq5ajubkZubm5XD62lqihuRlstBc8dgzzv3M2IsEA3t7xAXKzM7k8+Xl5NIndC116iAcBIqN40FJAXkYKHq+P04TN/XQtTOy6Zz+1GsBk7v1ZfsEZ85Gemd6PjNxuN6ZNmYKIBti//yA398NkjhqVjoaG+igZ9Wz6Lbf8AqtWPY6iIju++urLnq/onhAYMQI0ZzRi6OQpyD6fW0fwCT0M/rN+X63NZjP+8ten8YMfXIwrrrgKm/+zkctSOq40JhG99sp6rF69isvDiI8uQkAoBKg3CYWkwuVouQFWfyWPH+/AhAmluOuOn+Pe+x/CRRdeiKkzZuCOO/p/qt/6zhb86NprYbPZUTAmHw1HG/oLpBRCYIQI9PjgO0IJVEzxCLCh3dHGetTWOsHue16vvrwBJ598Mqqqa1BcUowt776L1auf5IZgXfnYWqRf//bX+P55S2Aym/Hu25uRnd17GNiVl34SAiNFgOaMRoqcSso5XS5semMT2o/5EfnmcaRrMrD4oqXRdUGMnFav/huqa75Akc2GiC6CR+5/hIsKyYZqeYX5OPTxAW7vGVsK8Nwzz6GoyIbzlixBZXk5vvyS5oxU0hUUryaRkeJNJL2CLS0teHLVaryz7R14vV5MnzoRV1xyJb5z7sLoCbFLllyA8vIKIiPpzZO0NRIZJa1pxW0YeUbi4puK0mnOKBWtTm0mBBSIAJGRAo2iBpW6NsqqQVfSUR0IEBmpw06K07JvDGzFKUgKqQ4BIiPVmUwZCpNnpAw7JJMWREbJZE0J20KekYRgp0hVREYpYmhqJiGgdASIjJRuIYXqR8M0hRpGxWoRGanYeHKqTsM0OdFPzrqJjJLTrqK3ijwj0SFOuQqIjFLO5MI0mDwjYXAkKd0IEBl1Y0F3hAAhICMCREYygq/mqmmYpmbrKVN3IiNl2kXxWtEwTfEmUp2CREaqM5kyFCbPSBl2SCYtiIySyZoStoU8IwnBTpGqiIxSxNDUTEJA6QgQGSndQgrVj4ZpCjWMitUiMlKx8eRUnYZpcqKfnHUTGSWnXUVvFXlGokOcchUQGaWcyYVpMHlGwuBIUroRIDLqxoLuCAFCQEYEiIxkBF/NVdMwTc3WU6buREbKtIvitaJhmuJNpDoFiYxUZzJlKEyekTLskExaaJOpMdSW/gi4mprwwvPP46vPP8fY4hLcdNONyMzM7JeR5XtmzVPweL24avnlmDJtWr88PRPIM+qJBt0LgQCRkRAoKlRGVXUVLjhrITB6FDqOh3D02Wfx1Jo1eHvrVpw6riSqdU11NS6+8GI88ec/obikGCuuWIFLLluG666/PpqHbggBsRGgYZrYCMskPxIJ48EH/ohVf/0LvvjiC9TVfoU///lPOHL0CG648bqoVh3HjuHKq1dg8eLzsWDBAtisNtz3wH24/c5fobrGEc3X94aGaX0RoedEESAyShRBhZY/Fgpj2bLz8f3zzotqeNNNP8Xc78xFRXkF2trauPSPyz7GRx9/hDlnzIvmmzVnBtLS0vDogw9F0/re0DCtLyL0nCgCREaJIqjQ8hl6PRYtuqCfdrNOmwZjmhH69HTu3e7te6HVaFH8rZOjeTUaLWw2Kz7etwfHw+Foes8b8ox6okH3QiBAc0ZCoKgiGQcPH8aKlSswSsub/tCBA5z2hoyMXq3INZuxf/cehI53YJQ2A/v378Xq1Wu4PJqIBmX796PJ68N1K1cgBC00kTA3OT5jxqxecuiBEBguAkRGw0UqCfJ9sr8MtY5a3Pzzn0db0+R2Q2vQwpCmj6axG+b5BAIBREIhIB1wO5vwzpZ3oNEAbIjm8/oQCoWxZcs2sPkpln7llVf2kkEPhEA8CBAZxYOWAvKyL2SfffrZoJpkZGTizAXzo94Py8yGW/c+cg+eeeYpZGTwQzSW/j/MQwr2H60zwtGwdxq+i3z/gu/hv9+vQTgCaDXA0iUXo7qqEmWfdeui6/S2BlWOXhICAyBAZDQAMEpNbmtpRdWXX3LqRb4OQ/NNLfr+zM0y4EzM79WExx5+GGfNPwfnLDy3V/qpkyZg98cfwe8PIDM7N/qutakFZpMJutGjuDQ2jzRa191dtJ33bG6KLkJACAS6e5cQ0kiG6AjMmjMH7F8819r16zFKq+PmdPqWmzd7Nv7y5JOoc7lhtdm412zY5XI6MWHKFIzWxSYbmsDuiyQ9J4pAf/88UYlUXlEIbH1nC8rKDuHW22+L6sXWFh3cu597Puucs2BIM6Ciojz6vrWtHV8d+QpXLb80mtb3hj7t90WEnhNFgMgoUQQVXP6f61/BihXXo9XdhBtWrsQN112PFStW4PQzv42jvmZOc4vZgnvvvRdPP/10tCX333s/fnjRRVi27JJoWt8b8oz6IkLPiSJAw7REEVRoeUd1NR5fsxomay4OlB/qpaXBmI3vzJwbTfvZLf+P+yJ24SUXI0efBVNRLv727LPR97FuyDOKhQqlJYLAN06cOHEiEQFUNjUROG/JElSWl+PLzsn01ESBWi0kAjRMExLNFJJFw7QUMrZETaVhmkRAJ1s1v/zlXfC2uJOtWdQeGRGgYZqM4FPVhAAh0I0ADdO6saA7QoAQkBEBIiMZwaeqCQFCoBsBIqNuLOiOECAEZESAyEhG8KlqQoAQ6EaAyKgbC7ojBAgBGREgMpIRfKqaECAEuhEgMurGgu4IAUJARgSIjGQEn6omBAiBbgSIjLqxoDtCgBCQEQEiIxnBp6oJAUKgGwEio24s6I4QIARkRIDISEbwqWpCgBDoRoDIqBsLuiMECAEZESAykhF8qpoQIAS6ESAy6saC7ggBQkBGBP4/9ahM+lnawN4AAAAASUVORK5CYII=">       <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>first derivative changes from positive to negative at  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = a">
  <mi>x</mi>
  <mo>=</mo>
  <mi>a</mi>
</math></span>      <em><strong>A1</strong></em></p>
<p>so local maximum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = a">
  <mi>x</mi>
  <mo>=</mo>
  <mi>a</mi>
</math></span>     <em><strong>A</strong><strong>G</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 2x + 2">
  <msup>
    <mi>f</mi>
    <mo>″</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>2</mn>
</math></span>     <em><strong>A1</strong></em></p>
<p>substituting <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
  <mi>b</mi>
</math></span> into <strong>their</strong> second derivative <em><strong>    (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( 3 \right) = 2 \times 3 + 2">
  <msup>
    <mi>f</mi>
    <mo>″</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mn>3</mn>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>2</mn>
  <mo>×</mo>
  <mn>3</mn>
  <mo>+</mo>
  <mn>2</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( b \right) = 8">
  <msup>
    <mi>f</mi>
    <mo>″</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>b</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>8</mn>
</math></span>     <em><strong>(A</strong><strong>1)</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="f''\left( b \right)">
  <msup>
    <mi>f</mi>
    <mo>″</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>b</mi>
    <mo>)</mo>
  </mrow>
</math></span> is positive so graph is concave up      <em><strong>R1</strong></em></p>
<p>so local minimum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = b">
  <mi>x</mi>
  <mo>=</mo>
  <mi>b</mi>
</math></span>       <em><strong>AG</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>normal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = a">
  <mi>x</mi>
  <mo>=</mo>
  <mi>a</mi>
</math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span> = −5 (seen anywhere)          <em><strong>(A1)</strong></em></p>
<p>attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
  <mi>y</mi>
</math></span>-coordinate at their value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
  <mi>b</mi>
</math></span>         <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 3 \right) = ">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mn>3</mn>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
</math></span> −10      <em><strong> (A1)</strong></em></p>
<p>tangent at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = b">
  <mi>x</mi>
  <mo>=</mo>
  <mi>b</mi>
</math></span> has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
  <mi>y</mi>
</math></span> = −10 (seen anywhere)        <em><strong> A1</strong></em></p>
<p>intersection at (−5, −10)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
  <mi>p</mi>
</math></span> = −5 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
  <mi>q</mi>
</math></span> = −10      <em><strong>  A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of a quadratic function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has its vertex at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>3</mn><mo>,</mo><mo>&#160;</mo><mn>4</mn><mo>)</mo></math>, and it passes through point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> as shown.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>

<div class="specification">
<p>The function can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>h</mi><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>k</mi></math>.</p>
</div>

<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math>.</p>
</div>

<div class="specification">
<p>Now consider another function <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>. The derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>&#8242;</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>d</mi></math>,&nbsp;where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>&#8712;</mo><mi mathvariant="normal">&#8477;</mi></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the axis of symmetry.</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>Write down the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>12</mn><mo>)</mo></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</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>Find the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is an increasing function.</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>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for which the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is concave-up.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math>            <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Must be an equation in the form “ <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ”. Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>=</mo><mn>3</mn></math>.</p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>k</mi><mo>=</mo><mn>4</mn></math>   (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn></math>)            <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math>             <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>5</mn><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>,</mo><mo> </mo><mo> </mo><mn>4</mn><mi>a</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>12</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn></math>             <em><strong>A1</strong></em></p>
<p> </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>recognize need to find derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>            <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced></math>  or  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>12</mn></math>             <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mn>5</mn></mfenced><mo>=</mo><mn>8</mn></math>  (may be seen as gradient in their equation)            <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>8</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced></math>  or  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>8</mn><mi>x</mi><mo>-</mo><mn>28</mn></math>             <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>28</mn></math>.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>Recognizing that for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> to be increasing, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mi>d</mi><mo>&gt;</mo><mn>0</mn></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>&gt;</mo><mn>0</mn></math>          <strong><em>(M1)</em></strong></p>
<p>The vertex must be above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mi>d</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>d</mi><mo>-</mo><mn>4</mn><mo>&lt;</mo><mn>0</mn></math>          <em><strong>(R1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>&lt;</mo><mn>4</mn></math>             <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempting to find discriminant of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo></math>          <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>12</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>2</mn></mfenced><mfenced><mrow><mn>22</mn><mo>-</mo><mi>d</mi></mrow></mfenced></math></p>
<p>recognizing discriminant must be negative          <em><strong>(R1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>32</mn><mo>+</mo><mn>8</mn><mi>d</mi><mo>&lt;</mo><mn>0</mn></math>   OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo>&lt;</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>&lt;</mo><mn>4</mn></math>             <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing that for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> to be concave up, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mo>&gt;</mo><mn>0</mn></math>          <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mo>&gt;</mo><mn>0</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>&gt;</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>&gt;</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>&gt;</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>&gt;</mo><mn>3</mn></math>          <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>In parts (a) and (b) of this question, a majority of candidates recognized the connection between the coordinates of the vertex and the axis of symmetry and the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>, and most candidates were able to successfully substitute the coordinates of point Q to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>. In part (c), the candidates who recognized the need to use the derivative to find the gradient of the tangent were generally successful in finding the equation of the line, although many did not give their equation in the proper form 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>, and instead wrote <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mn>8</mn><mi>x</mi><mo>-</mo><mn>28</mn></math>, thus losing the final mark. Parts (d) and (e) were much more challenging for candidates. Although a good number of candidates recognized that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>&gt;</mo><mn>0</mn></math> in part (d), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>&gt;</mo><mn>0</mn></math> in part (e), very few were able to proceed beyond this point and find the correct inequalities for their final answers.</p>
<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>
<br><hr><br><div class="specification">
<p>A function, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, has its derivative given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>&#8242;</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>12</mn><mi>x</mi><mo>+</mo><mi>p</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#8712;</mo><mi mathvariant="normal">&#8477;</mi></math>. The following&nbsp;diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>&#8242;</mo></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>&#8242;</mo></math> has an axis of symmetry <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>q</mi></math>.</p>
</div>

<div class="specification">
<p>The vertex of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>&#8242;</mo></math> lies on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</p>
</div>

<div class="specification">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a point of inflexion at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>a</mi></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of the discriminant of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo></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>Hence or otherwise, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</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>Find the value of the gradient of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo></math> at <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">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>″</mo></math>, the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>. Indicate clearly the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept.</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>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for which the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is concave-down. Justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math>          <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>3</mn></mrow></mfrac></math></p>
<p><br><strong>OR</strong></p>
<p>attempt to complete the square          <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>12</mn><mo>+</mo><mi>p</mi></math></p>
<p><br><strong>OR</strong></p>
<p>attempt to differentiate and equate to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math>          <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>2</mn></math>         <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>discriminant <math xmlns="http://www.w3.org/1998/Math/MathML"><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">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt to substitute into <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>a</mi><mi>c</mi></math>        <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>12</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mi>p</mi><mo>=</mo><mn>0</mn></math>        <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mn>2</mn><mo>)</mo><mo>=</mo><mn>0</mn></math>       <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>12</mn><mo>+</mo><mi>p</mi><mo>=</mo><mn>0</mn></math>        <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>12</mn></math>        <em><strong>A1</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"><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>12</mn></math>        <em><strong>A1</strong></em></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mn>0</mn></mfenced></math>        <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo>×</mo><mn>0</mn><mo>-</mo><mn>12</mn></math></p>
<p>gradient <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>12</mn></math>        <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 style="text-align:left;padding-left:120px;"><img src="data:image/png;base64,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">        <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for line with positive gradient, <em><strong>A1</strong> </em>for correct intercepts.</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 style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn></math>        <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&lt;</mo><mn>2</mn></math>        <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>&lt;</mo><mn>0</mn></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&lt;</mo><mn>2</mn></math>)  OR  the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo></math> is below the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&lt;</mo><mn>2</mn></math>)</p>
<p style="text-align:left;">OR   <img src="data:image/png;base64,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"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo></math>  (sign diagram must be labelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo></math>)        <em><strong>R1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Candidates did score well on this question. As always, candidates are encouraged to read the questions carefully for key words such as 'value' as opposed to 'expression'. So, if asked for the value of the discriminant, their answer should be a number and not an expression found from <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>a</mi><mi>c</mi></math>. As such the value of the discriminant in (b)(i) was often seen in (b)(ii). Please ask students to use a straight edge when sketching a straight line! Overall, the reasoning mark for determining where the graph of <em>f</em> is concave-down, was an improvement on previous years. Sign diagrams were typically well labelled, and the description contained clarity regarding which function was being referred to.</p>
<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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \log k(6x - 3{x^2})">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>log</mi>
  <mo>⁡</mo>
  <mi>k</mi>
  <mo stretchy="false">(</mo>
  <mn>6</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mn>3</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo stretchy="false">)</mo>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 &lt; x &lt; 2">
  <mn>0</mn>
  <mo>&lt;</mo>
  <mi>x</mi>
  <mo>&lt;</mo>
  <mn>2</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k &gt; 0">
  <mi>k</mi>
  <mo>&gt;</mo>
  <mn>0</mn>
</math></span>.</p>
<p>The equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mn>2</mn>
</math></span> has exactly one solution. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
  <mi>k</mi>
</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><strong>METHOD 1 – using discriminant</strong></p>
<p>correct equation without logs     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6x - 3{x^2} = {k^2}">
  <mn>6</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mn>3</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span></p>
<p>valid approach     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3{x^2} + 6x - {k^2} = 0,{\text{ }}3{x^2} - 6x + {k^2} = 0">
  <mo>−</mo>
  <mn>3</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>6</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>0</mn>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mn>3</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>6</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p>recognizing discriminant must be zero (seen anywhere)     <strong><em>M1</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Delta  = 0">
  <mi mathvariant="normal">Δ</mi>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p>correct discriminant     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{6^2} - 4( - 3)( - {k^2}),{\text{ }}36 - 12{k^2} = 0">
  <mrow>
    <msup>
      <mn>6</mn>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>4</mn>
  <mo stretchy="false">(</mo>
  <mo>−</mo>
  <mn>3</mn>
  <mo stretchy="false">)</mo>
  <mo stretchy="false">(</mo>
  <mo>−</mo>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo stretchy="false">)</mo>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mn>36</mn>
  <mo>−</mo>
  <mn>12</mn>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p>correct working     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12{k^2} = 36,{\text{ }}{k^2} = 3">
  <mn>12</mn>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>36</mn>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \sqrt 3 ">
  <mi>k</mi>
  <mo>=</mo>
  <msqrt>
    <mn>3</mn>
  </msqrt>
</math></span>     <strong><em>A2     N2</em></strong></p>
<p><strong>METHOD 2 – completing the square</strong></p>
<p>correct equation without logs     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6x - 3{x^2} = {k^2}">
  <mn>6</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mn>3</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span></p>
<p>valid approach to complete the square     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3({x^2} - 2x + 1) =  - {k^2} + 3,{\text{ }}{x^2} - 2x + 1 - 1 + \frac{{{k^2}}}{3} = 0">
  <mn>3</mn>
  <mo stretchy="false">(</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>1</mn>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mo>−</mo>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>3</mn>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>1</mn>
  <mo>−</mo>
  <mn>1</mn>
  <mo>+</mo>
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mi>k</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
    <mn>3</mn>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p>correct working     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{(x - 1)^2} =  - {k^2} + 3,{\text{ }}{(x - 1)^2} - 1 + \frac{{{k^2}}}{3} = 0">
  <mn>3</mn>
  <mrow>
    <mo stretchy="false">(</mo>
    <mi>x</mi>
    <mo>−</mo>
    <mn>1</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mo>−</mo>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>3</mn>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <mo stretchy="false">(</mo>
    <mi>x</mi>
    <mo>−</mo>
    <mn>1</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>1</mn>
  <mo>+</mo>
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mi>k</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
    <mn>3</mn>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p>recognizing conditions for one solution     <strong><em>M1</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(x - 1)^2} = 0,{\text{ }} - 1 + \frac{{{k^2}}}{3} = 0">
  <mrow>
    <mo stretchy="false">(</mo>
    <mi>x</mi>
    <mo>−</mo>
    <mn>1</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>0</mn>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mo>−</mo>
  <mn>1</mn>
  <mo>+</mo>
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mi>k</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
    <mn>3</mn>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p>correct working     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{k^2}}}{3} = 1,{\text{ }}{k^2} = 3">
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mi>k</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
    <mn>3</mn>
  </mfrac>
  <mo>=</mo>
  <mn>1</mn>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>k</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \sqrt 3 ">
  <mi>k</mi>
  <mo>=</mo>
  <msqrt>
    <mn>3</mn>
  </msqrt>
</math></span>    <strong> <em>A2     N2</em></strong></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="specification">
<p>A factory produces shirts. The cost, <em>C</em>, in Fijian dollars (FJD), of producing<em> x</em> shirts can be modelled by</p>
<p style="text-align: center;"><em>C</em>(<em>x</em>) = (<em>x</em>&nbsp;− 75)<sup>2</sup> + 100.</p>
</div>

<div class="specification">
<p>The cost of production should not exceed 500 FJD. To do this the factory needs to produce at least 55 shirts and at most <em>s</em> shirts.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the cost of producing 70 shirts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>s</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of shirts produced when the cost of production is lowest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(70 − 75)<sup>2</sup> + 100     <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituting in <em>x</em> = 70.</p>
<p>125     <em><strong>(A1) (C2)</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>(<em>s</em> − 75)<sup>2</sup> + 100 = 500     <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for equating <em>C</em>(<em>x</em>) to 500. Accept an inequality instead of =.</p>
<p><strong>OR</strong></p>
<p><img src="data:image/png;base64,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">     <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for sketching correct graph(s).</p>
<p>(<em>s</em> =) 95    <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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">     <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for an attempt at finding the minimum point using graph.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{95 + 55}}{2}">
  <mfrac>
    <mrow>
      <mn>95</mn>
      <mo>+</mo>
      <mn>55</mn>
    </mrow>
    <mn>2</mn>
  </mfrac>
</math></span>     <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for attempting to find the mid-point between their part (b) and 55.</p>
<p><strong>OR</strong></p>
<p>(<em>C'</em>(<em>x</em>) =) 2<em>x</em> − 150 = 0     <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for an attempt at differentiation that is correctly equated to zero.</p>
<p>75     <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The equation of a curve is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{2}{x^4} - \frac{3}{2}{x^2} + 7">
  <mi>y</mi>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>4</mn>
    </msup>
  </mrow>
  <mo>−<!-- − --></mo>
  <mfrac>
    <mn>3</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>7</mn>
</math></span>.</p>
</div>

<div class="specification">
<p>The gradient of the tangent to the curve at a point P is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 10">
  <mo>−<!-- − --></mo>
  <mn>10</mn>
</math></span>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\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>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of P.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^3} - 3x">
  <mn>2</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>3</mn>
  <mi>x</mi>
</math></span>     <strong><em>(A1)(A1)     (C2)</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="2{x^3}">
  <mn>2</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>3</mn>
    </msup>
  </mrow>
</math></span>, award <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3x">
  <mo>−</mo>
  <mn>3</mn>
  <mi>x</mi>
</math></span>.</p>
<p>Award at most <strong><em>(A1)(A0) </em></strong>if there are any extra terms.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^3} - 3x =  - 10">
  <mn>2</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>3</mn>
  <mi>x</mi>
  <mo>=</mo>
  <mo>−</mo>
  <mn>10</mn>
</math></span>    <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(M1) </em></strong>for equating their answer to part (a) to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 10">
  <mo>−</mo>
  <mn>10</mn>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x =  - 2">
  <mi>x</mi>
  <mo>=</mo>
  <mo>−</mo>
  <mn>2</mn>
</math></span>    <strong><em>(A1)</em>(ft)</strong></p>
<p> </p>
<p><strong>Note:     </strong>Follow through from part (a). Award <strong><em>(M0)(A0) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2">
  <mo>−</mo>
  <mn>2</mn>
</math></span> seen without working.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{2}{( - 2)^4} - \frac{3}{2}{( - 2)^2} + 7">
  <mi>y</mi>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <mo stretchy="false">(</mo>
    <mo>−</mo>
    <mn>2</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mn>4</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <mo stretchy="false">(</mo>
    <mo>−</mo>
    <mn>2</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>7</mn>
</math></span>    <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(M1) </em></strong>substituting their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2">
  <mo>−</mo>
  <mn>2</mn>
</math></span> into the original function.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 9">
  <mi>y</mi>
  <mo>=</mo>
  <mn>9</mn>
</math></span>    <strong><em>(A1)</em>(ft)     <em>(C4)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 2,{\text{ }}9)">
  <mo stretchy="false">(</mo>
  <mo>−</mo>
  <mn>2</mn>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mn>9</mn>
  <mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider a function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> with domain <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>&#60;</mo><mi>x</mi><mo>&#60;</mo><mi>b</mi></math>. The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math>,&nbsp;the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math>, the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, has <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><mi>p</mi><mo>,</mo><mo>&#160;</mo><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math> . There are&nbsp;local maximum points at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math> and a local minimum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find all the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is increasing. Justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a local maximum.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a local minimum. Justify your answer.</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>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has points of inflexion. Justify your answer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The total area of the region enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math>, the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>p</mi></mfenced><mo>+</mo><mi>f</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>4</mn></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Special note:</strong> In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> or the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> to earn the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> increases when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>0</mn></math><strong>                <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> increases when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>&gt;</mo><mn>0</mn></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> is above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis<strong>                <em>R1</em></strong></p>
<p><br><strong>Note:</strong> Do not award <em><strong>A0R1</strong></em>.</p>
<p> </p>
<p><em><strong>[2</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Special note:</strong> In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> or the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> to earn the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math><strong>               <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1</strong></em><em><strong> mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Special note:</strong> In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> or the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> to earn the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is minimum when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math>               <strong><em>A1</em></strong></p>
<p>because <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>p</mi></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>&lt;</mo><mn>0</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&lt;</mo><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>&gt;</mo><mn>0</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mi>p</mi></math></p>
<p>(may be seen in a sign diagram clearly labelled as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math>)</p>
<p>OR because <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> changes from negative to positive at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math></p>
<p>OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>p</mi></mfenced><mo>=</mo><mn>0</mn></math> and slope of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> is positive at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math>               <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Do not award<em><strong> A0 R1</strong></em></p>
<p> </p>
<p><em><strong>[2</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Special note:</strong> In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> or the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> to earn the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has points of inflexion when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>q</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math>               <strong><em>A2</em></strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> has turning points at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>q</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math></p>
<p>OR</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>q</mi></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>f</mi><mo>''</mo><mfenced><mi>r</mi></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> changes from increasing to decreasing or vice versa at each of these <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-values (may be seen in a sign diagram clearly labelled as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math>)              <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A0</strong></em> if any incorrect answers are given. Do not award<em><strong> A0R1</strong></em></p>
<p> </p>
<p><em><strong>[3</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Special note:</strong> In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> or the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> to earn the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p>recognizing area from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> (seen anywhere)               <strong><em>M1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>p</mi><mi>t</mi></munderover><mfenced open="|" close="|"><mrow><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p>recognizing to negate integral for area below <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis               <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>p</mi><mn>0</mn></munderover><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>-</mo><munderover><mo>∫</mo><mn>0</mn><mi>t</mi></munderover><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>p</mi><mn>0</mn></munderover><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>+</mo><munderover><mo>∫</mo><mi>t</mi><mn>0</mn></munderover><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>m</mi><mi>n</mi></munderover><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mi>f</mi><mfenced><mi>n</mi></mfenced><mo>-</mo><mi>f</mi><mfenced><mi>m</mi></mfenced></math> (for any integral)               <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>-</mo><mi>f</mi><mfenced><mi>p</mi></mfenced><mo>-</mo><mfenced open="[" close="]"><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced><mo>-</mo><mi>f</mi><mfenced><mn>0</mn></mfenced></mrow></mfenced></math>   OR   <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>-</mo><mi>f</mi><mfenced><mi>p</mi></mfenced><mo>+</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>-</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math>               <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>-</mo><mfenced open="[" close="]"><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced><mo>+</mo><mi>f</mi><mfenced><mi>p</mi></mfenced></mrow></mfenced><mo>=</mo><mn>20</mn><mo>,</mo><mo> </mo><mo> </mo><mn>2</mn><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>-</mo><mn>4</mn><mo>=</mo><mn>20</mn></math>               <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>12</mn></math>               <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[6</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {x^2} - x">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−<!-- − --></mo>
  <mi>x</mi>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
  <mi>x</mi>
  <mo>∈<!-- ∈ --></mo>
  <mrow>
    <mi mathvariant="double-struck">R</mi>
  </mrow>
</math></span>. The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.25.10.png" alt="N17/5/MATME/SP1/ENG/TZ0/08"></p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> crosses the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span>-axis at the origin and at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(1,{\text{ }}0)">
  <mrow>
    <mtext>P</mtext>
  </mrow>
  <mo stretchy="false">(</mo>
  <mn>1</mn>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;</mtext>
  </mrow>
  <mn>0</mn>
  <mo stretchy="false">)</mo>
</math></span>.</p>
</div>

<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
  <mi>L</mi>
</math></span> intersects the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> at another point Q, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.27.48.png" alt="N17/5/MATME/SP1/ENG/TZ0/08.c.d"></p>
</div>

<div class="question">
<p>Find the area of the region enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>valid approach     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {L - f,{\text{ }}\int_{ - 1}^1 {(1 - {x^2}){\text{d}}x} } "> <mo>∫</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>f</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span>, splitting area into triangles and integrals</p>
<p>correct integration     <strong><em>(A1)(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {x - \frac{{{x^3}}}{3}} \right]_{ - 1}^1,{\text{ }} - \frac{{{x^3}}}{3} - \frac{{{x^2}}}{2} + \frac{{{x^2}}}{2} + x"> <msubsup> <mrow> <mo>[</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <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>2</mn> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mi>x</mi> </math></span></p>
<p>substituting <strong>their</strong> limits into <strong>their</strong> integrated function and subtracting (in any order)     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - \frac{1}{3} - \left( { - 1 - \frac{{ - 1}}{3}} \right)"> <mn>1</mn> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><strong>Note:</strong>     Award <strong><em>M0 </em></strong>for substituting into original or differentiated function.</p>
<p> </p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{4}{3}"> <mo>=</mo> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span>     <strong><em>A2     N3</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{1}{{\sqrt {2x - 1} }}">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mrow>
      <msqrt>
        <mn>2</mn>
        <mi>x</mi>
        <mo>−<!-- − --></mo>
        <mn>1</mn>
      </msqrt>
    </mrow>
  </mfrac>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > \frac{1}{2}">
  <mi>x</mi>
  <mo>&gt;</mo>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
</math></span>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x} ">
  <mo>∫</mo>
  <mrow>
    <mrow>
      <msup>
        <mrow>
          <mrow>
            <mo>(</mo>
            <mrow>
              <mi>f</mi>
              <mrow>
                <mo>(</mo>
                <mi>x</mi>
                <mo>)</mo>
              </mrow>
            </mrow>
            <mo>)</mo>
          </mrow>
        </mrow>
        <mn>2</mn>
      </msup>
    </mrow>
    <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>Part of the graph of <em>f</em> is shown in the following diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The shaded region&nbsp;<em>R</em> is enclosed by the graph of <em>f</em>, the <em>x</em>-axis, and the lines <em>x</em> = 1 and <em>x</em> = 9 . Find the volume of the solid formed when <em>R</em> is revolved 360°&nbsp;about the <em>x</em>-axis.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>correct working      <em><strong>(A1)</strong></em></p>
<p>eg   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{2x - 1}}{\text{d}}x,\,\,\int {{{\left( {2x - 1} \right)}^{ - 1}},\,\,\frac{1}{{2x - 1}},\,\,\int {{{\left( {\frac{1}{{\sqrt u }}} \right)}^2}\frac{{{\text{d}}u}}{2}} } } ">
  <mo>∫</mo>
  <mrow>
    <mfrac>
      <mn>1</mn>
      <mrow>
        <mn>2</mn>
        <mi>x</mi>
        <mo>−</mo>
        <mn>1</mn>
      </mrow>
    </mfrac>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
    <mo>,</mo>
    <mspace width="thinmathspace"></mspace>
    <mspace width="thinmathspace"></mspace>
    <mo>∫</mo>
    <mrow>
      <mrow>
        <msup>
          <mrow>
            <mrow>
              <mo>(</mo>
              <mrow>
                <mn>2</mn>
                <mi>x</mi>
                <mo>−</mo>
                <mn>1</mn>
              </mrow>
              <mo>)</mo>
            </mrow>
          </mrow>
          <mrow>
            <mo>−</mo>
            <mn>1</mn>
          </mrow>
        </msup>
      </mrow>
      <mo>,</mo>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mfrac>
        <mn>1</mn>
        <mrow>
          <mn>2</mn>
          <mi>x</mi>
          <mo>−</mo>
          <mn>1</mn>
        </mrow>
      </mfrac>
      <mo>,</mo>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mo>∫</mo>
      <mrow>
        <mrow>
          <msup>
            <mrow>
              <mrow>
                <mo>(</mo>
                <mrow>
                  <mfrac>
                    <mn>1</mn>
                    <mrow>
                      <msqrt>
                        <mi>u</mi>
                      </msqrt>
                    </mrow>
                  </mfrac>
                </mrow>
                <mo>)</mo>
              </mrow>
            </mrow>
            <mn>2</mn>
          </msup>
        </mrow>
        <mfrac>
          <mrow>
            <mrow>
              <mtext>d</mtext>
            </mrow>
            <mi>u</mi>
          </mrow>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </mrow>
  </mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\int {\left( {f\left( x \right)} \right)} ^2}{\text{d}}x = \frac{1}{2}{\text{ln}}\left( {2x - 1} \right) + c">
  <mrow>
    <mo>∫</mo>
    <msup>
      <mrow>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mi>f</mi>
            <mrow>
              <mo>(</mo>
              <mi>x</mi>
              <mo>)</mo>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mn>2</mn>
    </msup>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <mtext>ln</mtext>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>2</mn>
      <mi>x</mi>
      <mo>−</mo>
      <mn>1</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>+</mo>
  <mi>c</mi>
</math></span>      <em><strong>A2 N3</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{\text{ln}}\left( {2x - 1} \right)">
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <mtext>ln</mtext>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>2</mn>
      <mi>x</mi>
      <mo>−</mo>
      <mn>1</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula involving <em>f </em><sup>2</sup> (accept absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
  <mi>π</mi>
</math></span> / d<em>x</em>)&nbsp; &nbsp; &nbsp;<strong><em>(M1)</em></strong></p>
<p><em>eg&nbsp; &nbsp;</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_1^9 {{y^2}{\text{d}}x,\,\,} \pi {\int {\left( {\frac{1}{{\sqrt {2x - 1} }}} \right)} ^2}{\text{d}}x,\,\,\left[ {\frac{1}{2}{\text{ln}}\left( {2x - 1} \right)} \right]_1^9">
  <msubsup>
    <mo>∫</mo>
    <mn>1</mn>
    <mn>9</mn>
  </msubsup>
  <mrow>
    <mrow>
      <msup>
        <mi>y</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
    <mo>,</mo>
    <mspace width="thinmathspace"></mspace>
    <mspace width="thinmathspace"></mspace>
  </mrow>
  <mi>π</mi>
  <mrow>
    <mo>∫</mo>
    <msup>
      <mrow>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mfrac>
              <mn>1</mn>
              <mrow>
                <msqrt>
                  <mn>2</mn>
                  <mi>x</mi>
                  <mo>−</mo>
                  <mn>1</mn>
                </msqrt>
              </mrow>
            </mfrac>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mn>2</mn>
    </msup>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
  <mo>,</mo>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <msubsup>
    <mrow>
      <mo>[</mo>
      <mrow>
        <mfrac>
          <mn>1</mn>
          <mn>2</mn>
        </mfrac>
        <mrow>
          <mtext>ln</mtext>
        </mrow>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mn>2</mn>
            <mi>x</mi>
            <mo>−</mo>
            <mn>1</mn>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>]</mo>
    </mrow>
    <mn>1</mn>
    <mn>9</mn>
  </msubsup>
</math></span></p>
<p>substituting limits into <strong>their</strong> integral and subtracting (in any order)&nbsp; &nbsp; &nbsp;<strong><em>(M1)</em></strong></p>
<p><em>eg</em>&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{2}\left( {{\text{ln}}\left( {17} \right) - {\text{ln}}\left( 1 \right)} \right),\,\,\pi \left( {0 - \frac{1}{2}{\text{ln}}\left( {2 \times 9 - 1} \right)} \right)">
  <mfrac>
    <mi>π</mi>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mn>17</mn>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mo>−</mo>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mrow>
        <mo>(</mo>
        <mn>1</mn>
        <mo>)</mo>
      </mrow>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>,</mo>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mi>π</mi>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>0</mn>
      <mo>−</mo>
      <mfrac>
        <mn>1</mn>
        <mn>2</mn>
      </mfrac>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mn>2</mn>
          <mo>×</mo>
          <mn>9</mn>
          <mo>−</mo>
          <mn>1</mn>
        </mrow>
        <mo>)</mo>
      </mrow>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span></p>
<p>correct working involving calculating a log value or using log law&nbsp; &nbsp; &nbsp;<em><strong>(A1)</strong></em></p>
<p><em>eg</em>&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( 1 \right) = 0,\,\,{\text{ln}}\left( {\frac{{17}}{1}} \right)">
  <mrow>
    <mtext>ln</mtext>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mn>1</mn>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>0</mn>
  <mo>,</mo>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>ln</mtext>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mrow>
          <mn>17</mn>
        </mrow>
        <mn>1</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{2}{\text{ln}}17\,\,\,\,\left( {{\text{accept }}\pi {\text{ln}}\sqrt {17} } \right)">
  <mfrac>
    <mi>π</mi>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <mtext>ln</mtext>
  </mrow>
  <mn>17</mn>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mrow>
        <mtext>accept&nbsp;</mtext>
      </mrow>
      <mi>π</mi>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <msqrt>
        <mn>17</mn>
      </msqrt>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>&nbsp;&nbsp; &nbsp;<em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> Full <em><strong>FT</strong></em> may be awarded as normal, from their incorrect answer in part (a), however, do not award the final two <em><strong>A</strong></em> marks unless they involve logarithms.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\left( {{x^3} + x} \right)^{\frac{3}{2}}}">
  <mi>y</mi>
  <mo>=</mo>
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>3</mn>
            </msup>
          </mrow>
          <mo>+</mo>
          <mi>x</mi>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mrow>
        <mfrac>
          <mn>3</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span>.</p>
</div>

<div class="specification">
<p>Consider the functions&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \sqrt {{x^3} + x} ">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <msqrt>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mi>x</mi>
  </msqrt>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 6 - 3{x^2}\sqrt {{x^3} + x} ">
  <mi>g</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>6</mn>
  <mo>−<!-- − --></mo>
  <mn>3</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <msqrt>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mi>x</mi>
  </msqrt>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span> ≥ 0.</p>
<p>The graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
  <mi>g</mi>
</math></span> are shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The shaded region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
  <mi>R</mi>
</math></span> is enclosed by the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
  <mi>g</mi>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
  <mi>y</mi>
</math></span>-axis and <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>Hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\left( {3{x^2} + 1} \right)\sqrt {{x^3} + x} } \,{\text{d}}x">
  <mo>∫</mo>
  <mrow>
    <mrow>
      <mo>(</mo>
      <mrow>
        <mn>3</mn>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mo>)</mo>
    </mrow>
    <msqrt>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>3</mn>
        </msup>
      </mrow>
      <mo>+</mo>
      <mi>x</mi>
    </msqrt>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
</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>Write down an expression for the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
  <mi>R</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
  <mi>R</mi>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>integrating by inspection from (a) or by substitution       <em><strong>(M1)</strong></em></p>
<p>eg   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}\int {\frac{3}{2}} \left( {3{x^2} + 1} \right)\sqrt {{x^3} + x} \,{\text{d}}x">
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mo>∫</mo>
  <mrow>
    <mfrac>
      <mn>3</mn>
      <mn>2</mn>
    </mfrac>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>3</mn>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
      <mo>+</mo>
      <mn>1</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
  <msqrt>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mi>x</mi>
  </msqrt>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^3} + x">
  <mi>u</mi>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>x</mi>
</math></span>, <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = 3{x^2} + 1">
  <mfrac>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>u</mi>
    </mrow>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>x</mi>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>3</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>1</mn>
</math></span></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{u^{\frac{1}{2}}}} ">
  <mo>∫</mo>
  <mrow>
    <mrow>
      <msup>
        <mi>u</mi>
        <mrow>
          <mfrac>
            <mn>1</mn>
            <mn>2</mn>
          </mfrac>
        </mrow>
      </msup>
    </mrow>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{u^{\frac{3}{2}}}}}{{1.5}}">
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mi>u</mi>
          <mrow>
            <mfrac>
              <mn>3</mn>
              <mn>2</mn>
            </mfrac>
          </mrow>
        </msup>
      </mrow>
    </mrow>
    <mrow>
      <mn>1.5</mn>
    </mrow>
  </mfrac>
</math></span></p>
<p>correct integrated expression in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span>       <em><strong>A2 N3</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}{\left( {{x^3} + x} \right)^{\frac{3}{2}}} + C">
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>3</mn>
            </msup>
          </mrow>
          <mo>+</mo>
          <mi>x</mi>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mrow>
        <mfrac>
          <mn>3</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>C</mi>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\left( {{x^3} + x} \right)}^{1.5}}}}{{1.5}} + C">
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mrow>
            <mrow>
              <mo>(</mo>
              <mrow>
                <mrow>
                  <msup>
                    <mi>x</mi>
                    <mn>3</mn>
                  </msup>
                </mrow>
                <mo>+</mo>
                <mi>x</mi>
              </mrow>
              <mo>)</mo>
            </mrow>
          </mrow>
          <mrow>
            <mn>1.5</mn>
          </mrow>
        </msup>
      </mrow>
    </mrow>
    <mrow>
      <mn>1.5</mn>
    </mrow>
  </mfrac>
  <mo>+</mo>
  <mi>C</mi>
</math></span></p>
<p><em><strong>[3 marks]</strong></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>integrating and subtracting functions (in any order)        <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {g - f} ">
  <mo>∫</mo>
  <mrow>
    <mi>g</mi>
    <mo>−</mo>
    <mi>f</mi>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f - \int g } ">
  <mo>∫</mo>
  <mrow>
    <mi>f</mi>
    <mo>−</mo>
    <mo>∫</mo>
    <mi>g</mi>
  </mrow>
</math></span></p>
<p>correct integral (including limits, accept absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{d}}x}">
  <mrow>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
  </mrow>
</math></span>)       <em><strong>A1 N2</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {\left( {g - f} \right)} \,{\text{d}}x">
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>1</mn>
  </msubsup>
  <mrow>
    <mrow>
      <mo>(</mo>
      <mrow>
        <mi>g</mi>
        <mo>−</mo>
        <mi>f</mi>
      </mrow>
      <mo>)</mo>
    </mrow>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {6 - 3{x^2}\sqrt {{x^3} + x}  - \sqrt {{x^3} + x} } \,{\text{d}}x">
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>1</mn>
  </msubsup>
  <mrow>
    <mn>6</mn>
    <mo>−</mo>
    <mn>3</mn>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <msqrt>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>3</mn>
        </msup>
      </mrow>
      <mo>+</mo>
      <mi>x</mi>
    </msqrt>
    <mo>−</mo>
    <msqrt>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>3</mn>
        </msup>
      </mrow>
      <mo>+</mo>
      <mi>x</mi>
    </msqrt>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {g\left( x \right) - } \int_0^1 {f\left( x \right)} ">
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>1</mn>
  </msubsup>
  <mrow>
    <mi>g</mi>
    <mrow>
      <mo>(</mo>
      <mi>x</mi>
      <mo>)</mo>
    </mrow>
    <mo>−</mo>
  </mrow>
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>1</mn>
  </msubsup>
  <mrow>
    <mi>f</mi>
    <mrow>
      <mo>(</mo>
      <mi>x</mi>
      <mo>)</mo>
    </mrow>
  </mrow>
</math></span></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>recognizing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{x^3} + x} ">
  <msqrt>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mi>x</mi>
  </msqrt>
</math></span> is a common factor (seen anywhere, may be seen in part (c))       <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 3{x^2} - 1} \right)\sqrt {{x^3} + x} ">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mo>−</mo>
      <mn>3</mn>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
      <mo>−</mo>
      <mn>1</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
  <msqrt>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mi>x</mi>
  </msqrt>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {6 - \left( {3{x^2} + 1} \right)} \sqrt {{x^3} + x} ">
  <mo>∫</mo>
  <mrow>
    <mn>6</mn>
    <mo>−</mo>
    <mrow>
      <mo>(</mo>
      <mrow>
        <mn>3</mn>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mo>)</mo>
    </mrow>
  </mrow>
  <msqrt>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mi>x</mi>
  </msqrt>
</math></span>,   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {3{x^2} - 1} \right)\sqrt {{x^3} + x} ">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>3</mn>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
      <mo>−</mo>
      <mn>1</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
  <msqrt>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mi>x</mi>
  </msqrt>
</math></span></p>
<p>correct integration      <em><strong>(A1)(A1)</strong></em></p>
<p>eg   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6x - \frac{2}{3}{\left( {{x^3} + x} \right)^{\frac{3}{2}}}">
  <mn>6</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>3</mn>
            </msup>
          </mrow>
          <mo>+</mo>
          <mi>x</mi>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mrow>
        <mfrac>
          <mn>3</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6x">
  <mn>6</mn>
  <mi>x</mi>
</math></span> and award <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{2}{3}{\left( {{x^3} + x} \right)^{\frac{3}{2}}}">
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>3</mn>
            </msup>
          </mrow>
          <mo>+</mo>
          <mi>x</mi>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mrow>
        <mfrac>
          <mn>3</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span>.</p>
<p>substituting limits into <strong>their</strong> integrated function and subtracting (in any order)       <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - \frac{2}{3}{\left( {{1^3} + 1} \right)^{\frac{3}{2}}}">
  <mn>6</mn>
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mrow>
            <msup>
              <mn>1</mn>
              <mn>3</mn>
            </msup>
          </mrow>
          <mo>+</mo>
          <mn>1</mn>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mrow>
        <mfrac>
          <mn>3</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 - \left[ {6 - \frac{2}{3}{{\left( {{1^3} + 1} \right)}^{\frac{3}{2}}}} \right]">
  <mn>0</mn>
  <mo>−</mo>
  <mrow>
    <mo>[</mo>
    <mrow>
      <mn>6</mn>
      <mo>−</mo>
      <mfrac>
        <mn>2</mn>
        <mn>3</mn>
      </mfrac>
      <mrow>
        <msup>
          <mrow>
            <mrow>
              <mo>(</mo>
              <mrow>
                <mrow>
                  <msup>
                    <mn>1</mn>
                    <mn>3</mn>
                  </msup>
                </mrow>
                <mo>+</mo>
                <mn>1</mn>
              </mrow>
              <mo>)</mo>
            </mrow>
          </mrow>
          <mrow>
            <mfrac>
              <mn>3</mn>
              <mn>2</mn>
            </mfrac>
          </mrow>
        </msup>
      </mrow>
    </mrow>
    <mo>]</mo>
  </mrow>
</math></span></p>
<p>correct working      <em><strong> (A1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - \frac{2}{3} \times 2\sqrt 2 ">
  <mn>6</mn>
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mo>×</mo>
  <mn>2</mn>
  <msqrt>
    <mn>2</mn>
  </msqrt>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - \frac{2}{3} \times \sqrt 4  \times \sqrt 2 ">
  <mn>6</mn>
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mo>×</mo>
  <msqrt>
    <mn>4</mn>
  </msqrt>
  <mo>×</mo>
  <msqrt>
    <mn>2</mn>
  </msqrt>
</math></span></p>
<p>area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R = 6 - \frac{{4\sqrt 2 }}{3}\,\,\,\left( { = 6 - \frac{2}{3}\sqrt 8 {\text{,}}\,\,\,6 - \frac{2}{3} \times {2^{\frac{3}{2}}}{\text{,}}\,\,\,\frac{{18 - 4\sqrt 2 }}{3}} \right)">
  <mi>R</mi>
  <mo>=</mo>
  <mn>6</mn>
  <mo>−</mo>
  <mfrac>
    <mrow>
      <mn>4</mn>
      <msqrt>
        <mn>2</mn>
      </msqrt>
    </mrow>
    <mn>3</mn>
  </mfrac>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mo>=</mo>
      <mn>6</mn>
      <mo>−</mo>
      <mfrac>
        <mn>2</mn>
        <mn>3</mn>
      </mfrac>
      <msqrt>
        <mn>8</mn>
      </msqrt>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>6</mn>
      <mo>−</mo>
      <mfrac>
        <mn>2</mn>
        <mn>3</mn>
      </mfrac>
      <mo>×</mo>
      <mrow>
        <msup>
          <mn>2</mn>
          <mrow>
            <mfrac>
              <mn>3</mn>
              <mn>2</mn>
            </mfrac>
          </mrow>
        </msup>
      </mrow>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mfrac>
        <mrow>
          <mn>18</mn>
          <mo>−</mo>
          <mn>4</mn>
          <msqrt>
            <mn>2</mn>
          </msqrt>
        </mrow>
        <mn>3</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>       <em><strong>A1  N3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {x^3} - 2{x^2} + ax + 6">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo>−<!-- − --></mo>
  <mn>2</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>a</mi>
  <mi>x</mi>
  <mo>+</mo>
  <mn>6</mn>
</math></span>. Part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> is 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 graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> crosses the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
  <mi>y</mi>
</math></span>-axis at the point P. The line <em>L</em> is tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> at P.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </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>Hence, find the equation of <em>L</em> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> has a local minimum at the point Q. The line <em>L</em> passes through Q.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f' = 3{x^2} - 4x + a"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mi>a</mi> </math></span> <em><strong>    A2 N2</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach      <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   <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></p>
<p>correct working      <em><strong>(A1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{\left( 0 \right)^2} - 4\left( 0 \right) + a"> <mn>3</mn> <mrow> <msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>+</mo> <mi>a</mi> </math></span>,  slope = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 0 \right) = a"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> </math></span></p>
<p>attempt to substitute gradient and coordinates into linear equation      <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 6 = a\left( {x - 0} \right)"> <mi>y</mi> <mo>−</mo> <mn>6</mn> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = a\left( {x - 6} \right)"> <mi>y</mi> <mo>−</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 = a\left( 0 \right) + c"> <mn>6</mn> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> </math></span>,  <em>L</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ax + 6"> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mn>6</mn> </math></span></p>
<p>correct equation      <em><strong>A1 N3</strong></em></p>
<p><em>eg</em>  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = ax + 6"> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mn>6</mn> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 6 = ax"> <mi>y</mi> <mo>−</mo> <mn>6</mn> <mo>=</mo> <mi>a</mi> <mi>x</mi> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 6 = a\left( {x - 0} \right)"> <mi>y</mi> <mo>−</mo> <mn>6</mn> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find intersection      <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = L"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>L</mi> </math></span></p>
<p>correct equation<em><strong>      (A1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} - 2{x^2} + ax + 6 = ax + 6"> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mn>6</mn> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mn>6</mn> </math></span></p>
<p>correct working<em><strong>      (A1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} - 2{x^2} = 0"> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}(x - 2) = 0"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </math></span> at Q      <em><strong>(A1)</strong></em></p>
<p> </p>
<p>valid approach to find minimum<em><strong>      (M1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <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>correct equation      <em><strong>(A1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{x^2} - 4x + a = 0"> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>substitution of <strong>their</strong> value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> at Q into <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> equation<em><strong>      (M1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{\left( 2 \right)^2} - 4\left( 2 \right) + a = 0"> <mn>3</mn> <mrow> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12 - 8 + a = 0"> <mn>12</mn> <mo>−</mo> <mn>8</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> = −4     <em><strong>A1 N0</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">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>
<br><hr><br><div class="specification">
<p>A closed cylindrical can with radius r centimetres and height h centimetres has a&nbsp;volume of 20<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
  <mi>π<!-- π --></mi>
</math></span> cm<sup>3</sup>.</p>
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"></p>
</div>

<div class="specification">
<p>The material for the base and top of the can costs 10 cents per cm<sup>2</sup> and the material for the curved side costs 8 cents per cm<sup>2</sup>. The total cost of the material, in cents, is <em>C</em>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>h</em> in terms of <em>r.</em></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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 20\pi {r^2} + \frac{{320\pi }}{r}">
  <mi>C</mi>
  <mo>=</mo>
  <mn>20</mn>
  <mi>π</mi>
  <mrow>
    <msup>
      <mi>r</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mfrac>
    <mrow>
      <mn>320</mn>
      <mi>π</mi>
    </mrow>
    <mi>r</mi>
  </mfrac>
</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>Given that there is a minimum value for <em>C</em>, find this minimum value in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
  <mi>π</mi>
</math></span>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>correct equation for volume      <strong><em>(A1)</em></strong><br><em>eg</em>  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi {r^2}h = 20\pi ">
  <mi>π</mi>
  <mrow>
    <msup>
      <mi>r</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mi>h</mi>
  <mo>=</mo>
  <mn>20</mn>
  <mi>π</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = \frac{{20}}{{{r^2}}}">
  <mi>h</mi>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>20</mn>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mi>r</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span>     <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find formula for cost of parts     <em><strong> (M1)</strong></em><br><em>eg </em> 10 × two circles, 8 × curved side</p>
<p>correct expression for cost of two circles in terms of <em>r</em> (seen anywhere)      <em><strong>A1</strong></em><br><em>eg  </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi {r^2} \times 10">
  <mn>2</mn>
  <mi>π</mi>
  <mrow>
    <msup>
      <mi>r</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>×</mo>
  <mn>10</mn>
</math></span></p>
<p>correct expression for cost of curved side (seen anywhere)      <em><strong>(A1)</strong></em><br><em>eg  </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi r \times h \times 8">
  <mn>2</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mo>×</mo>
  <mi>h</mi>
  <mo>×</mo>
  <mn>8</mn>
</math></span></p>
<p>correct expression for cost of curved side in terms of <em>r </em>     <em><strong>A1</strong></em><br>eg  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 \times 2\pi r \times \frac{{20}}{{{r^2}}},\,\,\frac{{320\pi }}{{{r^2}}}">
  <mn>8</mn>
  <mo>×</mo>
  <mn>2</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mo>×</mo>
  <mfrac>
    <mrow>
      <mn>20</mn>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mi>r</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
  <mo>,</mo>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mfrac>
    <mrow>
      <mn>320</mn>
      <mi>π</mi>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mi>r</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="C = 20\pi {r^2} + \frac{{320\pi }}{r}">
  <mi>C</mi>
  <mo>=</mo>
  <mn>20</mn>
  <mi>π</mi>
  <mrow>
    <msup>
      <mi>r</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mfrac>
    <mrow>
      <mn>320</mn>
      <mi>π</mi>
    </mrow>
    <mi>r</mi>
  </mfrac>
</math></span>      <em><strong>AG N0</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>recognize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C' = 0">
  <msup>
    <mi>C</mi>
    <mo>′</mo>
  </msup>
  <mo>=</mo>
  <mn>0</mn>
</math></span> at minimum       <em><strong>(R1)</strong></em><br>eg  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C' = 0,\,\,\frac{{{\text{d}}C}}{{{\text{d}}r}} = 0">
  <msup>
    <mi>C</mi>
    <mo>′</mo>
  </msup>
  <mo>=</mo>
  <mn>0</mn>
  <mo>,</mo>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mfrac>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>C</mi>
    </mrow>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>r</mi>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p>correct differentiation (may be seen in equation)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C' = 40\pi r - \frac{{320\pi }}{{{r^2}}}">
  <msup>
    <mi>C</mi>
    <mo>′</mo>
  </msup>
  <mo>=</mo>
  <mn>40</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mo>−</mo>
  <mfrac>
    <mrow>
      <mn>320</mn>
      <mi>π</mi>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mi>r</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span>        <em><strong>A1A1</strong></em></p>
<p>correct equation      <em><strong>A1</strong></em><br>eg  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="40\pi r - \frac{{320\pi }}{{{r^2}}} = 0,\,\,40\pi r\frac{{320\pi }}{{{r^2}}}">
  <mn>40</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mo>−</mo>
  <mfrac>
    <mrow>
      <mn>320</mn>
      <mi>π</mi>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mi>r</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
  <mo>,</mo>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mn>40</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mfrac>
    <mrow>
      <mn>320</mn>
      <mi>π</mi>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mi>r</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span></p>
<p>correct working     <em><strong>(A1)</strong></em><br>eg  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="40{r^3} = 320,\,\,{r^3} = 8">
  <mn>40</mn>
  <mrow>
    <msup>
      <mi>r</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>320</mn>
  <mo>,</mo>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <msup>
      <mi>r</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>8</mn>
</math></span></p>
<p><em>r</em> = 2 (m)     <em><strong>A1</strong></em></p>
<p>attempt to substitute <strong>their</strong> value of <em>r</em> into <em>C</em><br>eg  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="20\pi  \times 4 + 320 \times \frac{\pi }{2}">
  <mn>20</mn>
  <mi>π</mi>
  <mo>×</mo>
  <mn>4</mn>
  <mo>+</mo>
  <mn>320</mn>
  <mo>×</mo>
  <mfrac>
    <mi>π</mi>
    <mn>2</mn>
  </mfrac>
</math></span>     <em><strong>(M1)</strong></em></p>
<p>correct working<br>eg  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="80\pi  + 160\pi ">
  <mn>80</mn>
  <mi>π</mi>
  <mo>+</mo>
  <mn>160</mn>
  <mi>π</mi>
</math></span>      <em><strong>  (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="240\pi ">
  <mn>240</mn>
  <mi>π</mi>
</math></span> (cents)      <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> Do not accept 753.6, 753.98 or 754, even if 240<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
  <mi>π</mi>
</math></span> is seen.</p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \cos x">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>cos</mi>
  <mo>⁡<!-- ⁡ --></mo>
  <mi>x</mi>
</math></span>.</p>
</div>

<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = {x^k}">
  <mi>g</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mi>k</mi>
    </msup>
  </mrow>
</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>.</p>
</div>

<div class="specification">
<p>Let&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 21">
  <mi>k</mi>
  <mo>=</mo>
  <mn>21</mn>
</math></span> and&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h(x) = \left( {{f^{(19)}}(x) \times {g^{(19)}}(x)} \right)">
  <mi>h</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mrow>
        <msup>
          <mi>f</mi>
          <mrow>
            <mo stretchy="false">(</mo>
            <mn>19</mn>
            <mo stretchy="false">)</mo>
          </mrow>
        </msup>
      </mrow>
      <mo stretchy="false">(</mo>
      <mi>x</mi>
      <mo stretchy="false">)</mo>
      <mo>×<!-- × --></mo>
      <mrow>
        <msup>
          <mi>g</mi>
          <mrow>
            <mo stretchy="false">(</mo>
            <mn>19</mn>
            <mo stretchy="false">)</mo>
          </mrow>
        </msup>
      </mrow>
      <mo stretchy="false">(</mo>
      <mi>x</mi>
      <mo stretchy="false">)</mo>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i)     Find the first four derivatives of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
</math></span>.</p>
<p>(ii)     Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{(19)}}(x)">
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i)     Find the first three derivatives of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x)">
  <mi>g</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
</math></span>.</p>
<p>(ii)     Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{(19)}}(x) = \frac{{k!}}{{(k - p)!}}({x^{k - 19}})">
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mi>k</mi>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mo stretchy="false">(</mo>
      <mi>k</mi>
      <mo>−</mo>
      <mi>p</mi>
      <mo stretchy="false">)</mo>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mo stretchy="false">(</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mrow>
        <mi>k</mi>
        <mo>−</mo>
        <mn>19</mn>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">)</mo>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
  <mi>p</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i)     Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(x)">
  <msup>
    <mi>h</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
</math></span>.</p>
<p>(ii)     Hence, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}">
  <msup>
    <mi>h</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>π</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mo>−</mo>
      <mn>21</mn>
      <mo>!</mo>
    </mrow>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>π</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(i)     <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'(x) =  - \sin x,{\text{ }}f''(x) =  - \cos x,{\text{ }}{f^{(3)}}(x) = \sin x,{\text{ }}{f^{(4)}}(x) = \cos x">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mo>−</mo>
  <mi>sin</mi>
  <mo>⁡</mo>
  <mi>x</mi>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <msup>
    <mi>f</mi>
    <mo>″</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mo>−</mo>
  <mi>cos</mi>
  <mo>⁡</mo>
  <mi>x</mi>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>3</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>sin</mi>
  <mo>⁡</mo>
  <mi>x</mi>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>4</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>cos</mi>
  <mo>⁡</mo>
  <mi>x</mi>
</math></span>     <strong><em>A2     N2</em></strong></p>
<p>(ii)     valid approach     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span>recognizing that 19 is one less than a multiple of 4, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{(19)}}(x) = {f^{(3)}}(x)">
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>3</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{(19)}}(x) = \sin x">
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>sin</mi>
  <mo>⁡</mo>
  <mi>x</mi>
</math></span>     <strong><em>A1     N2</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>(i)     <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = k{x^{k - 1}}">
  <msup>
    <mi>g</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>k</mi>
  <mrow>
    <msup>
      <mi>x</mi>
      <mrow>
        <mi>k</mi>
        <mo>−</mo>
        <mn>1</mn>
      </mrow>
    </msup>
  </mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g''(x) = k(k - 1){x^{k - 2}},{\text{ }}{g^{(3)}}(x) = k(k - 1)(k - 2){x^{k - 3}}">
  <msup>
    <mi>g</mi>
    <mo>″</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>k</mi>
  <mo stretchy="false">(</mo>
  <mi>k</mi>
  <mo>−</mo>
  <mn>1</mn>
  <mo stretchy="false">)</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mrow>
        <mi>k</mi>
        <mo>−</mo>
        <mn>2</mn>
      </mrow>
    </msup>
  </mrow>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>3</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>k</mi>
  <mo stretchy="false">(</mo>
  <mi>k</mi>
  <mo>−</mo>
  <mn>1</mn>
  <mo stretchy="false">)</mo>
  <mo stretchy="false">(</mo>
  <mi>k</mi>
  <mo>−</mo>
  <mn>2</mn>
  <mo stretchy="false">)</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mrow>
        <mi>k</mi>
        <mo>−</mo>
        <mn>3</mn>
      </mrow>
    </msup>
  </mrow>
</math></span>     <strong><em>A1A1     N2</em></strong></p>
<p>(ii)     <strong>METHOD 1</strong></p>
<p>correct working that leads to the correct answer, involving the correct expression for the 19th derivative     <strong><em>A2</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k(k - 1)(k - 2) \ldots (k - 18) \times \frac{{(k - 19)!}}{{(k - 19)!}},{{\text{ }}_k}{P_{19}}">
  <mi>k</mi>
  <mo stretchy="false">(</mo>
  <mi>k</mi>
  <mo>−</mo>
  <mn>1</mn>
  <mo stretchy="false">)</mo>
  <mo stretchy="false">(</mo>
  <mi>k</mi>
  <mo>−</mo>
  <mn>2</mn>
  <mo stretchy="false">)</mo>
  <mo>…</mo>
  <mo stretchy="false">(</mo>
  <mi>k</mi>
  <mo>−</mo>
  <mn>18</mn>
  <mo stretchy="false">)</mo>
  <mo>×</mo>
  <mfrac>
    <mrow>
      <mo stretchy="false">(</mo>
      <mi>k</mi>
      <mo>−</mo>
      <mn>19</mn>
      <mo stretchy="false">)</mo>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mo stretchy="false">(</mo>
      <mi>k</mi>
      <mo>−</mo>
      <mn>19</mn>
      <mo stretchy="false">)</mo>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mo>,</mo>
  <mrow>
    <msub>
      <mrow>
        <mtext> </mtext>
      </mrow>
      <mi>k</mi>
    </msub>
  </mrow>
  <mrow>
    <msub>
      <mi>P</mi>
      <mrow>
        <mn>19</mn>
      </mrow>
    </msub>
  </mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 19">
  <mi>p</mi>
  <mo>=</mo>
  <mn>19</mn>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{k!}}{{(k - 19)!}}{x^{k - 19}}">
  <mfrac>
    <mrow>
      <mi>k</mi>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mo stretchy="false">(</mo>
      <mi>k</mi>
      <mo>−</mo>
      <mn>19</mn>
      <mo stretchy="false">)</mo>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mrow>
    <msup>
      <mi>x</mi>
      <mrow>
        <mi>k</mi>
        <mo>−</mo>
        <mn>19</mn>
      </mrow>
    </msup>
  </mrow>
</math></span>)     <strong><em>A1     N1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>correct working involving recognizing patterns in coefficients of first three derivatives (may be seen in part (b)(i)) leading to a general rule for 19th coefficient     <strong><em>A2</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'' = 2!\left( {\begin{array}{*{20}{c}} k \\ 2 \end{array}} \right),{\text{ }}k(k - 1)(k - 2) = \frac{{k!}}{{(k - 3)!}},{\text{ }}{g^{(3)}}(x){ = _k}{P_3}({x^{k - 3}})">
  <msup>
    <mi>g</mi>
    <mo>″</mo>
  </msup>
  <mo>=</mo>
  <mn>2</mn>
  <mo>!</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mtable rowspacing="4pt" columnspacing="1em">
        <mtr>
          <mtd>
            <mi>k</mi>
          </mtd>
        </mtr>
        <mtr>
          <mtd>
            <mn>2</mn>
          </mtd>
        </mtr>
      </mtable>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mi>k</mi>
  <mo stretchy="false">(</mo>
  <mi>k</mi>
  <mo>−</mo>
  <mn>1</mn>
  <mo stretchy="false">)</mo>
  <mo stretchy="false">(</mo>
  <mi>k</mi>
  <mo>−</mo>
  <mn>2</mn>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mi>k</mi>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mo stretchy="false">(</mo>
      <mi>k</mi>
      <mo>−</mo>
      <mn>3</mn>
      <mo stretchy="false">)</mo>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>3</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mrow>
    <msub>
      <mo>=</mo>
      <mi>k</mi>
    </msub>
  </mrow>
  <mrow>
    <msub>
      <mi>P</mi>
      <mn>3</mn>
    </msub>
  </mrow>
  <mo stretchy="false">(</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mrow>
        <mi>k</mi>
        <mo>−</mo>
        <mn>3</mn>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{(19)}}(x) = 19!\left( {\begin{array}{*{20}{c}} k \\ {19} \end{array}} \right),{\text{ }}19! \times \frac{{k!}}{{(k - 19)! \times 19!}},{{\text{ }}_k}{P_{19}}">
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mn>19</mn>
  <mo>!</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mtable rowspacing="4pt" columnspacing="1em">
        <mtr>
          <mtd>
            <mi>k</mi>
          </mtd>
        </mtr>
        <mtr>
          <mtd>
            <mrow>
              <mn>19</mn>
            </mrow>
          </mtd>
        </mtr>
      </mtable>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mn>19</mn>
  <mo>!</mo>
  <mo>×</mo>
  <mfrac>
    <mrow>
      <mi>k</mi>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mo stretchy="false">(</mo>
      <mi>k</mi>
      <mo>−</mo>
      <mn>19</mn>
      <mo stretchy="false">)</mo>
      <mo>!</mo>
      <mo>×</mo>
      <mn>19</mn>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mo>,</mo>
  <mrow>
    <msub>
      <mrow>
        <mtext> </mtext>
      </mrow>
      <mi>k</mi>
    </msub>
  </mrow>
  <mrow>
    <msub>
      <mi>P</mi>
      <mrow>
        <mn>19</mn>
      </mrow>
    </msub>
  </mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 19">
  <mi>p</mi>
  <mo>=</mo>
  <mn>19</mn>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{k!}}{{(k - 19)!}}{x^{k - 19}}">
  <mfrac>
    <mrow>
      <mi>k</mi>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mo stretchy="false">(</mo>
      <mi>k</mi>
      <mo>−</mo>
      <mn>19</mn>
      <mo stretchy="false">)</mo>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mrow>
    <msup>
      <mi>x</mi>
      <mrow>
        <mi>k</mi>
        <mo>−</mo>
        <mn>19</mn>
      </mrow>
    </msup>
  </mrow>
</math></span>)     <strong><em>A1     N1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i)     valid approach using product rule     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="uv' + vu',{\text{ }}{f^{(19)}}{g^{(20)}} + {f^{(20)}}{g^{(19)}}">
  <mi>u</mi>
  <msup>
    <mi>v</mi>
    <mo>′</mo>
  </msup>
  <mo>+</mo>
  <mi>v</mi>
  <msup>
    <mi>u</mi>
    <mo>′</mo>
  </msup>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>20</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>20</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
</math></span></p>
<p>correct 20th derivatives (must be seen in product rule)     <strong><em>(A1)(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{(20)}}(x) = \frac{{21!}}{{(21 - 20)!}}x,{\text{ }}{f^{(20)}}(x) = \cos x">
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>20</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>21</mn>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mo stretchy="false">(</mo>
      <mn>21</mn>
      <mo>−</mo>
      <mn>20</mn>
      <mo stretchy="false">)</mo>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mi>x</mi>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>20</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>cos</mi>
  <mo>⁡</mo>
  <mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(x) = \sin x(21!x) + \cos x\left( {\frac{{21!}}{2}{x^2}} \right){\text{ }}\left( {{\text{accept }}\sin x\left( {\frac{{21!}}{{1!}}x} \right) + \cos x\left( {\frac{{21!}}{{2!}}{x^2}} \right)} \right)">
  <msup>
    <mi>h</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>sin</mi>
  <mo>⁡</mo>
  <mi>x</mi>
  <mo stretchy="false">(</mo>
  <mn>21</mn>
  <mo>!</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>+</mo>
  <mi>cos</mi>
  <mo>⁡</mo>
  <mi>x</mi>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mrow>
          <mn>21</mn>
          <mo>!</mo>
        </mrow>
        <mn>2</mn>
      </mfrac>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mrow>
        <mtext>accept </mtext>
      </mrow>
      <mi>sin</mi>
      <mo>⁡</mo>
      <mi>x</mi>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mfrac>
            <mrow>
              <mn>21</mn>
              <mo>!</mo>
            </mrow>
            <mrow>
              <mn>1</mn>
              <mo>!</mo>
            </mrow>
          </mfrac>
          <mi>x</mi>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mo>+</mo>
      <mi>cos</mi>
      <mo>⁡</mo>
      <mi>x</mi>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mfrac>
            <mrow>
              <mn>21</mn>
              <mo>!</mo>
            </mrow>
            <mrow>
              <mn>2</mn>
              <mo>!</mo>
            </mrow>
          </mfrac>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>2</mn>
            </msup>
          </mrow>
        </mrow>
        <mo>)</mo>
      </mrow>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>    <strong><em>A1     N3</em></strong></p>
<p>(ii)     substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi ">
  <mi>x</mi>
  <mo>=</mo>
  <mi>π</mi>
</math></span> (seen anywhere)     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{(19)}}(\pi ){g^{(20)}}(\pi ) + {f^{(20)}}(\pi ){g^{(19)}}(\pi ),{\text{ }}\sin \pi \frac{{21!}}{{1!}}\pi  + \cos \pi \frac{{21!}}{{2!}}{\pi ^2}">
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>π</mi>
  <mo stretchy="false">)</mo>
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>20</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>π</mi>
  <mo stretchy="false">)</mo>
  <mo>+</mo>
  <mrow>
    <msup>
      <mi>f</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>20</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>π</mi>
  <mo stretchy="false">)</mo>
  <mrow>
    <msup>
      <mi>g</mi>
      <mrow>
        <mo stretchy="false">(</mo>
        <mn>19</mn>
        <mo stretchy="false">)</mo>
      </mrow>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mi>π</mi>
  <mo stretchy="false">)</mo>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mi>sin</mi>
  <mo>⁡</mo>
  <mi>π</mi>
  <mfrac>
    <mrow>
      <mn>21</mn>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mn>1</mn>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mi>π</mi>
  <mo>+</mo>
  <mi>cos</mi>
  <mo>⁡</mo>
  <mi>π</mi>
  <mfrac>
    <mrow>
      <mn>21</mn>
      <mo>!</mo>
    </mrow>
    <mrow>
      <mn>2</mn>
      <mo>!</mo>
    </mrow>
  </mfrac>
  <mrow>
    <msup>
      <mi>π</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span></p>
<p>evidence of one correct value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \pi ">
  <mi>sin</mi>
  <mo>⁡</mo>
  <mi>π</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \pi ">
  <mi>cos</mi>
  <mo>⁡</mo>
  <mi>π</mi>
</math></span> (seen anywhere)     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \pi  = 0,{\text{ }}\cos \pi  =  - 1">
  <mi>sin</mi>
  <mo>⁡</mo>
  <mi>π</mi>
  <mo>=</mo>
  <mn>0</mn>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mi>cos</mi>
  <mo>⁡</mo>
  <mi>π</mi>
  <mo>=</mo>
  <mo>−</mo>
  <mn>1</mn>
</math></span></p>
<p>evidence of correct values substituted into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'(\pi )">
  <msup>
    <mi>h</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>π</mi>
  <mo stretchy="false">)</mo>
</math></span>     <strong><em>A1</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="21!(\pi )\left( {0 - \frac{\pi }{{2!}}} \right),{\text{ }}21!(\pi )\left( { - \frac{\pi }{2}} \right),{\text{ }}0 + ( - 1)\frac{{21!}}{2}{\pi ^2}">
  <mn>21</mn>
  <mo>!</mo>
  <mo stretchy="false">(</mo>
  <mi>π</mi>
  <mo stretchy="false">)</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>0</mn>
      <mo>−</mo>
      <mfrac>
        <mi>π</mi>
        <mrow>
          <mn>2</mn>
          <mo>!</mo>
        </mrow>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mn>21</mn>
  <mo>!</mo>
  <mo stretchy="false">(</mo>
  <mi>π</mi>
  <mo stretchy="false">)</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mo>−</mo>
      <mfrac>
        <mi>π</mi>
        <mn>2</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mn>0</mn>
  <mo>+</mo>
  <mo stretchy="false">(</mo>
  <mo>−</mo>
  <mn>1</mn>
  <mo stretchy="false">)</mo>
  <mfrac>
    <mrow>
      <mn>21</mn>
      <mo>!</mo>
    </mrow>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>π</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span></p>
<p> </p>
<p><strong>Note: </strong>If candidates write only the first line followed by the answer, award <strong><em>A1A0A0</em></strong>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - 21!}}{2}{\pi ^2}">
  <mfrac>
    <mrow>
      <mo>−</mo>
      <mn>21</mn>
      <mo>!</mo>
    </mrow>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>π</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span>     <strong><em>AG     N0</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mn>5</mn><msup><mi>x</mi><mi>p</mi></msup></math>. Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</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>Hence, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>9</mn></msubsup><mfenced><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></mfenced><mo>d</mo><mi>x</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>=</mo><mn>3</mn><mo>-</mo><mn>5</mn><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math>       <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>10</mn><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math>             <em><strong>A1A1</strong></em></p>
<p>substituting limits into their integrated function and subtracting             <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mn>9</mn></mfenced><mo>-</mo><mn>10</mn><msup><mfenced><mn>9</mn></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>-</mo><mfenced><mrow><mn>3</mn><mfenced><mn>1</mn></mfenced><mo>-</mo><mn>10</mn><msup><mfenced><mn>1</mn></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></mfenced></math>  OR  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>-</mo><mn>10</mn><mo>×</mo><mn>3</mn><mo>-</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>10</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math>             <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many candidates could give the value of <em>p</em> correctly. However, many did struggle with the integration, including substituting limits into the integrand, without integrating at all. An incorrect value of <em>p</em> often resulted in arithmetic of greater complexity.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question, all lengths are in metres and time is in seconds.</strong></p>
<p>Consider two particles, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math>, which start to move at the same time.</p>
<p>Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> moves in a straight line such that its displacement from a fixed-point is given&nbsp;by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>10</mn><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup></math>, for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</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>Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> also moves in a straight line. The position of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math>.</p>
<p>The speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> is greater than the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>&gt;</mo><mi>q</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</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>recognizing velocity is derivative of displacement     <em><strong>(M1)</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mrow><mtext>d</mtext><mi>s</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo> </mo><mo>,</mo><mo> </mo><mfrac><mtext>d</mtext><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mfenced><mrow><mn>10</mn><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p>velocity<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>14</mn><mn>4</mn></mfrac><mi>t</mi><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced></math>        <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math>     <em><strong>(M1)</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mfenced><mo> </mo><mo>,</mo><mo> </mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math> , velocity<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p>correct speed     <em><strong>(A1)</strong></em></p>
<p><em>eg   </em><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p>recognizing relationship between speed and velocity (may be seen in inequality/equation)        <em><strong>R1</strong></em></p>
<p><em>eg</em>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced></math> , speed = | velocity | , graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> speed , <img src="data:image/png;base64,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"> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>P</mi><mn>2</mn></msub></math> velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>5</mn></math></p>
<p>correct inequality or equation that compares speed or velocity (accept any variable for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>)      <em><strong>A1</strong></em></p>
<p><em>eg</em>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced><mo>&gt;</mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo>&lt;</mo><mo>-</mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo>&gt;</mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo>=</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math> (seconds) (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>&gt;</mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math> , do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math>)       <em><strong>A1   N2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award the last two <em><strong>A1</strong></em> marks without the <em><strong>R1</strong></em>.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></math>, for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mo>&nbsp;</mo><mi>k</mi><mo>&gt;</mo><mn>0</mn></math>.</p>
<p>Let&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>p</mi><mo>,</mo><mo>&nbsp;</mo><mfrac><mi>k</mi><mi>p</mi></mfrac></mrow></mfenced></math>&nbsp;be any point on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>. Line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> is the tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>&nbsp;at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<p style="text-align: center;"><img 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"></p>
</div>

<div class="specification">
<p>Line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>2</mn><mi>p</mi><mo>,</mo><mo>&nbsp;</mo><mn>0</mn></mrow></mfenced></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis at point B.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>p</mi></mfenced></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>x</mi><mo>+</mo><msup><mi>p</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>=</mo><mn>0</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>Find the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AOB</mtext></math> in terms 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">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is translated by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> to give the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>.<br>In the following diagram:</p>
<ul>
<li>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> lies on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>
</li>
<li>points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext></math> lie on the vertical asymptote of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>
</li>
<li>points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math> lie on the horizontal asymptote of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>
</li>
<li>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>G</mtext></math> lies on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>FG</mtext></math> is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>DC</mtext></math>.</li>
</ul>
<p>Line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> is the tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math>, and passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math>.</p>
<p><img src="data:image/png;base64,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"></p>
<p>Given that triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>EDF</mtext></math> and rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CDFG</mtext></math> have equal areas, find the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>k</mi><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>       <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>p</mi></mfenced><mo>=</mo><mo>-</mo><mi>k</mi><msup><mi>p</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac></mrow></mfenced></math>    <em><strong> A1     N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use point and gradient to find equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math>        <em><strong>M1</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>=</mo><mo>-</mo><mi>k</mi><msup><mi>p</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>p</mi></mrow></mfenced><mo>,</mo><mo> </mo><mo> </mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mfenced><mi>p</mi></mfenced><mo>+</mo><mi>b</mi></math></p>
<p>correct working leading to answer       <em><strong> A1</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mi>k</mi><mi>p</mi><mo>=</mo><mo>-</mo><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mi>p</mi><mo>,</mo><mo> </mo><mo> </mo><mi>y</mi><mo>-</mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mi>x</mi><mo>+</mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>,</mo><mo> </mo><mo> </mo><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>x</mi><mo>+</mo><msup><mi>p</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>=</mo><mn>0</mn></math>    <em><strong> AG     N0</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 – area of a triangle</strong></p>
<p>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>       <em><strong>(M1)</strong></em></p>
<p>correct working to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of null<em><strong>       (A1)</strong></em></p>
<p><em>eg</em>   <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mn>2</mn><mi>p</mi><mi>k</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of null at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></math> (may be seen in area formula)       <em><strong> A1</strong></em></p>
<p>correct substitution to find area of triangle<em><strong>       (A1)</strong></em></p>
<p><em>eg</em>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mi>p</mi></mrow></mfenced><mfenced><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></mfenced><mo>,</mo><mo> </mo><mo> </mo><mi>p</mi><mo>×</mo><mfenced><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></mfenced></math></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AOB</mtext><mo>=</mo><mn>2</mn><mi>k</mi></math>    <em><strong> A1     N3</strong></em></p>
<p> </p>
<p><strong>METHOD 2 – integration</strong></p>
<p>recognizing to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>p</mi></math>       <em><strong>(M1)</strong></em></p>
<p><em>eg </em>  <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>p</mi></mrow></msubsup><msub><mi>L</mi><mrow><mn>1</mn><mo> </mo></mrow></msub><mo>d</mo><mi>x</mi><mo> </mo><mo>,</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>p</mi></mrow></msubsup><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></math></p>
<p>correct integration of <strong>both</strong> terms       <em><strong> A1</strong></em></p>
<p><em>eg </em>  <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi><mi>x</mi></mrow><mi>p</mi></mfrac><mo> </mo><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mi>k</mi><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac><mi>x</mi><mo>+</mo><mi>c</mi><mo> </mo><mo>,</mo><mo> </mo><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mi>k</mi><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac><mi>x</mi></mrow></mfenced><mn>0</mn><mrow><mn>2</mn><mi>p</mi></mrow></msubsup></math></p>
<p>substituting limits into <strong>their</strong> integrated function and subtracting (in either order)       <em><strong>(M1)</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mi>k</mi><msup><mfenced><mrow><mn>2</mn><mi>p</mi></mrow></mfenced><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>k</mi><mfenced><mrow><mn>2</mn><mi>p</mi></mrow></mfenced></mrow><mi>p</mi></mfrac><mo>-</mo><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mrow><mn>4</mn><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>k</mi><mi>p</mi></mrow><mi>p</mi></mfrac></math></p>
<p>correct working<em><strong>       (A1)</strong></em></p>
<p><em>eg </em>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>4</mn><mi>k</mi></math></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AOB</mtext><mo>=</mo><mn>2</mn><mi>k</mi></math>    <em><strong> A1     N3</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In this question, the second <em><strong>M</strong></em> mark may be awarded independently of the other marks, so it is possible to award <em><strong>(M0)(A0)M1(A0)(A0)A0</strong></em>.</p>
<p> </p>
<p>recognizing use of transformation      <em><strong>(M1)</strong></em></p>
<p><em>eg</em>   area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AOB</mtext></math> = area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>DEF</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mi>k</mi><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>3</mn><mo>,</mo></math> gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub><mo>=</mo></math> gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced><mtext>, 2p+4, </mtext></math> one correct shift</p>
<p>correct working<em><strong>       (A1)</strong></em></p>
<p><em>eg</em>   area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>DEF</mtext><mo>=</mo><mn>2</mn><mi>k</mi><mo>,</mo><mo> </mo><mtext>CD</mtext><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mtext>DF</mtext><mo>=</mo><mn>2</mn><mi>p</mi><mo>,</mo><mo> </mo><mtext>CG</mtext><mo>=</mo><mn>2</mn><mi>p</mi><mo>,</mo><mo> </mo><mtext>E</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac><mo>+</mo><mn>3</mn></mrow></mfenced><mo>,</mo><mo> </mo><mtext>F</mtext><mfenced><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced><mo>,</mo><mo> </mo><mtext>Q</mtext><mfenced><mrow><mi>p</mi><mo>+</mo><mn>4</mn><mo>,</mo><mo> </mo><mfrac><mi>k</mi><mi>p</mi></mfrac><mo>+</mo><mn>3</mn></mrow></mfenced><mo>,</mo></math> </p>
<p>gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mi>p</mi><mn>2</mn></msup></mfrac><mo>,</mo><mo> </mo><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mo>,</mo></math> area of rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CDFG</mtext><mo>=</mo><mn>2</mn><mi>k</mi></math></p>
<p>valid approach      <em><strong>(M1)</strong></em></p>
<p><em>eg   </em><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>ED</mtext><mo>×</mo><mtext>DF</mtext></mrow><mn>2</mn></mfrac><mo>=</mo><mtext>CD</mtext><mo>×</mo><mtext>DF</mtext><mo>,</mo><mo> </mo><mn>2</mn><mi>p</mi><mo>·</mo><mn>3</mn><mo>=</mo><mn>2</mn><mi>k</mi><mo> </mo><mo>,</mo><mo> </mo><mtext>ED</mtext><mo>=</mo><mn>2</mn><mtext>CD</mtext><mo> </mo><mo>,</mo><mo> </mo><msubsup><mo>∫</mo><mn>4</mn><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></msubsup><msub><mi>L</mi><mn>2</mn></msub><mo> </mo><mtext>d</mtext><mi>x</mi><mo>=</mo><mn>4</mn><mi>k</mi></math></p>
<p>correct working<em>      <strong>(A1)</strong></em></p>
<p><em>eg</em>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ED</mtext><mo>=</mo><mn>6</mn><mo>,</mo><mo> </mo><mtext>E</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>9</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>k</mi><mo>=</mo><mn>3</mn><mi>p</mi><mo>,</mo><mo> </mo><mtext>gradient</mtext><mo>=</mo><mfrac><mrow><mn>3</mn><mo>-</mo><mfenced><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mi>p</mi></mfrac></mstyle><mo>+</mo><mn>3</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>-</mo><mn>4</mn></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mo>-</mo><mn>6</mn></mrow><mfenced><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mn>3</mn></mfrac></mstyle></mfenced></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>9</mn><mi>k</mi></mfrac></math></p>
<p>correct expression for gradient (in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>)<em><strong>       (A1)</strong></em></p>
<p><em>eg</em>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>6</mn></mrow><mrow><mn>2</mn><mi>p</mi></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>9</mn><mo>-</mo><mn>3</mn></mrow><mrow><mn>4</mn><mo>-</mo><mfenced><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></mfenced></mrow></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mrow><mn>3</mn><mi>p</mi></mrow><msup><mi>p</mi><mn>2</mn></msup></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mo>-</mo><mfenced><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mfenced><mrow><mn>3</mn><mi>p</mi></mrow></mfenced></mrow><mi>p</mi></mfrac></mstyle><mo>+</mo><mn>3</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mstyle displaystyle="true"><mo>-</mo></mstyle><mstyle displaystyle="true"><mn>4</mn></mstyle></mrow></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>9</mn><mrow><mn>3</mn><mi>p</mi></mrow></mfrac></math></p>
<p>gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>3</mn><mi>p</mi></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>3</mn><msup><mi>p</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math>    <em><strong> A1     N3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math> and rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ORST</mi></math>. The rectangle has a vertex at the origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">O</mi></math>, a vertex on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mi>y</mi></mrow></mfenced></math>, a vertex on 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"><mi mathvariant="normal">T</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> and a vertex at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi></mrow></mfenced></math> on the graph.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> represent the perimeter of rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ORST</mi></math>.</p>
</div>

<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> represent the area of rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ORST</mi></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>8</mn></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>Find the dimensions of rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ORST</mi></math> that has maximum perimeter and determine the value of the maximum perimeter.</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 an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></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>Find the dimensions of rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ORST</mi></math> that has maximum area.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the maximum area of rectangle&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ORST</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(A1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mfenced><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>so&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>8</mn></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>AG</strong></p>
<p>&nbsp;</p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><strong>EITHER</strong></p>
<p>uses the axis of symmetry of a quadratic&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mrow><mn>2</mn><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></math></p>
<p>&nbsp;</p>
<p><strong>OR</strong></p>
<p>forms&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></p>
<p>&nbsp;</p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>substitutes their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> into&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>15</mn><mn>4</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>so the dimensions of rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ORST</mi></math> of maximum perimeter are&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>&nbsp;by&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>15</mn><mn>4</mn></mfrac></math></p>
<p>&nbsp;</p>
<p><strong>EITHER</strong></p>
<p>substitutes their value of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>&nbsp;into&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>8</mn></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>8</mn></math></p>
<p>&nbsp;</p>
<p><strong>OR</strong></p>
<p>substitutes their values of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>&nbsp;into&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>2</mn><mfenced><mfrac><mn>15</mn><mn>4</mn></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mn>17</mn><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>so the maximum perimeter is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>17</mn><mn>2</mn></mfrac></math></p>
<p>&nbsp;</p>
<p><strong>METHOD 2</strong></p>
<p>attempts to complete the square&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mfrac><mn>17</mn><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>substitutes their value of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>&nbsp;into&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>15</mn><mn>4</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>so the dimensions of rectangle&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ORST</mi></math>&nbsp;of maximum perimeter are&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>&nbsp;by&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>15</mn><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mn>17</mn><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>so the maximum perimeter is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>17</mn><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</p>
<p>&nbsp;</p>
<p><strong>[6 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></math>&nbsp;into&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>x</mi><mi>y</mi></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>x</mi><mfenced><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mo>&nbsp;</mo><mo>&nbsp;</mo><mfenced><mrow><mo>=</mo><mn>4</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>&nbsp;</p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>A</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>attempts to solve their&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>A</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>&nbsp;for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><msqrt><mn>3</mn></msqrt></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mn>3</mn></mfrac></mrow></mfenced><mo>&nbsp;</mo><mfenced><mrow><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>substitutes their (positive) value of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>&nbsp;into&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><msup><mfenced><mfrac><mn>2</mn><msqrt><mn>3</mn></msqrt></mfrac></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>8</mn><mn>3</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>&nbsp;</p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>16</mn><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>16</mn><msqrt><mn>3</mn></msqrt></mrow><mn>9</mn></mfrac></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>A1</strong></p>
<p>&nbsp;</p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>12</mn><mo>-</mo><mn>2</mn><mi>x</mi></msqrt><mo>,</mo><mo>&nbsp;</mo><mi>x</mi><mo>≤</mo><mi>a</mi></math>.&nbsp;The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<p>The shaded region is enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></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"><mfenced><mrow><mi>a</mi><mo>,</mo><mo>&nbsp;</mo><mn>0</mn></mrow></mfenced></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the volume of the solid formed when the shaded region is revolved <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</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>recognize <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math>       <em><strong>(M1)</strong></em></p>
<p><em>eg </em>  <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>12</mn><mo>-</mo><mn>2</mn><mi>x</mi></msqrt><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>12</mn></math> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>6</mn></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>6</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>6</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>)    <em><strong>A1  N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either <strong>their</strong> limits or the function into volume formula (must involve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><msup><mo> </mo><mn>2</mn></msup></math>)      <em><strong>(M1)</strong></em></p>
<p><em>eg </em>  <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>6</mn></msubsup><mi>f</mi><msup><mo> </mo><mn>2</mn></msup><mo>d</mo><mi>x</mi><mo> </mo><mo>,</mo><mo> </mo><mi mathvariant="normal">π</mi><mo>∫</mo><msup><mfenced><msqrt><mn>12</mn><mo>-</mo><mn>2</mn><mi>x</mi></msqrt></mfenced><mn>2</mn></msup><mo> </mo><mo>,</mo><mo> </mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>0</mn><mn>6</mn></msubsup><mn>12</mn><mo>-</mo><mn>2</mn><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> </p>
<p>correct integration of each term      <em><strong>A1  A1</strong></em></p>
<p><em>eg </em>  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>,</mo><mo> </mo><mn>12</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo> </mo><mo>,</mo><mo> </mo><msubsup><mfenced open="[" close="]"><mrow><mn>12</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>0</mn><mn>6</mn></msubsup></math></p>
<p>substituting limits into <strong>their integrated</strong> function and subtracting (in any order)      <em><strong>(M1)</strong></em></p>
<p><em>eg </em>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mn>12</mn><mfenced><mn>6</mn></mfenced><mo>-</mo><msup><mfenced><mn>6</mn></mfenced><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mi mathvariant="normal">π</mi><mfenced><mn>0</mn></mfenced><mo> </mo><mo>,</mo><mo> </mo><mn>72</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>36</mn><mi mathvariant="normal">π</mi><mo> </mo><mo>,</mo><mo> </mo><mfenced><mrow><mn>12</mn><mfenced><mn>6</mn></mfenced><mo>-</mo><msup><mfenced><mn>6</mn></mfenced><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mfenced><mn>0</mn></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> if candidate has substituted into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>,</mo><mo> </mo><mi>f</mi><msup><mo> </mo><mn>2</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math>.</p>
<p> </p>
<p>volume<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>36</mn><mi mathvariant="normal">π</mi></math>      <em><strong>A1  N2</strong></em>      </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The graph of a function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>ln</mi><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mn>20</mn></mrow></mfenced></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></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>evidence of integration      <em><strong>(M1)</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo> </mo><mtext>d</mtext><mi>x</mi><mo> </mo><mo>,</mo><mo> </mo><mo>∫</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup></math></p>
<p>correct integration (accept missing <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>c</mi></math>)      <em><strong>(A1)</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo> </mo><mo>,</mo><mo> </mo><mn>3</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><mi>c</mi></math></p>
<p>substituting initial condition into <strong>their</strong> integrated expression (must have <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>c</mi></math>)       <em><strong>M1</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mo>×</mo><mi>ln</mi><mo> </mo><mn>4</mn></mrow></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>20</mn></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> if candidate has substituted into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo></math>.</p>
<p> </p>
<p>correct application of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><msup><mi>a</mi><mi>b</mi></msup></mfenced><mo>=</mo><mi>b</mi><mo> </mo><mi>log</mi><mo> </mo><mi>a</mi></math> rule (seen anywhere)      <em><strong>(A1)</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>4</mn><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn><mo> </mo><mo>,</mo><mo> </mo><msup><mtext>e</mtext><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></msup><mo> </mo><mo>,</mo><mo> </mo><mi>ln</mi><mo> </mo><msup><mn>4</mn><mn>2</mn></msup></math></p>
<p>correct application of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>ln</mi><mo> </mo><mi>a</mi></mrow></msup><mtext>=</mtext><mtext mathvariant="italic">a</mtext></math> rule (seen anywhere)      <em><strong>(A1)</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></msup><mo>=</mo><mn>16</mn><mo> </mo><mo>,</mo><mo> </mo><msup><mfenced><msup><mtext>e</mtext><mrow><mi>ln</mi><mo> </mo><mn>4</mn></mrow></msup></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>4</mn><mn>2</mn></msup></math></p>
<p>correct working      <em><strong>(A1)</strong></em></p>
<p><em>eg</em>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>16</mn><mo>+</mo><mi>c</mi><mo>=</mo><mn>20</mn><mo> </mo><mo>,</mo><mo> </mo><mn>3</mn><mo>×</mo><msup><mfenced><mn>4</mn></mfenced><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>20</mn><mo> </mo><mo>,</mo><mo> </mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>28</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>-</mo><mn>28</mn></math>        <em><strong>A1 N4</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the functions&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>k</mi></math>&nbsp;where&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>

<div class="specification">
<p>The graphs of <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> have a common tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>.</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>Show that&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><mrow><mtext>e</mtext><mo>+</mo><mn>6</mn></mrow><mn>2</mn></mfrac></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>Hence, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mtext>e</mtext><mo>+</mo><mfrac><msup><mtext>e</mtext><mn>2</mn></msup><mn>4</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>&nbsp;</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></msup></math>&nbsp; OR&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mn>3</mn></mfenced><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mn>3</mn><mo>-</mo><mn>2</mn></mrow></msup></math>&nbsp;(may be seen anywhere)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>&nbsp;</p>
<p><strong>Note:</strong> The derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> must be explicitly seen, either in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
<p>&nbsp;</p>
<p>recognizing&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mn>3</mn></mfenced><mo>=</mo><mi>g</mi><mo>'</mo><mfenced><mn>3</mn></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mo>-</mo><mi>h</mi></mrow></mfenced><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mn>3</mn><mo>-</mo><mn>2</mn></mrow></msup><mo>&nbsp;</mo><mo>&nbsp;</mo><mfenced><mrow><mo>=</mo><mi>e</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>+</mo><mn>2</mn><mi>h</mi><mo>=</mo><mtext>e</mtext></math>&nbsp; OR&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mi>h</mi><mo>=</mo><mo>-</mo><mfrac><mtext>e</mtext><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>&nbsp;</p>
<p><strong>Note:</strong> The final<em><strong> A1</strong></em> is dependent on one of the previous marks being awarded.</p>
<p>&nbsp;</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><mrow><mtext>e</mtext><mo>+</mo><mn>6</mn></mrow><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;AG</strong></em></p>
<p>&nbsp;</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>3</mn></mfenced><mo>=</mo><mi>g</mi><mfenced><mn>3</mn></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mfenced><mrow><mn>3</mn><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mn>3</mn><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>k</mi></math></p>
<p>correct equation in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></p>
<p>&nbsp;</p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mfenced><mrow><mn>3</mn><mo>-</mo><mfrac><mrow><mtext>e</mtext><mo>+</mo><mn>6</mn></mrow><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mn>3</mn><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>k</mi></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mtext>e</mtext><mo>+</mo><msup><mfenced><mfrac><mrow><mn>6</mn><mo>-</mo><mtext>e</mtext><mo>-</mo><mn>6</mn></mrow><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>&nbsp;</mo><mfenced><mrow><mo>=</mo><mtext>e</mtext><mo>+</mo><msup><mfenced><mfrac><mrow><mo>-</mo><mtext>e</mtext></mrow><mn>2</mn></mfrac></mfenced><mn>2</mn></msup></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>&nbsp;</p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mtext>e</mtext><mo>+</mo><msup><mfenced><mrow><mn>3</mn><mo>-</mo><mfrac><mrow><mtext>e</mtext><mo>+</mo><mn>6</mn></mrow><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mtext>e</mtext><mo>+</mo><mn>9</mn><mo>-</mo><mn>3</mn><mtext>e</mtext><mo>-</mo><mn>18</mn><mo>+</mo><mfrac><mrow><msup><mtext>e</mtext><mn>2</mn></msup><mo>+</mo><mn>12</mn><mtext>e</mtext><mo>+</mo><mn>36</mn></mrow><mn>4</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>&nbsp;</p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mtext>e</mtext><mo>+</mo><mfrac><msup><mtext>e</mtext><mn>2</mn></msup><mn>4</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>&nbsp;AG</strong></em></p>
<p>&nbsp;</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><msup><mi>x</mi><mn>4</mn></msup></mfrac></math>&nbsp;for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>0</mn></math>.</p>
</div>

<div class="specification">
<p>Consider the function defined by&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><msup><mi>x</mi><mn>4</mn></msup></mfrac></math>&nbsp;for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>0</mn></math>&nbsp;and its graph&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that&nbsp;<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>1</mn><mo>-</mo><mn>4</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow><msup><mi>x</mi><mn>5</mn></msup></mfrac></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>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a horizontal tangent at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>. Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>20</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>9</mn></mrow><msup><mi>x</mi><mn>6</mn></msup></mfrac></math>,&nbsp;show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is a local maximum point.</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>Solve&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>&gt;</mo><mn>0</mn></math>&nbsp;for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</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>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, showing clearly the value of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept and the&nbsp;approximate position of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient or product rule<em><strong>&nbsp; &nbsp; &nbsp; &nbsp; (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><mfrac><mrow><msup><mi>x</mi><mn>4</mn></msup><mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><mi>x</mi></mfrac></mstyle></mfenced><mo>-</mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></mrow><msup><mfenced><msup><mi>x</mi><mn>4</mn></msup></mfenced><mn>2</mn></msup></mfrac></math>&nbsp; OR&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>4</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>5</mn></mrow></msup></mrow></mfenced><mo>+</mo><mfenced><msup><mi>x</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup></mfenced><mfenced><mfrac><mn>1</mn><mi>x</mi></mfrac></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>correct working&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mn>4</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow><msup><mi>x</mi><mn>8</mn></msup></mfrac></math>&nbsp; OR&nbsp;&nbsp;cancelling&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math>&nbsp; OR&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>4</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow><msup><mi>x</mi><mn>5</mn></msup></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>5</mn></msup></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mn>4</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow><msup><mi>x</mi><mn>5</mn></msup></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;AG</strong></em></p>
<p>&nbsp;</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"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><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>&nbsp; &nbsp; &nbsp; &nbsp; (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>1</mn><mo>-</mo><mn>4</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow><msup><mi>x</mi><mn>5</mn></msup></mfrac><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math><em><strong>&nbsp; &nbsp; &nbsp; &nbsp; (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mtext>e</mtext><mfrac><mn>1</mn><mn>4</mn></mfrac></msup></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>substitution of their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math><em><strong>&nbsp; &nbsp; &nbsp; &nbsp; (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></msup></mrow><msup><mfenced><msup><mtext>e</mtext><mfrac><mn>1</mn><mn>4</mn></mfrac></msup></mfenced><mn>4</mn></msup></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mtext>e</mtext></mrow></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msup><mtext>e</mtext><mfrac><mn>1</mn><mn>4</mn></mfrac></msup><mo>,</mo><mo> </mo><mfrac><mn>1</mn><mrow><mn>4</mn><mtext>e</mtext></mrow></mfrac></mrow></mfenced></math></p>
<p>&nbsp;</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><msup><mtext>e</mtext><mfrac><mn>1</mn><mn>4</mn></mfrac></msup></mfenced><mo>=</mo><mfrac><mrow><mn>20</mn><mo> </mo><mi>ln</mi><mo> </mo><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></msup><mo>-</mo><mn>9</mn></mrow><msup><mfenced><msup><mtext>e</mtext><mfrac><mn>1</mn><mn>4</mn></mfrac></msup></mfenced><mn>6</mn></msup></mfrac></math><em><strong>&nbsp; &nbsp; &nbsp; &nbsp; (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>5</mn><mo>-</mo><mn>9</mn></mrow><msup><mtext>e</mtext><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></msup></mfrac><mo>&nbsp;</mo><mo>&nbsp;</mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>4</mn><msup><mtext>e</mtext><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></msup></mfrac></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>which is negative&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;R1</strong></em></p>
<p>hence&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>&nbsp;is a local maximum&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;AG</strong></em></p>
<p>&nbsp;</p>
<p><strong>Note:</strong> The <em><strong>R1</strong></em> is dependent on the previous <em><strong>A1</strong></em> being awarded.</p>
<p>&nbsp;</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>&gt;</mo><mn>0</mn></math><em><strong>&nbsp; &nbsp; &nbsp; &nbsp; (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>1</mn></math>&nbsp;&nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;A1</strong></em></p>
<p>&nbsp;</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 style="padding-left:90px;"><img src="data:image/png;base64,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">&nbsp;&nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><strong>Note:&nbsp;</strong>Award <em><strong>A1</strong></em> for one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept only, located at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></p>
<p><em><strong>&nbsp; &nbsp; &nbsp;A1</strong></em> for local maximum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, in approximately correct position<br><em><strong>&nbsp; &nbsp; &nbsp;A1</strong></em> for curve approaching <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>∞</mo></math> (including change in concavity).</p>
<p>&nbsp;</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = \frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>3</mn>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mrow>
            <mo stretchy="false">(</mo>
            <mrow>
              <msup>
                <mi>x</mi>
                <mn>3</mn>
              </msup>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
            <mo stretchy="false">)</mo>
          </mrow>
          <mn>5</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(0) = 1">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mn>0</mn>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mn>1</mn>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</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>valid approach     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f'{\text{d}}x,{\text{ }}\int {\frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}{\text{d}}x} } ">
  <mo>∫</mo>
  <mrow>
    <msup>
      <mi>f</mi>
      <mo>′</mo>
    </msup>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
    <mo>,</mo>
    <mrow>
      <mtext> </mtext>
    </mrow>
    <mo>∫</mo>
    <mrow>
      <mfrac>
        <mrow>
          <mn>3</mn>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>2</mn>
            </msup>
          </mrow>
        </mrow>
        <mrow>
          <mrow>
            <msup>
              <mrow>
                <mo stretchy="false">(</mo>
                <mrow>
                  <msup>
                    <mi>x</mi>
                    <mn>3</mn>
                  </msup>
                </mrow>
                <mo>+</mo>
                <mn>1</mn>
                <mo stretchy="false">)</mo>
              </mrow>
              <mn>5</mn>
            </msup>
          </mrow>
        </mrow>
      </mfrac>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>x</mi>
    </mrow>
  </mrow>
</math></span></p>
<p>correct integration by substitution/inspection     <strong><em>A2</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) =  - \frac{1}{4}{({x^3} + 1)^{ - 4}} + c,{\text{ }}\frac{{ - 1}}{{4{{({x^3} + 1)}^4}}}">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>1</mn>
    <mn>4</mn>
  </mfrac>
  <mrow>
    <mo stretchy="false">(</mo>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mn>1</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mrow>
        <mo>−</mo>
        <mn>4</mn>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>c</mi>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mfrac>
    <mrow>
      <mo>−</mo>
      <mn>1</mn>
    </mrow>
    <mrow>
      <mn>4</mn>
      <mrow>
        <msup>
          <mrow>
            <mo stretchy="false">(</mo>
            <mrow>
              <msup>
                <mi>x</mi>
                <mn>3</mn>
              </msup>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
            <mo stretchy="false">)</mo>
          </mrow>
          <mn>4</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span></p>
<p>correct substitution into <strong>their </strong>integrated function (must include <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
  <mi>c</mi>
</math></span>)     <strong><em>M1</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = \frac{{ - 1}}{{4{{({0^3} + 1)}^4}}} + c,{\text{ }} - \frac{1}{4} + c = 1">
  <mn>1</mn>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mo>−</mo>
      <mn>1</mn>
    </mrow>
    <mrow>
      <mn>4</mn>
      <mrow>
        <msup>
          <mrow>
            <mo stretchy="false">(</mo>
            <mrow>
              <msup>
                <mn>0</mn>
                <mn>3</mn>
              </msup>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
            <mo stretchy="false">)</mo>
          </mrow>
          <mn>4</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
  <mo>+</mo>
  <mi>c</mi>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mo>−</mo>
  <mfrac>
    <mn>1</mn>
    <mn>4</mn>
  </mfrac>
  <mo>+</mo>
  <mi>c</mi>
  <mo>=</mo>
  <mn>1</mn>
</math></span></p>
<p> </p>
<p><strong>Note:</strong>     Award <strong><em>M0 </em></strong>if candidates substitute into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’’">
  <msup>
    <mi>f</mi>
    <mo>″</mo>
  </msup>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = \frac{5}{4}">
  <mi>c</mi>
  <mo>=</mo>
  <mfrac>
    <mn>5</mn>
    <mn>4</mn>
  </mfrac>
</math></span>     <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) =  - \frac{1}{4}{({x^3} + 1)^{ - 4}} + \frac{5}{4}{\text{ }}\left( { = \frac{{ - 1}}{{4{{({x^3} + 1)}^4}}} + \frac{5}{4},{\text{ }}\frac{{5{{({x^3} + 1)}^4} - 1}}{{4{{({x^3} + 1)}^4}}}} \right)">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>1</mn>
    <mn>4</mn>
  </mfrac>
  <mrow>
    <mo stretchy="false">(</mo>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>3</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mn>1</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mrow>
        <mo>−</mo>
        <mn>4</mn>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mfrac>
    <mn>5</mn>
    <mn>4</mn>
  </mfrac>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mo>=</mo>
      <mfrac>
        <mrow>
          <mo>−</mo>
          <mn>1</mn>
        </mrow>
        <mrow>
          <mn>4</mn>
          <mrow>
            <msup>
              <mrow>
                <mo stretchy="false">(</mo>
                <mrow>
                  <msup>
                    <mi>x</mi>
                    <mn>3</mn>
                  </msup>
                </mrow>
                <mo>+</mo>
                <mn>1</mn>
                <mo stretchy="false">)</mo>
              </mrow>
              <mn>4</mn>
            </msup>
          </mrow>
        </mrow>
      </mfrac>
      <mo>+</mo>
      <mfrac>
        <mn>5</mn>
        <mn>4</mn>
      </mfrac>
      <mo>,</mo>
      <mrow>
        <mtext> </mtext>
      </mrow>
      <mfrac>
        <mrow>
          <mn>5</mn>
          <mrow>
            <msup>
              <mrow>
                <mo stretchy="false">(</mo>
                <mrow>
                  <msup>
                    <mi>x</mi>
                    <mn>3</mn>
                  </msup>
                </mrow>
                <mo>+</mo>
                <mn>1</mn>
                <mo stretchy="false">)</mo>
              </mrow>
              <mn>4</mn>
            </msup>
          </mrow>
          <mo>−</mo>
          <mn>1</mn>
        </mrow>
        <mrow>
          <mn>4</mn>
          <mrow>
            <msup>
              <mrow>
                <mo stretchy="false">(</mo>
                <mrow>
                  <msup>
                    <mi>x</mi>
                    <mn>3</mn>
                  </msup>
                </mrow>
                <mo>+</mo>
                <mn>1</mn>
                <mo stretchy="false">)</mo>
              </mrow>
              <mn>4</mn>
            </msup>
          </mrow>
        </mrow>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>     <strong><em>A1</em></strong>     <strong><em>N4</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A particle P starts from point O and moves along a straight line. The graph of its velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
  <mi>v</mi>
</math></span> ms<sup>−1</sup>&nbsp;after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
  <mi>t</mi>
</math></span> seconds, for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
  <mi>t</mi>
</math></span> ≤ 6 , is shown in the following diagram.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
  <mi>v</mi>
</math></span> has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
  <mi>t</mi>
</math></span>-intercepts when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
  <mi>t</mi>
</math></span> = 0, 2 and 4.</p>
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( t \right)">
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mi>t</mi>
    <mo>)</mo>
  </mrow>
</math></span> represents the displacement of P from O after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
  <mi>t</mi>
</math></span> seconds.</p>
<p>It is known that P travels a distance of 15 metres in the first 2 seconds. It is also known that&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 2 \right) = s\left( 5 \right)">
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>2</mn>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>5</mn>
    <mo>)</mo>
  </mrow>
</math></span> and&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_2^4 {v\,{\text{d}}t}&nbsp; = 9">
  <msubsup>
    <mo>∫<!-- ∫ --></mo>
    <mn>2</mn>
    <mn>4</mn>
  </msubsup>
  <mrow>
    <mi>v</mi>
    <mspace width="thinmathspace"></mspace>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>t</mi>
  </mrow>
  <mo>=</mo>
  <mn>9</mn>
</math></span>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 4 \right) - s\left( 2 \right)">
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>4</mn>
    <mo>)</mo>
  </mrow>
  <mo>−</mo>
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>2</mn>
    <mo>)</mo>
  </mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled in the first 5 seconds.</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>recognizing relationship between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
  <mi>v</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
  <mi>s</mi>
</math></span>      <em><strong>(M1)</strong></em></p>
<p><em>eg     </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {v = s} ">
  <mo>∫</mo>
  <mrow>
    <mi>v</mi>
    <mo>=</mo>
    <mi>s</mi>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s' = v">
  <msup>
    <mi>s</mi>
    <mo>′</mo>
  </msup>
  <mo>=</mo>
  <mi>v</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 4 \right) - s\left( 2 \right) = 9">
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>4</mn>
    <mo>)</mo>
  </mrow>
  <mo>−</mo>
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>2</mn>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>9</mn>
</math></span>      <em><strong>A1  N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correctly interpreting distance travelled in first 2 seconds (seen anywhere, including part (a) or the area of 15 indicated on diagram)        <em><strong>(A1)</strong></em></p>
<p><em>eg</em>    <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^2 {v = 15} ">
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>2</mn>
  </msubsup>
  <mrow>
    <mi>v</mi>
    <mo>=</mo>
    <mn>15</mn>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 2 \right) = 15">
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>2</mn>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>15</mn>
</math></span></p>
<p>valid approach to find total distance travelled       <em><strong>(M1)</strong></em></p>
<p><em>eg</em>    sum of 3 areas,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^4 {v + } \int_4^5 v ">
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>4</mn>
  </msubsup>
  <mrow>
    <mi>v</mi>
    <mo>+</mo>
  </mrow>
  <msubsup>
    <mo>∫</mo>
    <mn>4</mn>
    <mn>5</mn>
  </msubsup>
  <mi>v</mi>
</math></span>,  shaded areas in diagram between 0 and 5</p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if only <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^5 {\left| v \right|} ">
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>5</mn>
  </msubsup>
  <mrow>
    <mrow>
      <mo>|</mo>
      <mi>v</mi>
      <mo>|</mo>
    </mrow>
  </mrow>
</math></span> is seen.</p>
<p>correct working towards finding distance travelled between 2 and 5 (seen anywhere including within total area expression or on diagram)       <em><strong>(A1)</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_2^4 {v - } \int_4^5 v ">
  <msubsup>
    <mo>∫</mo>
    <mn>2</mn>
    <mn>4</mn>
  </msubsup>
  <mrow>
    <mi>v</mi>
    <mo>−</mo>
  </mrow>
  <msubsup>
    <mo>∫</mo>
    <mn>4</mn>
    <mn>5</mn>
  </msubsup>
  <mi>v</mi>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_2^4 {v = } \int_4^5 {\left| v \right|} ">
  <msubsup>
    <mo>∫</mo>
    <mn>2</mn>
    <mn>4</mn>
  </msubsup>
  <mrow>
    <mi>v</mi>
    <mo>=</mo>
  </mrow>
  <msubsup>
    <mo>∫</mo>
    <mn>4</mn>
    <mn>5</mn>
  </msubsup>
  <mrow>
    <mrow>
      <mo>|</mo>
      <mi>v</mi>
      <mo>|</mo>
    </mrow>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_4^5 {v\,{\text{d}}t}  =  - 9">
  <msubsup>
    <mo>∫</mo>
    <mn>4</mn>
    <mn>5</mn>
  </msubsup>
  <mrow>
    <mi>v</mi>
    <mspace width="thinmathspace"></mspace>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>t</mi>
  </mrow>
  <mo>=</mo>
  <mo>−</mo>
  <mn>9</mn>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 4 \right) - s\left( 2 \right) - \left[ {s\left( 5 \right) - s\left( 4 \right)} \right]">
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>4</mn>
    <mo>)</mo>
  </mrow>
  <mo>−</mo>
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>2</mn>
    <mo>)</mo>
  </mrow>
  <mo>−</mo>
  <mrow>
    <mo>[</mo>
    <mrow>
      <mi>s</mi>
      <mrow>
        <mo>(</mo>
        <mn>5</mn>
        <mo>)</mo>
      </mrow>
      <mo>−</mo>
      <mi>s</mi>
      <mrow>
        <mo>(</mo>
        <mn>4</mn>
        <mo>)</mo>
      </mrow>
    </mrow>
    <mo>]</mo>
  </mrow>
</math></span>,</p>
<p>equal areas <img src="data:image/png;base64,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"></p>
<p>correct working using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( 5 \right) = s\left( 2 \right)">
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>5</mn>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>2</mn>
    <mo>)</mo>
  </mrow>
</math></span>      <em><strong>(A1)</strong></em></p>
<p><em>eg   </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 + 9 - \left( { - 9} \right)">
  <mn>15</mn>
  <mo>+</mo>
  <mn>9</mn>
  <mo>−</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mo>−</mo>
      <mn>9</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 + 2\left[ {s\left( 4 \right) - s\left( 2 \right)} \right]">
  <mn>15</mn>
  <mo>+</mo>
  <mn>2</mn>
  <mrow>
    <mo>[</mo>
    <mrow>
      <mi>s</mi>
      <mrow>
        <mo>(</mo>
        <mn>4</mn>
        <mo>)</mo>
      </mrow>
      <mo>−</mo>
      <mi>s</mi>
      <mrow>
        <mo>(</mo>
        <mn>2</mn>
        <mo>)</mo>
      </mrow>
    </mrow>
    <mo>]</mo>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 + 2\left( 9 \right)">
  <mn>15</mn>
  <mo>+</mo>
  <mn>2</mn>
  <mrow>
    <mo>(</mo>
    <mn>9</mn>
    <mo>)</mo>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times s\left( 4 \right) - s\left( 2 \right)">
  <mn>2</mn>
  <mo>×</mo>
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>4</mn>
    <mo>)</mo>
  </mrow>
  <mo>−</mo>
  <mi>s</mi>
  <mrow>
    <mo>(</mo>
    <mn>2</mn>
    <mo>)</mo>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="48 - 15">
  <mn>48</mn>
  <mo>−</mo>
  <mn>15</mn>
</math></span></p>
<p>total distance travelled = 33 (m)        <em><strong>A1   N2</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'(x) = {\sin ^3}(2x)\cos (2x)">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>sin</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mi>cos</mi>
  <mo>⁡</mo>
  <mo stretchy="false">(</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
</math></span>. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
</math></span>, given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {\frac{\pi }{4}} \right) = 1">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mi>π</mi>
        <mn>4</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>1</mn>
</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>evidence of integration     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f'(x){\text{d}}x} ">
  <mo>∫</mo>
  <mrow>
    <msup>
      <mi>f</mi>
      <mo>′</mo>
    </msup>
    <mo stretchy="false">(</mo>
    <mi>x</mi>
    <mo stretchy="false">)</mo>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
  </mrow>
</math></span></p>
<p>correct integration (accept missing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
  <mi>C</mi>
</math></span>)     <strong><em>(A2)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times \frac{{{{\sin }^4}(2x)}}{4},{\text{ }}\frac{1}{8}{\sin ^4}(2x) + C">
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mo>×</mo>
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mrow>
            <mi>sin</mi>
          </mrow>
          <mn>4</mn>
        </msup>
      </mrow>
      <mo stretchy="false">(</mo>
      <mn>2</mn>
      <mi>x</mi>
      <mo stretchy="false">)</mo>
    </mrow>
    <mn>4</mn>
  </mfrac>
  <mo>,</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mfrac>
    <mn>1</mn>
    <mn>8</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>sin</mi>
      <mn>4</mn>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>+</mo>
  <mi>C</mi>
</math></span></p>
<p>substituting initial condition into their integrated expression (must have <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + C">
  <mo>+</mo>
  <mi>C</mi>
</math></span>)     <strong><em>M1</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = \frac{1}{8}{\sin ^4}\left( {\frac{\pi }{2}} \right) + C">
  <mn>1</mn>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>8</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>sin</mi>
      <mn>4</mn>
    </msup>
  </mrow>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mi>π</mi>
        <mn>2</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>+</mo>
  <mi>C</mi>
</math></span></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M0 </em></strong>if they substitute into the original or differentiated function.</p>
<p> </p>
<p>recognizing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \left( {\frac{\pi }{2}} \right) = 1">
  <mi>sin</mi>
  <mo>⁡</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mi>π</mi>
        <mn>2</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>1</mn>
</math></span>     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = \frac{1}{8}{(1)^4} + C">
  <mn>1</mn>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>8</mn>
  </mfrac>
  <mrow>
    <mo stretchy="false">(</mo>
    <mn>1</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mn>4</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>C</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = \frac{7}{8}">
  <mi>C</mi>
  <mo>=</mo>
  <mfrac>
    <mn>7</mn>
    <mn>8</mn>
  </mfrac>
</math></span>    <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{8}{\sin ^4}(2x) + \frac{7}{8}">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>8</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>sin</mi>
      <mn>4</mn>
    </msup>
  </mrow>
  <mo stretchy="false">(</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>+</mo>
  <mfrac>
    <mn>7</mn>
    <mn>8</mn>
  </mfrac>
</math></span>     <strong><em>A1     N5</em></strong></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="specification">
<p>A quadratic function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = a{x^2} + bx + c">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>a</mi>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>b</mi>
  <mi>x</mi>
  <mo>+</mo>
  <mi>c</mi>
</math></span>. The points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}5)">
  <mo stretchy="false">(</mo>
  <mn>0</mn>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;</mtext>
  </mrow>
  <mn>5</mn>
  <mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 4,{\text{ }}5)">
  <mo stretchy="false">(</mo>
  <mo>−<!-- − --></mo>
  <mn>4</mn>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;</mtext>
  </mrow>
  <mn>5</mn>
  <mo stretchy="false">)</mo>
</math></span> lie on the graph of <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>

<div class="specification">
<p>The <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
  <mi>y</mi>
</math></span>-coordinate of the minimum of the graph is 3.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the axis of symmetry of the graph of <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">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
  <mi>c</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
  <mi>a</mi>
</math></span> and of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
  <mi>b</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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="x =  - 2">
  <mi>x</mi>
  <mo>=</mo>
  <mo>−</mo>
  <mn>2</mn>
</math></span>     <strong><em>(A1)(A1)     (C2)</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="x = ">
  <mi>x</mi>
  <mo>=</mo>
</math></span> (a constant) and <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2">
  <mo>−</mo>
  <mn>2</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(c = ){\text{ }}5">
  <mo stretchy="false">(</mo>
  <mi>c</mi>
  <mo>=</mo>
  <mo stretchy="false">)</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mn>5</mn>
</math></span>     <strong><em>(A1)     (C1)</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{b}{{2a}} =  - 2">
  <mo>−</mo>
  <mfrac>
    <mi>b</mi>
    <mrow>
      <mn>2</mn>
      <mi>a</mi>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mo>−</mo>
  <mn>2</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{( - 2)^2} - 2b + 5 = 3">
  <mi>a</mi>
  <mrow>
    <mo stretchy="false">(</mo>
    <mo>−</mo>
    <mn>2</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>2</mn>
  <mi>b</mi>
  <mo>+</mo>
  <mn>5</mn>
  <mo>=</mo>
  <mn>3</mn>
</math></span> or equivalent</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{( - 4)^2} - 4b + 5 = 5">
  <mi>a</mi>
  <mrow>
    <mo stretchy="false">(</mo>
    <mo>−</mo>
    <mn>4</mn>
    <msup>
      <mo stretchy="false">)</mo>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>4</mn>
  <mi>b</mi>
  <mo>+</mo>
  <mn>5</mn>
  <mo>=</mo>
  <mn>5</mn>
</math></span> or equivalent</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2a( - 2) + b = 0">
  <mn>2</mn>
  <mi>a</mi>
  <mo stretchy="false">(</mo>
  <mo>−</mo>
  <mn>2</mn>
  <mo stretchy="false">)</mo>
  <mo>+</mo>
  <mi>b</mi>
  <mo>=</mo>
  <mn>0</mn>
</math></span> or equivalent     <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(M1) </em></strong>for two of the above equations.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 0.5">
  <mi>a</mi>
  <mo>=</mo>
  <mn>0.5</mn>
</math></span>     <strong><em>(A1)</em>(ft)</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 2">
  <mi>b</mi>
  <mo>=</mo>
  <mn>2</mn>
</math></span>     <strong><em>(A1)</em>(ft)     <em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award at most <strong><em>(M1)(A1)</em>(ft)<em>(A0) </em></strong>if the answers are reversed.</p>
<p>Follow through from parts (a) and (b).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Let&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{6 - 2x}}{{\sqrt {16 + 6x - {x^2}} }}">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>6</mn>
      <mo>−</mo>
      <mn>2</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <msqrt>
        <mn>16</mn>
        <mo>+</mo>
        <mn>6</mn>
        <mi>x</mi>
        <mo>−</mo>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
      </msqrt>
    </mrow>
  </mfrac>
</math></span>. The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The region <em>R</em> is enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span>-axis, and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
  <mi>y</mi>
</math></span>-axis. Find the area of <em>R</em>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong> (limits in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span>)</p>
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span>-intercept&nbsp; &nbsp; &nbsp; <em><strong>(M1)</strong></em></p>
<p><em>eg</em>&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 0">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>0</mn>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6 - 2x}}{{\sqrt {16 + 6x - {x^2}} }} = 0">
  <mfrac>
    <mrow>
      <mn>6</mn>
      <mo>−</mo>
      <mn>2</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <msqrt>
        <mn>16</mn>
        <mo>+</mo>
        <mn>6</mn>
        <mi>x</mi>
        <mo>−</mo>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
      </msqrt>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - 2x = 0">
  <mn>6</mn>
  <mo>−</mo>
  <mn>2</mn>
  <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="x">
  <mi>x</mi>
</math></span>-intercept is 3&nbsp; &nbsp; &nbsp; <em><strong>(A1)</strong></em></p>
<p>valid approach using substitution or inspection&nbsp; &nbsp; &nbsp; <em><strong>(M1)</strong></em></p>
<p><em>eg</em>&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 16 + 6x&nbsp;- {x^2}">
  <mi>u</mi>
  <mo>=</mo>
  <mn>16</mn>
  <mo>+</mo>
  <mn>6</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^3 {\frac{{6 - 2x}}{{\sqrt u }}} {\text{d}}x">
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>3</mn>
  </msubsup>
  <mrow>
    <mfrac>
      <mrow>
        <mn>6</mn>
        <mo>−</mo>
        <mn>2</mn>
        <mi>x</mi>
      </mrow>
      <mrow>
        <msqrt>
          <mi>u</mi>
        </msqrt>
      </mrow>
    </mfrac>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}u = 6 - 2x">
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
  <mo>=</mo>
  <mn>6</mn>
  <mo>−</mo>
  <mn>2</mn>
  <mi>x</mi>
</math></span>,&nbsp; <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{\sqrt u }}} ">
  <mo>∫</mo>
  <mrow>
    <mfrac>
      <mn>1</mn>
      <mrow>
        <msqrt>
          <mi>u</mi>
        </msqrt>
      </mrow>
    </mfrac>
  </mrow>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{u^{\frac{1}{2}}}">
  <mn>2</mn>
  <mrow>
    <msup>
      <mi>u</mi>
      <mrow>
        <mfrac>
          <mn>1</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span>,</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \sqrt {16 + 6x&nbsp;- {x^2}} ">
  <mi>u</mi>
  <mo>=</mo>
  <msqrt>
    <mn>16</mn>
    <mo>+</mo>
    <mn>6</mn>
    <mi>x</mi>
    <mo>−</mo>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>2</mn>
      </msup>
    </mrow>
  </msqrt>
</math></span>,&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = \left( {6 - 2x} \right)\frac{1}{2}{\left( {16 + 6x - {x^2}} \right)^{ - \frac{1}{2}}}">
  <mfrac>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>u</mi>
    </mrow>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>x</mi>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>6</mn>
      <mo>−</mo>
      <mn>2</mn>
      <mi>x</mi>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mn>16</mn>
          <mo>+</mo>
          <mn>6</mn>
          <mi>x</mi>
          <mo>−</mo>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>2</mn>
            </msup>
          </mrow>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mrow>
        <mo>−</mo>
        <mfrac>
          <mn>1</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int 2 \,{\text{d}}u">
  <mo>∫</mo>
  <mn>2</mn>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2u">
  <mn>2</mn>
  <mi>u</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f\left( x \right)} \,{\text{d}}x = 2\sqrt {16 + 6x - {x^2}} ">
  <mo>∫</mo>
  <mrow>
    <mi>f</mi>
    <mrow>
      <mo>(</mo>
      <mi>x</mi>
      <mo>)</mo>
    </mrow>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
  <mo>=</mo>
  <mn>2</mn>
  <msqrt>
    <mn>16</mn>
    <mo>+</mo>
    <mn>6</mn>
    <mi>x</mi>
    <mo>−</mo>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>2</mn>
      </msup>
    </mrow>
  </msqrt>
</math></span>&nbsp; &nbsp; &nbsp;&nbsp;<em><strong>(A2)</strong></em></p>
<p>substituting <strong>both</strong> of <strong>their</strong> limits into <strong>their</strong> integrated function and subtracting&nbsp; &nbsp; &nbsp; <em><strong>(M1)</strong></em></p>
<p><em>eg&nbsp;</em>&nbsp; <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sqrt {16 + 6\left( 3 \right) - {3^2}}&nbsp; - 2\sqrt {16 + 6{{\left( 0 \right)}^2} - {0^2}} ">
  <mn>2</mn>
  <msqrt>
    <mn>16</mn>
    <mo>+</mo>
    <mn>6</mn>
    <mrow>
      <mo>(</mo>
      <mn>3</mn>
      <mo>)</mo>
    </mrow>
    <mo>−</mo>
    <mrow>
      <msup>
        <mn>3</mn>
        <mn>2</mn>
      </msup>
    </mrow>
  </msqrt>
  <mo>−</mo>
  <mn>2</mn>
  <msqrt>
    <mn>16</mn>
    <mo>+</mo>
    <mn>6</mn>
    <mrow>
      <msup>
        <mrow>
          <mrow>
            <mo>(</mo>
            <mn>0</mn>
            <mo>)</mo>
          </mrow>
        </mrow>
        <mn>2</mn>
      </msup>
    </mrow>
    <mo>−</mo>
    <mrow>
      <msup>
        <mn>0</mn>
        <mn>2</mn>
      </msup>
    </mrow>
  </msqrt>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sqrt {16 + 18 - 9}&nbsp; - 2\sqrt {16} ">
  <mn>2</mn>
  <msqrt>
    <mn>16</mn>
    <mo>+</mo>
    <mn>18</mn>
    <mo>−</mo>
    <mn>9</mn>
  </msqrt>
  <mo>−</mo>
  <mn>2</mn>
  <msqrt>
    <mn>16</mn>
  </msqrt>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> if they substitute into original or differentiated function. Do not accept only “– 0” as evidence of substituting lower limit.</p>
<p>&nbsp;</p>
<p>correct working&nbsp; &nbsp; &nbsp;&nbsp;<em><strong>(A1)</strong></em></p>
<p><em>eg</em>&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sqrt {25}&nbsp; - 2\sqrt {16} ">
  <mn>2</mn>
  <msqrt>
    <mn>25</mn>
  </msqrt>
  <mo>−</mo>
  <mn>2</mn>
  <msqrt>
    <mn>16</mn>
  </msqrt>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10 - 8">
  <mn>10</mn>
  <mo>−</mo>
  <mn>8</mn>
</math></span></p>
<p>area = 2&nbsp; &nbsp; &nbsp; <em><strong>A1 N2</strong></em></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><strong>METHOD 2</strong>&nbsp;(limits in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
  <mi>u</mi>
</math></span>)</p>
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span>-intercept&nbsp; &nbsp; &nbsp;&nbsp;<em><strong>(M1)</strong></em></p>
<p><em>eg</em>&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 0">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>0</mn>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6 - 2x}}{{\sqrt {16 + 6x - {x^2}} }} = 0">
  <mfrac>
    <mrow>
      <mn>6</mn>
      <mo>−</mo>
      <mn>2</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <msqrt>
        <mn>16</mn>
        <mo>+</mo>
        <mn>6</mn>
        <mi>x</mi>
        <mo>−</mo>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
      </msqrt>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - 2x = 0">
  <mn>6</mn>
  <mo>−</mo>
  <mn>2</mn>
  <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="x">
  <mi>x</mi>
</math></span>-intercept is 3&nbsp; &nbsp; &nbsp;&nbsp;<em><strong>(A1)</strong></em></p>
<p>valid approach using substitution or inspection&nbsp; &nbsp; &nbsp;&nbsp;<em><strong>(M1)</strong></em></p>
<p><em>eg</em>&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 16 + 6x&nbsp;- {x^2}">
  <mi>u</mi>
  <mo>=</mo>
  <mn>16</mn>
  <mo>+</mo>
  <mn>6</mn>
  <mi>x</mi>
  <mo>−</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^3 {\frac{{6 - 2x}}{{\sqrt u }}} {\text{d}}x">
  <msubsup>
    <mo>∫</mo>
    <mn>0</mn>
    <mn>3</mn>
  </msubsup>
  <mrow>
    <mfrac>
      <mrow>
        <mn>6</mn>
        <mo>−</mo>
        <mn>2</mn>
        <mi>x</mi>
      </mrow>
      <mrow>
        <msqrt>
          <mi>u</mi>
        </msqrt>
      </mrow>
    </mfrac>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}u = 6 - 2x">
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
  <mo>=</mo>
  <mn>6</mn>
  <mo>−</mo>
  <mn>2</mn>
  <mi>x</mi>
</math></span>,&nbsp; <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{\sqrt u }}} ">
  <mo>∫</mo>
  <mrow>
    <mfrac>
      <mn>1</mn>
      <mrow>
        <msqrt>
          <mi>u</mi>
        </msqrt>
      </mrow>
    </mfrac>
  </mrow>
</math></span>,&nbsp;</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \sqrt {16 + 6x&nbsp;- {x^2}} ">
  <mi>u</mi>
  <mo>=</mo>
  <msqrt>
    <mn>16</mn>
    <mo>+</mo>
    <mn>6</mn>
    <mi>x</mi>
    <mo>−</mo>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>2</mn>
      </msup>
    </mrow>
  </msqrt>
</math></span>,&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = \left( {6 - 2x} \right)\frac{1}{2}{\left( {16 + 6x - {x^2}} \right)^{ - \frac{1}{2}}}">
  <mfrac>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>u</mi>
    </mrow>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>x</mi>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>6</mn>
      <mo>−</mo>
      <mn>2</mn>
      <mi>x</mi>
    </mrow>
    <mo>)</mo>
  </mrow>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mn>16</mn>
          <mo>+</mo>
          <mn>6</mn>
          <mi>x</mi>
          <mo>−</mo>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>2</mn>
            </msup>
          </mrow>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mrow>
        <mo>−</mo>
        <mfrac>
          <mn>1</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int 2 \,{\text{d}}u">
  <mo>∫</mo>
  <mn>2</mn>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
</math></span></p>
<p>correct integration&nbsp; &nbsp; &nbsp; <em><strong>(A2)</strong></em></p>
<p><em>eg</em>&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{\sqrt u }}} \,{\text{d}}u = 2{u^{\frac{1}{2}}}">
  <mo>∫</mo>
  <mrow>
    <mfrac>
      <mn>1</mn>
      <mrow>
        <msqrt>
          <mi>u</mi>
        </msqrt>
      </mrow>
    </mfrac>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mrow>
    <msup>
      <mi>u</mi>
      <mrow>
        <mfrac>
          <mn>1</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span>,&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int 2 \,{\text{d}}u = 2u">
  <mo>∫</mo>
  <mn>2</mn>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mi>u</mi>
</math></span></p>
<p>both correct limits for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
  <mi>u</mi>
</math></span>&nbsp; &nbsp; &nbsp; <em><strong>(A1)</strong></em></p>
<p><em>eg</em>&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
  <mi>u</mi>
</math></span> = 16 and&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
  <mi>u</mi>
</math></span> = 25,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{16}^{25} {\frac{1}{{\sqrt u }}{\text{d}}u} ">
  <msubsup>
    <mo>∫</mo>
    <mrow>
      <mn>16</mn>
    </mrow>
    <mrow>
      <mn>25</mn>
    </mrow>
  </msubsup>
  <mrow>
    <mfrac>
      <mn>1</mn>
      <mrow>
        <msqrt>
          <mi>u</mi>
        </msqrt>
      </mrow>
    </mfrac>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>u</mi>
  </mrow>
</math></span>,&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {2{u^{\frac{1}{2}}}} \right]_{16}^{25}">
  <msubsup>
    <mrow>
      <mo>[</mo>
      <mrow>
        <mn>2</mn>
        <mrow>
          <msup>
            <mi>u</mi>
            <mrow>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </mrow>
          </msup>
        </mrow>
      </mrow>
      <mo>]</mo>
    </mrow>
    <mrow>
      <mn>16</mn>
    </mrow>
    <mrow>
      <mn>25</mn>
    </mrow>
  </msubsup>
</math></span>,&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
  <mi>u</mi>
</math></span> =&nbsp;4 and&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
  <mi>u</mi>
</math></span> = 5,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_4^5 2 \,{\text{d}}u">
  <msubsup>
    <mo>∫</mo>
    <mn>4</mn>
    <mn>5</mn>
  </msubsup>
  <mn>2</mn>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
</math></span>,&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {2u} \right]_4^5">
  <msubsup>
    <mrow>
      <mo>[</mo>
      <mrow>
        <mn>2</mn>
        <mi>u</mi>
      </mrow>
      <mo>]</mo>
    </mrow>
    <mn>4</mn>
    <mn>5</mn>
  </msubsup>
</math></span></p>
<p>substituting <strong>both</strong> of <strong>their</strong> limits for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
  <mi>u</mi>
</math></span> (do not accept 0 and 3) into <strong>their</strong> integrated function and subtracting&nbsp; &nbsp; &nbsp;<em><strong>(M1)</strong></em></p>
<p><em>eg</em>&nbsp; &nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sqrt {25}&nbsp; - 2\sqrt {16} ">
  <mn>2</mn>
  <msqrt>
    <mn>25</mn>
  </msqrt>
  <mo>−</mo>
  <mn>2</mn>
  <msqrt>
    <mn>16</mn>
  </msqrt>
</math></span>,&nbsp;&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10 - 8">
  <mn>10</mn>
  <mo>−</mo>
  <mn>8</mn>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if they substitute into original or differentiated function, or if they have not attempted to find limits for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
  <mi>u</mi>
</math></span>.</p>
<p>&nbsp;</p>
<p>area = 2&nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1 N2</strong></em></p>
<p>&nbsp;</p>
<p><strong><em>[8 marks]</em></strong></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 graph of the function&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></math>.</p>
</div>

<div class="specification">
<p>The equation of the tangent to the graph of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>&nbsp;at&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn></math>&nbsp;is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mo>=</mo><mn>4</mn><mo>-</mo><mn>5</mn><mi>x</mi></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <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">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the gradient of this tangent.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. It appeared in a paper that permitted the use of a calculator, and so might not be suitable for all forms of practice.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>+</mo><mfrac><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>     <em><strong>(A1)(A1)(A1)    (C3)<br></strong></em></p>
<p><strong><br></strong><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi></math>, <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>k</mi></math>, and <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>.<br>Award at most <em><strong>(A1)(A1)(A0)</strong></em> if additional terms are seen.</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"><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mo> </mo><mfenced><mfrac><mrow><mo>-</mo><mn>5</mn></mrow><mn>2</mn></mfrac></mfenced></math>     <em><strong>(A1)    (C1)<br></strong></em></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"><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>=</mo><mn>2</mn><mo>×</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mfrac><mi>k</mi><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math>       <em><strong>(M1)<br></strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for equating their gradient from part (b) to their substituted derivative from part (a).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>6</mn></math>      <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>    (C2)</strong></em></p>
<p><strong><br>Note:</strong> Follow through from parts (a) and (b).</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the curve with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>The tangent to the curve at the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> is parallel to the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup><mi>x</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>evidence of using product rule           <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><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mi>k</mi><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>+</mo><mn>2</mn><mo>×</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup><mfenced><mrow><mn>2</mn><mi>k</mi><mi>x</mi><mo>-</mo><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math>            <em><strong>A1</strong></em></p>
<p>correct working for one of (seen anywhere)            <em><strong>A1</strong></em></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>⇒</mo><mi>k</mi><msup><mtext>e</mtext><mi>k</mi></msup><mo>+</mo><mn>2</mn><msup><mtext>e</mtext><mi>k</mi></msup></math></p>
<p style="padding-left:30px;"><br><strong>OR</strong></p>
<p style="padding-left:30px;">slope of tangent is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup></math></p>
<p><br>their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> equals the <em>slope</em> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup></mrow></mfenced></math> (seen anywhere)           <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><msup><mtext>e</mtext><mi>k</mi></msup><mo>+</mo><mn>2</mn><msup><mtext>e</mtext><mi>k</mi></msup><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>3</mn></math>            <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>The product rule was well recognised and used with 𝑥=1 properly substituted into this expression. Although the majority of the candidates tried equating the derivative to the slope of the tangent line, the slope of the tangent line was not correctly identified; many candidates incorrectly substituted 𝑥=1 into the tangent equation, thus finding the <em>y</em>-coordinate instead of the slope.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{{\text{ln}}\,5x}}{{kx}}">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <mi>k</mi>
      <mi>x</mi>
    </mrow>
  </mfrac>
</math></span> where&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
  <mi>x</mi>
  <mo>&gt;</mo>
  <mn>0</mn>
</math></span>,&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in {\mathbb{R}^ + }">
  <mi>k</mi>
  <mo>∈<!-- ∈ --></mo>
  <mrow>
    <msup>
      <mrow>
        <mi mathvariant="double-struck">R</mi>
      </mrow>
      <mo>+</mo>
    </msup>
  </mrow>
</math></span>.</p>
</div>

<div class="specification">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> has exactly one maximum point P.</p>
</div>

<div class="specification">
<p>The second derivative of&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> is given by&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = \frac{{2\,{\text{ln}}\,5x - 3}}{{k{x^3}}}">
  <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>ln</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
      <mi>x</mi>
      <mo>−<!-- − --></mo>
      <mn>3</mn>
    </mrow>
    <mrow>
      <mi>k</mi>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>3</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span>.&nbsp;The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> has exactly one&nbsp;point of inflexion Q.</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( x \right) = \frac{{1 - {\text{ln}}\,5x}}{{k{x^2}}}">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>1</mn>
      <mo>−</mo>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <mi>k</mi>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</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>Find the <em>x</em>-coordinate of 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>Show that the <em>x</em>-coordinate of Q is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{1}}}{5}{{\text{e}}^{\frac{3}{2}}}">
  <mfrac>
    <mrow>
      <mtext>1</mtext>
    </mrow>
    <mn>5</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mfrac>
          <mn>3</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </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>The region <em>R</em> is enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span>, the <em>x</em>-axis, and the vertical lines through the maximum point P and the point of inflexion Q.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Given that the area of <em>R</em> is 3, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
  <mi>k</mi>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule      <em><strong> (M1)</strong></em></p>
<p>correct substitution into quotient rule</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{5kx\left( {\frac{1}{{5x}}} \right) - k\,{\text{ln}}\,5x}}{{{{\left( {kx} \right)}^2}}}">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>5</mn>
      <mi>k</mi>
      <mi>x</mi>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mn>5</mn>
              <mi>x</mi>
            </mrow>
          </mfrac>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mo>−</mo>
      <mi>k</mi>
      <mspace width="thinmathspace"></mspace>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mrow>
            <mrow>
              <mo>(</mo>
              <mrow>
                <mi>k</mi>
                <mi>x</mi>
              </mrow>
              <mo>)</mo>
            </mrow>
          </mrow>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span> (or equivalent)        <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{k - k\,{\text{ln}}\,5x}}{{{k^2}{x^2}}}">
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mi>k</mi>
      <mo>−</mo>
      <mi>k</mi>
      <mspace width="thinmathspace"></mspace>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <mrow>
        <msup>
          <mi>k</mi>
          <mn>2</mn>
        </msup>
      </mrow>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span>, <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {k \in {\mathbb{R}^ + }} \right)">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mi>k</mi>
      <mo>∈</mo>
      <mrow>
        <msup>
          <mrow>
            <mi mathvariant="double-struck">R</mi>
          </mrow>
          <mo>+</mo>
        </msup>
      </mrow>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span></span>        <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{1 - {\text{ln}}\,5x}}{{k{x^2}}}">
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>1</mn>
      <mo>−</mo>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <mi>k</mi>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
</math></span>        <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0">
  <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> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 - {\text{ln}}\,5x}}{{k{x^2}}} = 0">
  <mfrac>
    <mrow>
      <mn>1</mn>
      <mo>−</mo>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
      <mi>x</mi>
    </mrow>
    <mrow>
      <mi>k</mi>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,5x = 1">
  <mrow>
    <mtext>ln</mtext>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mn>5</mn>
  <mi>x</mi>
  <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{e}}}{5}">
  <mi>x</mi>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mtext>e</mtext>
    </mrow>
    <mn>5</mn>
  </mfrac>
</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 0">
  <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> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\,{\text{ln}}\,5x - 3}}{{k{x^3}}} = 0">
  <mfrac>
    <mrow>
      <mn>2</mn>
      <mspace width="thinmathspace"></mspace>
      <mrow>
        <mtext>ln</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
      <mi>x</mi>
      <mo>−</mo>
      <mn>3</mn>
    </mrow>
    <mrow>
      <mi>k</mi>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>3</mn>
        </msup>
      </mrow>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,5x = \frac{3}{2}">
  <mrow>
    <mtext>ln</mtext>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mn>5</mn>
  <mi>x</mi>
  <mo>=</mo>
  <mfrac>
    <mn>3</mn>
    <mn>2</mn>
  </mfrac>
</math></span>     <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5x = {{\text{e}}^{\frac{3}{2}}}">
  <mn>5</mn>
  <mi>x</mi>
  <mo>=</mo>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mfrac>
          <mn>3</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math></span>     <em><strong>A1</strong></em></p>
<p>so the point of inflexion occurs at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{\text{1}}}{5}{{\text{e}}^{\frac{3}{2}}}">
  <mi>x</mi>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mtext>1</mtext>
    </mrow>
    <mn>5</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mfrac>
          <mn>3</mn>
          <mn>2</mn>
        </mfrac>
      </mrow>
    </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>attempt to integrate  <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{ln}}\,5x \Rightarrow \frac{{{\text{d}}u}}{{{\text{d}}x}} = \frac{1}{x}">
  <mi>u</mi>
  <mo>=</mo>
  <mrow>
    <mtext>ln</mtext>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mn>5</mn>
  <mi>x</mi>
  <mo stretchy="false">⇒</mo>
  <mfrac>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>u</mi>
    </mrow>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>x</mi>
    </mrow>
  </mfrac>
  <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="\int {\frac{{{\text{ln}}\,5x}}{{kx}}} {\text{d}}x = \frac{1}{k}\int {u\,} {\text{d}}u">
  <mo>∫</mo>
  <mrow>
    <mfrac>
      <mrow>
        <mrow>
          <mtext>ln</mtext>
        </mrow>
        <mspace width="thinmathspace"></mspace>
        <mn>5</mn>
        <mi>x</mi>
      </mrow>
      <mrow>
        <mi>k</mi>
        <mi>x</mi>
      </mrow>
    </mfrac>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mi>k</mi>
  </mfrac>
  <mo>∫</mo>
  <mrow>
    <mi>u</mi>
    <mspace width="thinmathspace"></mspace>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
</math></span>     <em><strong>(</strong><strong>A1)</strong></em></p>
<p><em><strong>EITHER</strong></em></p>
<p>=<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{u^2}}}{{2k}}">
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mi>u</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
    <mrow>
      <mn>2</mn>
      <mi>k</mi>
    </mrow>
  </mfrac>
</math></span>     <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{k}\int\limits_1^{\frac{3}{2}} {u\,{\text{d}}u = } \left[ {\frac{{{u^2}}}{{2k}}} \right]_1^{\frac{3}{2}}">
  <mfrac>
    <mn>1</mn>
    <mi>k</mi>
  </mfrac>
  <munderover>
    <mo>∫</mo>
    <mn>1</mn>
    <mrow>
      <mfrac>
        <mn>3</mn>
        <mn>2</mn>
      </mfrac>
    </mrow>
  </munderover>
  <mrow>
    <mi>u</mi>
    <mspace width="thinmathspace"></mspace>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>u</mi>
    <mo>=</mo>
  </mrow>
  <msubsup>
    <mrow>
      <mo>[</mo>
      <mrow>
        <mfrac>
          <mrow>
            <mrow>
              <msup>
                <mi>u</mi>
                <mn>2</mn>
              </msup>
            </mrow>
          </mrow>
          <mrow>
            <mn>2</mn>
            <mi>k</mi>
          </mrow>
        </mfrac>
      </mrow>
      <mo>]</mo>
    </mrow>
    <mn>1</mn>
    <mrow>
      <mfrac>
        <mn>3</mn>
        <mn>2</mn>
      </mfrac>
    </mrow>
  </msubsup>
</math></span>     <em><strong>A1</strong></em></p>
<p><em><strong>OR</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{{\left( {{\text{ln}}\,5x} \right)}^2}}}{{2k}}">
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mrow>
        <msup>
          <mrow>
            <mrow>
              <mo>(</mo>
              <mrow>
                <mrow>
                  <mtext>ln</mtext>
                </mrow>
                <mspace width="thinmathspace"></mspace>
                <mn>5</mn>
                <mi>x</mi>
              </mrow>
              <mo>)</mo>
            </mrow>
          </mrow>
          <mn>2</mn>
        </msup>
      </mrow>
    </mrow>
    <mrow>
      <mn>2</mn>
      <mi>k</mi>
    </mrow>
  </mfrac>
</math></span>     <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_{\frac{{\text{e}}}{5}}^{\frac{{\text{1}}}{5}{{\text{e}}^{\frac{3}{2}}}} {\frac{{{\text{ln}}\,5x}}{{kx}}} {\text{d}}x = \left[ {\frac{{{{\left( {{\text{ln}}\,5x} \right)}^2}}}{{2k}}} \right]_{\frac{{\text{e}}}{5}}^{\frac{{\text{1}}}{5}{{\text{e}}^{\frac{3}{2}}}}">
  <munderover>
    <mo>∫</mo>
    <mrow>
      <mfrac>
        <mrow>
          <mtext>e</mtext>
        </mrow>
        <mn>5</mn>
      </mfrac>
    </mrow>
    <mrow>
      <mfrac>
        <mrow>
          <mtext>1</mtext>
        </mrow>
        <mn>5</mn>
      </mfrac>
      <mrow>
        <msup>
          <mrow>
            <mtext>e</mtext>
          </mrow>
          <mrow>
            <mfrac>
              <mn>3</mn>
              <mn>2</mn>
            </mfrac>
          </mrow>
        </msup>
      </mrow>
    </mrow>
  </munderover>
  <mrow>
    <mfrac>
      <mrow>
        <mrow>
          <mtext>ln</mtext>
        </mrow>
        <mspace width="thinmathspace"></mspace>
        <mn>5</mn>
        <mi>x</mi>
      </mrow>
      <mrow>
        <mi>k</mi>
        <mi>x</mi>
      </mrow>
    </mfrac>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
  <mo>=</mo>
  <msubsup>
    <mrow>
      <mo>[</mo>
      <mrow>
        <mfrac>
          <mrow>
            <mrow>
              <msup>
                <mrow>
                  <mrow>
                    <mo>(</mo>
                    <mrow>
                      <mrow>
                        <mtext>ln</mtext>
                      </mrow>
                      <mspace width="thinmathspace"></mspace>
                      <mn>5</mn>
                      <mi>x</mi>
                    </mrow>
                    <mo>)</mo>
                  </mrow>
                </mrow>
                <mn>2</mn>
              </msup>
            </mrow>
          </mrow>
          <mrow>
            <mn>2</mn>
            <mi>k</mi>
          </mrow>
        </mfrac>
      </mrow>
      <mo>]</mo>
    </mrow>
    <mrow>
      <mfrac>
        <mrow>
          <mtext>e</mtext>
        </mrow>
        <mn>5</mn>
      </mfrac>
    </mrow>
    <mrow>
      <mfrac>
        <mrow>
          <mtext>1</mtext>
        </mrow>
        <mn>5</mn>
      </mfrac>
      <mrow>
        <msup>
          <mrow>
            <mtext>e</mtext>
          </mrow>
          <mrow>
            <mfrac>
              <mn>3</mn>
              <mn>2</mn>
            </mfrac>
          </mrow>
        </msup>
      </mrow>
    </mrow>
  </msubsup>
</math></span>     <em><strong>A1</strong></em></p>
<p><em><strong>THEN</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{2k}}\left( {\frac{9}{4} - 1} \right)">
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mrow>
      <mn>2</mn>
      <mi>k</mi>
    </mrow>
  </mfrac>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>9</mn>
        <mn>4</mn>
      </mfrac>
      <mo>−</mo>
      <mn>1</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{5}{{8k}}">
  <mo>=</mo>
  <mfrac>
    <mn>5</mn>
    <mrow>
      <mn>8</mn>
      <mi>k</mi>
    </mrow>
  </mfrac>
</math></span>     <em><strong>A1</strong></em></p>
<p>setting <strong>their</strong> expression for area equal to 3       <em><strong>M</strong><strong>1</strong></em> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{{8k}} = 3">
  <mfrac>
    <mn>5</mn>
    <mrow>
      <mn>8</mn>
      <mi>k</mi>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{5}{{24}}">
  <mi>k</mi>
  <mo>=</mo>
  <mfrac>
    <mn>5</mn>
    <mrow>
      <mn>24</mn>
    </mrow>
  </mfrac>
</math></span>     <em><strong>A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find&nbsp;<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {x{{\text{e}}^{{x^2} - 1}}{\text{d}}x} ">
  <mo>∫</mo>
  <mrow>
    <mi>x</mi>
    <mrow>
      <msup>
        <mrow>
          <mtext>e</mtext>
        </mrow>
        <mrow>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>2</mn>
            </msup>
          </mrow>
          <mo>−</mo>
          <mn>1</mn>
        </mrow>
      </msup>
    </mrow>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
  </mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
</math></span>, given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = x{{\text{e}}^{{x^2} - 1}}">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>x</mi>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>−</mo>
        <mn>1</mn>
      </mrow>
    </msup>
  </mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f( - 1) = 3">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mo>−</mo>
  <mn>1</mn>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mn>3</mn>
</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>valid approach to set up integration by substitution/inspection &nbsp; &nbsp; <strong><em>(M1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^2} - 1,{\text{ d}}u = 2x,{\text{ }}\int {2x{{\text{e}}^{{x^2} - 1}}{\text{d}}x} ">
  <mi>u</mi>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−</mo>
  <mn>1</mn>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;d</mtext>
  </mrow>
  <mi>u</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;</mtext>
  </mrow>
  <mo>∫</mo>
  <mrow>
    <mn>2</mn>
    <mi>x</mi>
    <mrow>
      <msup>
        <mrow>
          <mtext>e</mtext>
        </mrow>
        <mrow>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>2</mn>
            </msup>
          </mrow>
          <mo>−</mo>
          <mn>1</mn>
        </mrow>
      </msup>
    </mrow>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
  </mrow>
</math></span></p>
<p>correct expression &nbsp; &nbsp; <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\int {2x{{\text{e}}^{{x^2} - 1}}{\text{d}}x,{\text{ }}\frac{1}{2}\int {{{\text{e}}^u}{\text{d}}u} } ">
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mo>∫</mo>
  <mrow>
    <mn>2</mn>
    <mi>x</mi>
    <mrow>
      <msup>
        <mrow>
          <mtext>e</mtext>
        </mrow>
        <mrow>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>2</mn>
            </msup>
          </mrow>
          <mo>−</mo>
          <mn>1</mn>
        </mrow>
      </msup>
    </mrow>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
    <mo>,</mo>
    <mrow>
      <mtext>&nbsp;</mtext>
    </mrow>
    <mfrac>
      <mn>1</mn>
      <mn>2</mn>
    </mfrac>
    <mo>∫</mo>
    <mrow>
      <mrow>
        <msup>
          <mrow>
            <mtext>e</mtext>
          </mrow>
          <mi>u</mi>
        </msup>
      </mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>u</mi>
    </mrow>
  </mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{{\text{e}}^{{x^2} - 1}} + c">
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>−</mo>
        <mn>1</mn>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>c</mi>
</math></span> &nbsp; &nbsp; <strong><em>A2</em></strong> &nbsp; &nbsp; <strong><em>N4</em></strong></p>
<p>&nbsp;</p>
<p><strong>Notes: </strong>Award <strong><em>A1</em> </strong>if missing “<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + c">
  <mo>+</mo>
  <mi>c</mi>
</math></span>”.</p>
<p>&nbsp;</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>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x =&nbsp; - 1">
  <mi>x</mi>
  <mo>=</mo>
  <mo>−</mo>
  <mn>1</mn>
</math></span> into <strong>their </strong>answer from (a) &nbsp; &nbsp; <strong><em>(M1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{{\text{e}}^0},{\text{ }}\frac{1}{2}{{\text{e}}^{1 - 1}} = 3">
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mn>0</mn>
    </msup>
  </mrow>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;</mtext>
  </mrow>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mn>1</mn>
        <mo>−</mo>
        <mn>1</mn>
      </mrow>
    </msup>
  </mrow>
  <mo>=</mo>
  <mn>3</mn>
</math></span></p>
<p>correct working &nbsp; &nbsp; <strong><em>(A1)</em></strong></p>
<p><em>eg</em><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>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} + c = 3,{\text{ }}c = 2.5">
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mo>+</mo>
  <mi>c</mi>
  <mo>=</mo>
  <mn>3</mn>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;</mtext>
  </mrow>
  <mi>c</mi>
  <mo>=</mo>
  <mn>2.5</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{2}{{\text{e}}^{{x^2} - 1}} + 2.5">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>−</mo>
        <mn>1</mn>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>2.5</mn>
</math></span> &nbsp; &nbsp; <strong><em>A1</em></strong> &nbsp; &nbsp; <strong><em>N2</em></strong></p>
<p><strong><em>[3 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>A cylinder with radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
  <mi>r</mi>
</math></span> and height <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
  <mi>h</mi>
</math></span> is 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 sum of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
  <mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
  <mi>h</mi>
</math></span> for this cylinder is 12 cm.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an equation for the area, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
  <mi>A</mi>
</math></span>, of the <strong>curved</strong> surface in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
  <mi>r</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}A}}{{{\text{d}}r}}">
  <mfrac>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>A</mi>
    </mrow>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>r</mi>
    </mrow>
  </mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
  <mi>r</mi>
</math></span> when the area of the curved surface is maximized.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2\pi r\left( {12 - r} \right)">
  <mi>A</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>12</mn>
      <mo>−</mo>
      <mi>r</mi>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>  <strong>OR</strong>  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 24\pi r - 2\pi {r^2}">
  <mi>A</mi>
  <mo>=</mo>
  <mn>24</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mo>−</mo>
  <mn>2</mn>
  <mi>π</mi>
  <mrow>
    <msup>
      <mi>r</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span>        <em><strong>(A1)(M1)  (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r + h = 12">
  <mi>r</mi>
  <mo>+</mo>
  <mi>h</mi>
  <mo>=</mo>
  <mn>12</mn>
</math></span>  or  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 12 - r">
  <mi>h</mi>
  <mo>=</mo>
  <mn>12</mn>
  <mo>−</mo>
  <mi>r</mi>
</math></span>  seen. Award <em><strong>(M1)</strong></em> for correctly substituting into curved surface area of a cylinder. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2\pi r\left( {12 - r} \right)">
  <mi>A</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>12</mn>
      <mo>−</mo>
      <mi>r</mi>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>  <strong>OR </strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 24\pi r - 2\pi {r^2}">
  <mi>A</mi>
  <mo>=</mo>
  <mn>24</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mo>−</mo>
  <mn>2</mn>
  <mi>π</mi>
  <mrow>
    <msup>
      <mi>r</mi>
      <mn>2</mn>
    </msup>
  </mrow>
</math></span>.</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi  - 4\pi r">
  <mn>24</mn>
  <mi>π</mi>
  <mo>−</mo>
  <mn>4</mn>
  <mi>π</mi>
  <mi>r</mi>
</math></span>       <em><strong>(A1)</strong></em><strong>(ft)<em>(A1)</em>(ft)</strong><em><strong>  (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(A1)</strong></em><strong>(ft)</strong> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi">
  <mn>24</mn>
  <mi>π</mi>
</math></span> and  <em><strong>(A1)</strong></em><strong>(ft)</strong> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4\pi r">
  <mo>−</mo>
  <mn>4</mn>
  <mi>π</mi>
  <mi>r</mi>
</math></span> . Follow through from part (a). Award at most <em><strong>(A1)</strong></em><strong>(ft)<em>(A0)</em></strong> if additional terms are seen.</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi  - 4\pi r = 0">
  <mn>24</mn>
  <mi>π</mi>
  <mo>−</mo>
  <mn>4</mn>
  <mi>π</mi>
  <mi>r</mi>
  <mo>=</mo>
  <mn>0</mn>
</math></span>       <strong><em>(M1)</em></strong></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for setting <em>their</em> part (b) equal to zero.</p>
<p style="text-align: left;">6 (cm)       <strong><em>(A1)</em>(ft)</strong><em><strong>  (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Follow through from part (b).</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A quadratic function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = a(x - p)(x - 3)">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mi>a</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo>−<!-- − --></mo>
  <mi>p</mi>
  <mo stretchy="false">)</mo>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo>−<!-- − --></mo>
  <mn>3</mn>
  <mo stretchy="false">)</mo>
</math></span>. The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> has axis of symmetry <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2.5">
  <mi>x</mi>
  <mo>=</mo>
  <mn>2.5</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
  <mi>y</mi>
</math></span>-intercept at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }} - 6)">
  <mo stretchy="false">(</mo>
  <mn>0</mn>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;</mtext>
  </mrow>
  <mo>−<!-- − --></mo>
  <mn>6</mn>
  <mo stretchy="false">)</mo>
</math></span></p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = kx - 5"> <mi>y</mi> <mo>=</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mn>5</mn> </math></span> is a tangent to the curve of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>. Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using <em>x</em>-intercept)</strong></p>
<p>determining that 3 is an <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercept     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - 3 = 0"> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>0</mn> </math></span>, <img src="images/Schermafbeelding_2017-08-11_om_13.55.43.png" alt="M17/5/MATME/SP1/ENG/TZ1/09.a/M"></p>
<p>valid approach     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 - 2.5,{\text{ }}\frac{{p + 3}}{2} = 2.5"> <mn>3</mn> <mo>−</mo> <mn>2.5</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mi>p</mi> <mo>+</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mn>2.5</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </math></span>     <strong><em>A1</em></strong>     <strong><em>N2</em></strong></p>
<p><strong>METHOD 2 (expanding <em>f </em>(<em>x</em>)) </strong></p>
<p>correct expansion (accept absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>)     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{x^2} - a(3 + p)x + 3ap,{\text{ }}{x^2} - (3 + p)x + 3p"> <mi>a</mi> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>a</mi> <mi>p</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>p</mi> </math></span></p>
<p>valid approach involving equation of axis of symmetry     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - b}}{{2a}} = 2.5,{\text{ }}\frac{{a(3 + p)}}{{2a}} = \frac{5}{2},{\text{ }}\frac{{3 + p}}{2} = \frac{5}{2}"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mo>+</mo> <mi>p</mi> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </math></span>     <strong><em>A1</em></strong>     <strong><em>N2</em></strong></p>
<p><strong>METHOD 3 (using derivative)</strong></p>
<p>correct derivative (accept absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>)     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a(2x - 3 - p),{\text{ }}2x - 3 - p"> <mi>a</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mi>p</mi> </math></span></p>
<p>valid approach     <strong>(<em>M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(2.5) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>2.5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </math></span>     <strong><em>A1</em></strong>     <strong><em>N2</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>attempt to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }} - 6)"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> </math></span>     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6 = a(0 - 2)(0 - 3),{\text{ }}0 = a( - 8)( - 9),{\text{ }}a{(0)^2} - 5a(0) + 6a =  - 6"><mo>−</mo><mn>6</mn><mo>=</mo><mi>a</mi><mo>(</mo><mn>0</mn><mo>−</mo><mn>2</mn><mo>)</mo><mo>(</mo><mn>0</mn><mo>−</mo><mn>3</mn><mo>)</mo><mo>,</mo><mtext> </mtext><mi>a</mi><mrow><mo>(</mo><mn>0</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mo>−</mo><mn>5</mn><mi>a</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mn>6</mn><mi>a</mi><mo>=</mo><mo>−</mo><mn>6</mn></math></span></p>
<p>correct working     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6 = 6a"> <mo>−</mo> <mn>6</mn> <mo>=</mo> <mn>6</mn> <mi>a</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a =  - 1"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>     <strong><em>A1</em></strong>     <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using discriminant)</strong></p>
<p>recognizing tangent intersects curve once     <strong><em>(M1)</em></strong></p>
<p>recognizing one solution when discriminant = 0     <strong><em>M1</em></strong></p>
<p>attempt to set up equation     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g = f,{\text{ }}kx - 5 =  - {x^2} + 5x - 6"> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>6</mn> </math></span></p>
<p>rearranging their equation to equal zero     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 5x + kx + 1 = 0"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>5</mn> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct discriminant (if seen explicitly, not just in quadratic formula)     <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(k - 5)^2} - 4,{\text{ }}25 - 10k + {k^2} - 4"> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>25</mn> <mo>−</mo> <mn>10</mn> <mi>k</mi> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> </math></span></p>
<p>correct working     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k - 5 =  \pm 2,{\text{ }}(k - 3)(k - 7) = 0,{\text{ }}\frac{{10 \pm \sqrt {100 - 4 \times 21} }}{2}"> <mi>k</mi> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mo>±</mo> <mn>2</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>10</mn> <mo>±</mo> <msqrt> <mn>100</mn> <mo>−</mo> <mn>4</mn> <mo>×</mo> <mn>21</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 3,{\text{ }}7"> <mi>k</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>7</mn> </math></span>     <strong><em>A1A1</em></strong>     <strong><em>N0</em></strong></p>
<p><strong>METHOD 2 (using derivatives)</strong></p>
<p>attempt to set up equation     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g = f,{\text{ }}kx - 5 =  - {x^2} + 5x - 6"> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>6</mn> </math></span></p>
<p>recognizing derivative/slope are equal     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’ = {m_T},{\text{ }}f' = k"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow> <msub> <mi>m</mi> <mi>T</mi> </msub> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>k</mi> </math></span></p>
<p>correct derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2x + 5"> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> </math></span></p>
<p>attempt to set up equation in terms of either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>     <strong><em>M1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 2x + 5)x - 5 =  - {x^2} + 5x - 6,{\text{ }}k\left( {\frac{{5 - k}}{2}} \right) - 5 =  - {\left( {\frac{{5 - k}}{2}} \right)^2} + 5\left( {\frac{{5 - k}}{2}} \right) - 6"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>6</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>5</mn> <mo>−</mo> <mi>k</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>5</mn> <mo>−</mo> <mi>k</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>5</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>5</mn> <mo>−</mo> <mi>k</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>6</mn> </math></span></p>
<p>rearranging their equation to equal zero     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 1 = 0,{\text{ }}{k^2} - 10k + 21 = 0"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>10</mn> <mi>k</mi> <mo>+</mo> <mn>21</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct working     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x =  \pm 1,{\text{ }}(k - 3)(k - 7) = 0,{\text{ }}\frac{{10 \pm \sqrt {100 - 4 \times 21} }}{2}"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>10</mn> <mo>±</mo> <msqrt> <mn>100</mn> <mo>−</mo> <mn>4</mn> <mo>×</mo> <mn>21</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 3,{\text{ }}7"> <mi>k</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>7</mn> </math></span>     <strong><em>A1A1</em></strong>     <strong><em>N0</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>ln</mi><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>4</mn></math>.</p>
<p>The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> which crosses the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, with&nbsp;coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo>&nbsp;</mo><mn>0</mn><mo>)</mo></math>. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the tangent to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</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>Given that the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math>,&nbsp;find 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>B</mtext></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <em><strong>&nbsp;(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mn>0</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn><mfenced><mrow><mo>=</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>17</mn></math>&nbsp; OR&nbsp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>±</mo><msqrt><mn>17</mn></msqrt></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msqrt><mn>17</mn></msqrt></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to differentiate&nbsp;(must include <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi></math> and/or&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn></mrow></mfrac></math>)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn></mrow></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>A1</strong></em></p>
<p>setting their derivative&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn><mo>=</mo><mn>6</mn><mi>x</mi></math>&nbsp; OR&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>=</mo><mn>0</mn></math>&nbsp;(or equivalent)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>A1</strong></em></p>
<p>valid attempt to solve their quadratic&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>&nbsp;(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>8</mn></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> if the candidate’s final answer includes additional solutions&nbsp;(such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>,</mo><mo>&nbsp;</mo><mn>8</mn></math>).</p>
<p>&nbsp;</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>Consider <em>f</em>(<em>x</em>), <em>g</em>(<em>x</em>) and <em>h</em>(<em>x</em>), for x∈<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{R}"> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span> where <em>h</em>(<em>x</em>) = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {f \circ g} \right)"> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mrow> <mo>)</mo> </mrow> </math></span>(<em>x</em>).</p>
<p>Given that <em>g</em>(3) = 7 , <em>g′</em> (3) = 4 and <em>f ′ </em>(7) = −5 , find the gradient of the normal to the curve of <em>h</em> at <em>x</em> = 3.</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>recognizing the need to find <em>h′</em>    <em><strong>  (M1)</strong></em></p>
<p>recognizing the need to find <em>h′ </em>(3) (seen anywhere)    <em><strong>  (M1)</strong></em></p>
<p>evidence of choosing chain rule      <em><strong>  (M1)</strong></em></p>
<p><em>eg   </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},\,\,f'\left( {g\left( 3 \right)} \right) \times g'\left( 3 \right),\,\,f'\left( g \right) \times g'"> <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> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>g</mi> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> </math></span></p>
<p>correct working       <em><strong>(A1)</strong></em></p>
<p><em>eg  </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 7 \right) \times 4,\,\, - 5 \times 4"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> <mo>×</mo> <mn>4</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mn>5</mn> <mo>×</mo> <mn>4</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h'\left( 3 \right) =  - 20"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>20</mn> </math></span>      <strong><em>(A1)</em></strong></p>
<p>evidence of taking <strong>their</strong> negative reciprocal for normal       <em><strong>(M1)</strong></em></p>
<p><em>eg  </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{{h'\left( 3 \right)}},\,\,{m_1}{m_2} =  - 1"> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <msup> <mi>h</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span></p>
<p>gradient of normal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{20}}"> <mfrac> <mn>1</mn> <mrow> <mn>20</mn> </mrow> </mfrac> </math></span>      <em><strong>A1 N4</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Particle A travels in a straight line such that its displacement, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> metres, from a fixed origin&nbsp;after <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>8</mn><mi>t</mi><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>10</mn></math>, 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;">Particle A starts at the origin and passes through the origin again when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>p</mi></math>.</p>
</div>

<div class="specification">
<p>Particle A changes direction when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>q</mi></math>.</p>
</div>

<div class="specification">
<p>The total distance travelled by particle A is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the displacement of particle A from the origin when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>q</mi></math>.</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>Find the distance of particle A from the origin when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn></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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></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>A second particle, particle B, travels along the same straight line such that its velocity is given&nbsp;by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>14</mn><mo>-</mo><mn>2</mn><mi>t</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>k</mi></math>, the distance travelled by particle B is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>setting&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>0</mn></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mi>t</mi><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mfenced><mrow><mn>8</mn><mo>-</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>8</mn></math>&nbsp; (accept&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>8</mn><mo>,</mo><mo>&nbsp;</mo><mfenced><mrow><mn>8</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>)&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> if the candidate’s final answer includes additional solutions&nbsp;(such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>8</mn></math>).</p>
<p>&nbsp;</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>recognition that when particle changes direction <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>0</mn></math> OR local maximum on&nbsp;graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> OR vertex of parabola&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>4</mn></math>&nbsp;(accept&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn></math>)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</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>substituting their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>t</mi></mfenced></math> OR integrating <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mi>t</mi></mfenced></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>displacement</mtext><mo>=</mo><mn>16</mn><mo> </mo><mo>(</mo><mtext>m</mtext><mo>)</mo></math> &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mn>10</mn></mfenced><mo>=</mo><mo>-</mo><mn>20</mn></math>&nbsp; OR&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>distance</mtext><mo>=</mo><mfenced open="|" close="|"><mrow><mtext>s</mtext><mfenced><mi>t</mi></mfenced></mrow></mfenced></math>&nbsp;OR&nbsp;integrating&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mi>t</mi></mfenced></math>&nbsp;from&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>&nbsp;to&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>distance</mtext><mo>=</mo><mn>20</mn><mo> </mo><mo>(</mo><mtext>m</mtext><mo>)</mo></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</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"><mn>16</mn></math> forward <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mo>&nbsp;</mo><mn>36</mn></math> backward&nbsp; OR&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mo>+</mo><mn>16</mn><mo>+</mo><mn>20</mn></math>&nbsp; OR&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>10</mn></munderover><mfenced open="|" close="|"><mrow><mi>v</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mo>d</mo><mi>t</mi></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>52</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</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><strong>METHOD 1</strong></p>
<p>graphical method with triangles on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mi>t</mi></mfenced></math> graph&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn><mo>+</mo><mfenced><mfrac><mrow><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>52</mn><mo>,</mo><mo>&nbsp;</mo><mi>x</mi><mo>=</mo><msqrt><mn>3</mn></msqrt></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>7</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</p>
<p><strong>METHOD 2</strong></p>
<p>recognition that distance&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>∫</mo><mfenced open="|" close="|"><mrow><mi>v</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mo>d</mo><mi>t</mi></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>7</mn></munderover><mfenced><mrow><mn>14</mn><mo>-</mo><mn>2</mn><mi>t</mi></mrow></mfenced><mo>d</mo><mi>t</mi><mo>+</mo><munderover><mo>∫</mo><mn>7</mn><mi>k</mi></munderover><mfenced><mrow><mn>2</mn><mi>t</mi><mo>-</mo><mn>14</mn></mrow></mfenced><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mn>14</mn><mi>t</mi><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mn>0</mn><mn>7</mn></msubsup><mo>+</mo><msubsup><mfenced open="[" close="]"><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>14</mn><mi>t</mi></mrow></mfenced><mn>7</mn><mi>k</mi></msubsup></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn><mfenced><mn>7</mn></mfenced><mo>-</mo><msup><mn>7</mn><mn>2</mn></msup><mo>+</mo><mfenced><mrow><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>14</mn><mi>k</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><msup><mn>7</mn><mn>2</mn></msup><mo>-</mo><mn>14</mn><mfenced><mn>7</mn></mfenced></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>52</mn></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>7</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></math>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.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>
<br><hr><br><div class="specification">
<p>Maria owns a cheese factory. The amount of cheese, in kilograms, Maria sells in one week, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q">
  <mi>Q</mi>
</math></span>, is given by</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q = 882 - 45p">
  <mi>Q</mi>
  <mo>=</mo>
  <mn>882</mn>
  <mo>−<!-- − --></mo>
  <mn>45</mn>
  <mi>p</mi>
</math></span>,</p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
  <mi>p</mi>
</math></span> is the price of a kilogram of cheese in euros (EUR).</p>
</div>

<div class="specification">
<p>Maria earns <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(p - 6.80){\text{ EUR}}">
  <mo stretchy="false">(</mo>
  <mi>p</mi>
  <mo>−<!-- − --></mo>
  <mn>6.80</mn>
  <mo stretchy="false">)</mo>
  <mrow>
    <mtext>&nbsp;EUR</mtext>
  </mrow>
</math></span> for each kilogram of cheese sold.</p>
</div>

<div class="specification">
<p>To calculate her weekly profit <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W">
  <mi>W</mi>
</math></span>, in EUR, Maria multiplies the amount of cheese she sells by the amount she earns per kilogram.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of cheese is 8 EUR.</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 how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W">
  <mi>W</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
  <mi>p</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the price, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
  <mi>p</mi>
</math></span>, that will give Maria the highest weekly profit.</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>522 (kg)     <strong><em>(A1)     (C1)</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="522(8 - 6.80)">
  <mn>522</mn>
  <mo stretchy="false">(</mo>
  <mn>8</mn>
  <mo>−</mo>
  <mn>6.80</mn>
  <mo stretchy="false">)</mo>
</math></span> or equivalent     <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(M1) </em></strong>for multiplying their answer to part (a) by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(8 - 6.80)">
  <mo stretchy="false">(</mo>
  <mn>8</mn>
  <mo>−</mo>
  <mn>6.80</mn>
  <mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p>626 (EUR) (626.40)     <strong><em>(A1)</em>(ft)     <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Follow through from part (a).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(W = ){\text{ }}(882 - 45p)(p - 6.80)">
  <mo stretchy="false">(</mo>
  <mi>W</mi>
  <mo>=</mo>
  <mo stretchy="false">)</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mo stretchy="false">(</mo>
  <mn>882</mn>
  <mo>−</mo>
  <mn>45</mn>
  <mi>p</mi>
  <mo stretchy="false">)</mo>
  <mo stretchy="false">(</mo>
  <mi>p</mi>
  <mo>−</mo>
  <mn>6.80</mn>
  <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="(W = ) - 45{p^2} + 1188p - 5997.6">
  <mo stretchy="false">(</mo>
  <mi>W</mi>
  <mo>=</mo>
  <mo stretchy="false">)</mo>
  <mo>−</mo>
  <mn>45</mn>
  <mrow>
    <msup>
      <mi>p</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>1188</mn>
  <mi>p</mi>
  <mo>−</mo>
  <mn>5997.6</mn>
</math></span>     <strong><em>(A1)     (C1)</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>sketch of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W">
  <mi>W</mi>
</math></span> with some indication of the maximum     <strong><em>(M1)</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 90p + 1188 = 0">
  <mo>−</mo>
  <mn>90</mn>
  <mi>p</mi>
  <mo>+</mo>
  <mn>1188</mn>
  <mo>=</mo>
  <mn>0</mn>
</math></span>     <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(M1) </em></strong>for equating the correct derivative of their part (c) to zero.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(p = ){\text{ }}\frac{{ - 1188}}{{2 \times ( - 45)}}">
  <mo stretchy="false">(</mo>
  <mi>p</mi>
  <mo>=</mo>
  <mo stretchy="false">)</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mfrac>
    <mrow>
      <mo>−</mo>
      <mn>1188</mn>
    </mrow>
    <mrow>
      <mn>2</mn>
      <mo>×</mo>
      <mo stretchy="false">(</mo>
      <mo>−</mo>
      <mn>45</mn>
      <mo stretchy="false">)</mo>
    </mrow>
  </mfrac>
</math></span>     <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Award <strong><em>(M1) </em></strong>for correct substitution into the formula for axis of symmetry.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(p = ){\text{ }}13.2{\text{ (EUR)}}">
  <mo stretchy="false">(</mo>
  <mi>p</mi>
  <mo>=</mo>
  <mo stretchy="false">)</mo>
  <mrow>
    <mtext> </mtext>
  </mrow>
  <mn>13.2</mn>
  <mrow>
    <mtext> (EUR)</mtext>
  </mrow>
</math></span>     <strong><em>(A1)</em>(ft)     <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:     </strong>Follow through from their part (c), if the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
  <mi>p</mi>
</math></span> is such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6.80 &lt; p &lt; 19.6">
  <mn>6.80</mn>
  <mo>&lt;</mo>
  <mi>p</mi>
  <mo>&lt;</mo>
  <mn>19.6</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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>The point A has coordinates (4 , −8) and the point B has coordinates (−2 , 4).</p>
</div>

<div class="specification">
<p>The point D has coordinates (−3 , 1).</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of C, the midpoint of line segment AB.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the gradient of the line DC.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the line DC. Write your answer in the form <em>ax</em> + <em>by</em> + <em>d</em> = 0 where <em>a</em> , <em>b</em> and <em>d</em> are integers.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(1, −2)    <em><strong>(A1)(A1) (C2)</strong></em><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 1 and <em><strong>(A1)</strong></em> for −2, seen as a coordinate pair.</p>
<p>Accept <em>x</em> = 1, <em>y</em> = −2. Award <em><strong>(A1)(A0)</strong></em> if <em>x</em> and <em>y</em> coordinates are reversed.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 - \left( { - 2} \right)}}{{ - 3 - 1}}">
  <mfrac>
    <mrow>
      <mn>1</mn>
      <mo>−</mo>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mo>−</mo>
          <mn>2</mn>
        </mrow>
        <mo>)</mo>
      </mrow>
    </mrow>
    <mrow>
      <mo>−</mo>
      <mn>3</mn>
      <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 correct substitution, of their part (a), into gradient formula.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" =  - \frac{3}{4}\,\,\,\left( { - 0.75} \right)">
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>4</mn>
  </mfrac>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mo>−</mo>
      <mn>0.75</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>     <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>  (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </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 - 1 =  - \frac{3}{4}\left( {x + 3} \right)">
  <mi>y</mi>
  <mo>−</mo>
  <mn>1</mn>
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>4</mn>
  </mfrac>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mi>x</mi>
      <mo>+</mo>
      <mn>3</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>  <em><strong>OR </strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y + 2 =  - \frac{3}{4}\left( {x - 1} \right)">
  <mi>y</mi>
  <mo>+</mo>
  <mn>2</mn>
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>4</mn>
  </mfrac>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mi>x</mi>
      <mo>−</mo>
      <mn>1</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>  <em><strong>OR</strong></em>  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y =  - \frac{3}{4}x - \frac{5}{4}">
  <mi>y</mi>
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>4</mn>
  </mfrac>
  <mi>x</mi>
  <mo>−</mo>
  <mfrac>
    <mn>5</mn>
    <mn>4</mn>
  </mfrac>
</math></span>      <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of their part (b) and a given point.</p>
<p><em><strong>OR</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 =  - \frac{3}{4} \times  - 3 + c">
  <mn>1</mn>
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>4</mn>
  </mfrac>
  <mo>×</mo>
  <mo>−</mo>
  <mn>3</mn>
  <mo>+</mo>
  <mi>c</mi>
</math></span>  <em><strong>OR</strong></em>  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 =  - \frac{3}{4} \times 1 + c">
  <mo>−</mo>
  <mn>2</mn>
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>4</mn>
  </mfrac>
  <mo>×</mo>
  <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 correct substitution of their part (b) and a given point.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x + 4y + 5 = 0">
  <mn>3</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>4</mn>
  <mi>y</mi>
  <mo>+</mo>
  <mn>5</mn>
  <mo>=</mo>
  <mn>0</mn>
</math></span>  (accept any integer multiple, including negative multiples)    <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from parts (a) and (b). Where the gradient in part (b) is found to be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{0}">
  <mfrac>
    <mn>5</mn>
    <mn>0</mn>
  </mfrac>
</math></span>, award at most <em><strong>(M1)(A0)</strong></em> for either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x =  - 3">
  <mi>x</mi>
  <mo>=</mo>
  <mo>−</mo>
  <mn>3</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 3 = 0">
  <mi>x</mi>
  <mo>+</mo>
  <mn>3</mn>
  <mo>=</mo>
  <mn>0</mn>
</math></span>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A potter sells <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span> vases per month.</p>
<p>His monthly profit in Australian dollars (AUD) can be modelled by</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="P\left( x \right) =&nbsp; - \frac{1}{5}{x^3} + 7{x^2} - 120{\text{,}}\,\,x \geqslant 0.">
  <mi>P</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mo>−<!-- − --></mo>
  <mfrac>
    <mn>1</mn>
    <mn>5</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>3</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>7</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>−<!-- − --></mo>
  <mn>120</mn>
  <mrow>
    <mtext>,</mtext>
  </mrow>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mi>x</mi>
  <mo>⩾<!-- ⩾ --></mo>
  <mn>0.</mn>
</math></span></p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
  <mi>P</mi>
</math></span> if no vases are sold.</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>Differentiate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( x \right)">
  <mi>P</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><strong>Hence</strong>, find the number of vases that will maximize the profit.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>−120 (AUD)       <em><strong>(A1)   (C1)</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{3}{5}{x^2} + 14x">
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>5</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>14</mn>
  <mi>x</mi>
</math></span>     <em><strong>(A1)(A1)     (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for each correct term. Award at most <em><strong>(A1)(A0)</strong></em> for extra terms seen.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{3}{5}{x^2} + 14x = 0">
  <mo>−</mo>
  <mfrac>
    <mn>3</mn>
    <mn>5</mn>
  </mfrac>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>14</mn>
  <mi>x</mi>
  <mo>=</mo>
  <mn>0</mn>
</math></span>     <strong><em><span style="font-family: 'Verdana',sans-serif;">(M1)</span></em></strong></span></p>
<p style="text-align: start;"><strong><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">Note:</span></strong><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"> Award <strong><em><span style="font-family: 'Verdana',sans-serif;">(M1)</span></em></strong> for equating their derivative to zero.</span></p>
<p style="text-align: start;"><strong><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">OR</span></strong></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">sketch of their derivative (approximately correct shape) with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
  <mi>x</mi>
</math></span>-intercept seen       <strong><em><span style="font-family: 'Verdana',sans-serif;">(M1)</span></em></strong></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="23\frac{1}{3}\,\,\,\left( {23.3{\text{,}}\,\,23.3333 \ldots {\text{,}}\,\,\frac{{70}}{3}} \right)">
  <mn>23</mn>
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>23.3</mn>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>23.3333</mn>
      <mo>…</mo>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mfrac>
        <mrow>
          <mn>70</mn>
        </mrow>
        <mn>3</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>      <strong><em>(A1)</em>(ft)</strong></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><strong>Note:</strong> Award <em><strong>(C2)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="23\frac{1}{3}\,\,\,\left( {23.3{\text{,}}\,\,23.3333 \ldots {\text{,}}\,\,\frac{{70}}{3}} \right)">
  <mn>23</mn>
  <mfrac>
    <mn>1</mn>
    <mn>3</mn>
  </mfrac>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mspace width="thinmathspace"></mspace>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mn>23.3</mn>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>23.3333</mn>
      <mo>…</mo>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mfrac>
        <mrow>
          <mn>70</mn>
        </mrow>
        <mn>3</mn>
      </mfrac>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span> seen without working.</span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">23      <strong><em>(A1)</em>(ft)   <em><span style="font-family: 'Verdana',sans-serif;">(C3)</span></em>  </strong>    </span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><strong>Note:</strong> Follow through from part (b).</span></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 15 - {x^2}"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>15</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}"> <mi>x</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>. The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and the rectangle OABC, where A is on the negative <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis, B is on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>, and C is on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis.</p>
<p><img src="images/Schermafbeelding_2018-02-11_om_13.13.04.png" alt="N17/5/MATME/SP1/ENG/TZ0/06"></p>
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of A that gives the maximum area of OABC.</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 find the area of OABC     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{OA}} \times {\text{OC, }}x \times f(x),{\text{ }}f(x) \times ( - x)"> <mrow> <mtext>OA</mtext> </mrow> <mo>×</mo> <mrow> <mtext>OC, </mtext> </mrow> <mi>x</mi> <mo>×</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span></p>
<p>correct expression for area in one variable     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area}} = x(15 - {x^2}),{\text{ }}15x - {x^3},{\text{ }}{x^3} - 15x"> <mrow> <mtext>area</mtext> </mrow> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>15</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>15</mn> <mi>x</mi> </math></span></p>
<p>valid approach to find maximum <strong>area</strong> (seen anywhere)     <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A’(x) = 0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct derivative     <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 - 3{x^2},{\text{ }}(15 - {x^2}) + x( - 2x) = 0,{\text{ }} - 15 + 3{x^2}"> <mn>15</mn> <mo>−</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mn>15</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>15</mn> <mo>+</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span></p>
<p>correct working     <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15 = 3{x^2},{\text{ }}{x^2} = 5,{\text{ }}x = \sqrt 5 "> <mn>15</mn> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>=</mo> <msqrt> <mn>5</mn> </msqrt> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x =  - \sqrt 5 {\text{ }}\left( {{\text{accept A}}\left( { - \sqrt 5 ,{\text{ }}0} \right)} \right)"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msqrt> <mn>5</mn> </msqrt> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>accept A</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msqrt> <mn>5</mn> </msqrt> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>     <strong><em>A2     N3</em></strong></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>The derivative of a function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'(x) = 2{{\text{e}}^{ - 3x}}">
  <msup>
    <mi>f</mi>
    <mo>′</mo>
  </msup>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mn>2</mn>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mo>−</mo>
        <mn>3</mn>
        <mi>x</mi>
      </mrow>
    </msup>
  </mrow>
</math></span>. The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
  <mi>f</mi>
</math></span> passes through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{3}{\text{,}}\,\,5} \right)">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>1</mn>
        <mn>3</mn>
      </mfrac>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>.</p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</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>recognizing to integrate   <em><strong>(M1)</strong></em></p>
<p><em>eg   </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f'} ">
  <mo>∫</mo>
  <mrow>
    <msup>
      <mi>f</mi>
      <mo>′</mo>
    </msup>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {2{{\text{e}}^{ - 3x}}{\text{d}}x} ">
  <mo>∫</mo>
  <mrow>
    <mn>2</mn>
    <mrow>
      <msup>
        <mrow>
          <mtext>e</mtext>
        </mrow>
        <mrow>
          <mo>−</mo>
          <mn>3</mn>
          <mi>x</mi>
        </mrow>
      </msup>
    </mrow>
    <mrow>
      <mtext>d</mtext>
    </mrow>
    <mi>x</mi>
  </mrow>
</math></span>,  <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}u =  - 3">
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
  <mo>=</mo>
  <mo>−</mo>
  <mn>3</mn>
</math></span></p>
<p>correct integral (do not penalize for missing +<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
  <mi>C</mi>
</math></span>)     <em><strong>(A2)</strong><br></em></p>
<p><em>eg   </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{2}{3}{{\text{e}}^{ - 3x}} + C">
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mo>−</mo>
        <mn>3</mn>
        <mi>x</mi>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>C</mi>
</math></span></p>
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{3}{\text{,}}\,\,5} \right)">
  <mrow>
    <mo>(</mo>
    <mrow>
      <mfrac>
        <mn>1</mn>
        <mn>3</mn>
      </mfrac>
      <mrow>
        <mtext>,</mtext>
      </mrow>
      <mspace width="thinmathspace"></mspace>
      <mspace width="thinmathspace"></mspace>
      <mn>5</mn>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span> (in any order) into <strong>their</strong> integrated expression (must have +<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
  <mi>C</mi>
</math></span>)      <em><strong>M1</strong></em></p>
<p><em>eg</em>   <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{2}{3}{{\text{e}}^{ - 3\left( {1/3} \right)}} + C = 5">
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mo>−</mo>
        <mn>3</mn>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mn>1</mn>
            <mrow>
              <mo>/</mo>
            </mrow>
            <mn>3</mn>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mi>C</mi>
  <mo>=</mo>
  <mn>5</mn>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if they substitute into original or differentiated function.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) =  - \frac{2}{3}{{\text{e}}^{ - 3x}} + 5 + \frac{2}{3}{{\text{e}}^{ - 1}}">
  <mi>f</mi>
  <mo stretchy="false">(</mo>
  <mi>x</mi>
  <mo stretchy="false">)</mo>
  <mo>=</mo>
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mo>−</mo>
        <mn>3</mn>
        <mi>x</mi>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>5</mn>
  <mo>+</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mo>−</mo>
        <mn>1</mn>
      </mrow>
    </msup>
  </mrow>
</math></span> (or <em>any</em> equivalent form, <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{2}{3}{{\text{e}}^{ - 3x}} + 5 - \frac{2}{{ - 3{\text{e}}}}">
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mn>3</mn>
  </mfrac>
  <mrow>
    <msup>
      <mrow>
        <mtext>e</mtext>
      </mrow>
      <mrow>
        <mo>−</mo>
        <mn>3</mn>
        <mi>x</mi>
      </mrow>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>5</mn>
  <mo>−</mo>
  <mfrac>
    <mn>2</mn>
    <mrow>
      <mo>−</mo>
      <mn>3</mn>
      <mrow>
        <mtext>e</mtext>
      </mrow>
    </mrow>
  </mfrac>
</math></span>)               <em><strong>A1   N4</strong></em></p>
<p><em><strong>[5 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>The diagram shows part of the graph of a function <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>. The graph passes through point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}(1,{\text{ }}3)">
  <mrow>
    <mtext>A</mtext>
  </mrow>
  <mo stretchy="false">(</mo>
  <mn>1</mn>
  <mo>,</mo>
  <mrow>
    <mtext>&nbsp;</mtext>
  </mrow>
  <mn>3</mn>
  <mo stretchy="false">)</mo>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_06.22.37.png" alt="M17/5/MATSD/SP1/ENG/TZ2/13"></p>
</div>

<div class="specification">
<p>The tangent to the graph of <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> at A has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y =&nbsp; - 2x + 5">
  <mi>y</mi>
  <mo>=</mo>
  <mo>−<!-- − --></mo>
  <mn>2</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>5</mn>
</math></span>. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N">
  <mi>N</mi>
</math></span> be the normal to the graph of <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> at A.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(1)"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </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 the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N"> <mi>N</mi> </math></span>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ax + by + d = 0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{Z}"> <mi>d</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </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>Draw the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N"> <mi>N</mi> </math></span> on the diagram above.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>3     <strong><em>(A1)</em></strong>     <strong><em>(C1)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong>     Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 3"> <mi>y</mi> <mo>=</mo> <mn>3</mn> </math></span></p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 = 0.5(1) + c"> <mn>3</mn> <mo>=</mo> <mn>0.5</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </math></span><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><strong>OR</strong><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 3 = 0.5(x - 1)"> <mi>y</mi> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>0.5</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span>     <strong><em>(A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong>     Award <strong><em>(A1) </em></strong>for correct gradient, <strong><em>(A1) </em></strong>for correct substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}(1,{\text{ }}3)"> <mrow> <mtext>A</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> </math></span> in the equation of line.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - 2y + 5 = 0"> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mn>5</mn> <mo>=</mo> <mn>0</mn> </math></span> or any integer multiple     <strong><em>(A1)</em>(ft)</strong>     <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong>     Award <strong><em>(A1)</em>(ft) </strong>for their equation correctly rearranged in the indicated form.</p>
<p>The candidate’s answer <strong>must </strong>be an equation for this mark.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-16_om_08.26.38.png" alt="M17/5/MATSD/SP1/ENG/TZ2/13.c/M">     <strong><em>(M1)(A1)</em>(ft)</strong>     <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong>     Award <strong><em>M1) </em></strong>for a straight line, with positive gradient, passing through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }}3)"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> </math></span>, <strong><em>(A1)</em>(ft) </strong>for line (or extension of their line) passing approximately through 2.5 or their intercept with the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>6</mn><mo>+</mo><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>4</mn><mi>π</mi></math>.</p>
<p>The following diagram shows 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>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> touches the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, as shown. The shaded region is&nbsp;enclosed by 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> and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, between the points&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
</div>

<div class="specification">
<p>The right cone in the following diagram has a total surface area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi mathvariant="normal">π</mi></math>, equal to the shaded&nbsp;area in the previous diagram.</p>
<p>The cone has a base radius of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>, height <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, and slant height <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></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>Show that the area of the shaded region is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi mathvariant="normal">π</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the volume of the cone.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>+</mo><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> (or setting their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math>)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> (or&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi mathvariant="normal">π</mi><mo>,</mo><mo>&nbsp;</mo><mi>x</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <em><strong>A1A1</strong></em></p>
<p>&nbsp;</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to integrate&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi mathvariant="normal">π</mi><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow></munderover><mfenced><mrow><mn>6</mn><mo>+</mo><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>d</mo><mi>x</mi></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced><mi mathvariant="normal">π</mi><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow></msubsup></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1A1</strong></em></p>
<p>substitute their limits into their integrated expression and subtract&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>18</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>3</mn><mi mathvariant="normal">π</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>6</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo> </mo><mi mathvariant="normal">π</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>6</mn><mfenced><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow></mfenced><mo>+</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>6</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>0</mn></mrow></mfenced><mo>&nbsp;</mo><mfenced><mrow><mo>=</mo><mn>18</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>6</mn><mi mathvariant="normal">π</mi></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>area<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo>=</mo><mn>12</mn><mi mathvariant="normal">π</mi></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>AG</strong></em></p>
<p>&nbsp;</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute into formula for surface area (including base)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><msup><mn>2</mn><mn>2</mn></msup></mfenced><mo>+</mo><mi mathvariant="normal">π</mi><mfenced><mn>2</mn></mfenced><mfenced><mi>l</mi></mfenced><mo>=</mo><mn>12</mn><mi mathvariant="normal">π</mi></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>2</mn><mtext>π</mtext><mi>l</mi><mo>=</mo><mn>12</mn><mi mathvariant="normal">π</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mtext>π</mtext><mi>l</mi><mo>=</mo><mn>8</mn><mi mathvariant="normal">π</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>=</mo><mn>4</mn></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</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>valid attempt to find the height of the cone &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>(M1)</strong></em></p>
<p>e.g.&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mtext>their</mtext><mo>&nbsp;</mo><mi>l</mi></mrow></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><msqrt><mn>12</mn></msqrt><mo>&nbsp;</mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<em><strong>(A1)</strong></em></p>
<p>attempt to use&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mi>h</mi></math>&nbsp;with their values substituted&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><mfenced><msup><mn>2</mn><mn>2</mn></msup></mfenced><mfenced><msqrt><mn>12</mn></msqrt></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mfrac><mrow><mn>4</mn><mi mathvariant="normal">π</mi><msqrt><mn>12</mn></msqrt></mrow><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>8</mn><mi mathvariant="normal">π</mi><msqrt><mn>3</mn></msqrt></mrow><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mn>8</mn><mi mathvariant="normal">π</mi></mrow><msqrt><mn>3</mn></msqrt></mfrac></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<em><strong>A1</strong></em></p>
<p>&nbsp;</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a ball attached to the end of a spring, which is suspended from a ceiling.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The height, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> metres, of the ball above the ground at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds after being released can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>

<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the height of the ball above the ground when it is released.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the minimum height of the ball above the ground.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the ball takes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> seconds to return to its initial height above the ground for the first time.</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>For the first 2 seconds of its motion, determine the amount of time that the ball is less than&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>8</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math>&nbsp;metres above the ground.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the rate of change of the ball’s height above the ground when&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math>.&nbsp;Give your answer in the form&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi mathvariant="normal">π</mi><msqrt><mi>q</mi></msqrt><mo> </mo><msup><mi>ms</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>&nbsp;where&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>attempts to find&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>0</mn></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>cos</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn><mfenced><mrow><mo>=</mo><mn>2</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>2</mn></math> (m)&nbsp;(above the ground)&nbsp; &nbsp; &nbsp; &nbsp;<strong>A1</strong></p>
<p>&nbsp;</p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>uses the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced></math> which is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math>&nbsp; &nbsp; &nbsp;&nbsp; <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn></math> (m)</p>
<p>&nbsp;</p>
<p><strong>OR</strong></p>
<p>the amplitude of motion is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn></math> (m) and the mean position is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>8</mn></math> (m)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<strong>M1</strong></p>
<p>&nbsp;</p>
<p><strong>OR</strong></p>
<p>finds <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi mathvariant="normal">π</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced></math>, attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and determines that the minimum height above the ground occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mo>…</mo></math>&nbsp; &nbsp; &nbsp; &nbsp; <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn></math> (m)</p>
<p>&nbsp;</p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>4</mn></math> (m) (above the ground)&nbsp; &nbsp; &nbsp; &nbsp;<strong>A1</strong></p>
<p>&nbsp;</p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>the ball is released from its maximum height and returns there a period later&nbsp; &nbsp; &nbsp; &nbsp;<strong>R1</strong></p>
<p>the period is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi mathvariant="normal">π</mi></mfrac><mfenced><mrow><mo>=</mo><mn>2</mn></mrow></mfenced><mo>&nbsp;</mo><mfenced><mi mathvariant="normal">s</mi></mfenced></math> &nbsp; &nbsp; &nbsp;<strong> A1</strong></p>
<p>&nbsp;</p>
<p><strong>OR</strong></p>
<p>attempts to solve&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>2</mn></math>&nbsp;for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>&nbsp;&nbsp; &nbsp; &nbsp;<strong>&nbsp;M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>…</mo></math>&nbsp;&nbsp; &nbsp; &nbsp;<strong>&nbsp;A1</strong></p>
<p>&nbsp;</p>
<p><strong>THEN</strong></p>
<p>so it takes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> seconds for the ball to return to its initial position for the first time&nbsp; &nbsp; &nbsp; &nbsp;<strong>AG</strong></p>
<p>&nbsp;</p>
<p><strong>[2 marks]</strong></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"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>8</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math>&nbsp; &nbsp; &nbsp; &nbsp;(M1)</strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;A1</strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mi>t</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math>&nbsp; &nbsp; &nbsp; &nbsp;(A1)</strong></p>
<p>&nbsp;</p>
<p><strong>Note: </strong>Accept extra correct positive solutions for <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mi>t</mi></math></strong>.</p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo>&nbsp;</mo><mfenced><mrow><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;A1</strong></p>
<p>&nbsp;</p>
<p><strong>Note:</strong> Do not award <strong>A1</strong> if solutions outside&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn></math>&nbsp;are also stated.</p>
<p>the ball is less than&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>8</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math>&nbsp;metres above the ground for&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></math>(s)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo></math>(s)&nbsp; &nbsp; &nbsp; &nbsp;<strong>A1</strong></p>
<p>&nbsp;&nbsp;</p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER<br></strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong> (M1)</strong></p>
<p>&nbsp;</p>
<p><strong>OR</strong></p>
<p>recognizes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced></math> is required&nbsp; &nbsp; &nbsp; &nbsp;<strong>&nbsp;(M1)</strong></p>
<p>&nbsp;</p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi mathvariant="normal">π</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; <strong>A1</strong></p>
<p>attempts to evaluate their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi mathvariant="normal">π</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi mathvariant="normal">π</mi><msqrt><mn>3</mn></msqrt><mo>&nbsp;</mo><mfenced><msup><mi>ms</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp; <strong>A1</strong></p>
<p>&nbsp;</p>
<p><strong>Note:</strong> Accept equivalent correct answer forms where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi mathvariant="normal">π</mi><msqrt><mn>3</mn></msqrt></math>.</p>
<p>&nbsp;</p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Given 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><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>recognition that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>∫</mo><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mo>d</mo><mi>x</mi></math>                 <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math>                 <em><strong>(A1)</strong></em></p>
<p>substitute both <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> values into their integrated expression including <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"><mn>2</mn><mo>=</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>+</mo><mi>c</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn></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"><munderover><mo>∫</mo><mn>2</mn><mi>y</mi></munderover><mo>d</mo><mi>y</mi><mo>=</mo><munderover><mo>∫</mo><mfrac><mstyle displaystyle="true"><mn>3</mn><mi mathvariant="normal">π</mi></mstyle><mn>4</mn></mfrac><mi>x</mi></munderover><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mo>d</mo><mi>x</mi></math>                 <em><strong>(M1)(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mo>-</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>                 <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn></math>                 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>The derivative of a function <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"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msqrt><mi>x</mi></msqrt></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>3</mn></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>4</mn></mfenced></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>correct working&nbsp; &nbsp; &nbsp; <em><strong>(A1)</strong></em></p>
<p><em>eg&nbsp; &nbsp;</em>−5 + (8 − 1)(3)</p>
<p><em>u</em><sub>8</sub>&nbsp;= 16&nbsp; &nbsp; &nbsp;<em><strong>A1 N2</strong></em></p>
<p>&nbsp;</p>
<p><em><strong>[2 marks]</strong></em></p>
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>∫</mo><mn>3</mn><msqrt><mi>x</mi></msqrt><mo> </mo><mi mathvariant="normal">d</mi><mi>x</mi></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>(A1)</strong></p>
<p>attempts to integrate&nbsp; &nbsp; &nbsp; &nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><msup><mi>x</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mi>C</mi><mo>&nbsp;</mo><mo>&nbsp;</mo><mfenced><mrow><mo>=</mo><mn>2</mn><mi>x</mi><msqrt><mi>x</mi></msqrt><mo>+</mo><mi>C</mi></mrow></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>A1</strong></p>
<p>uses&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>3</mn></math>&nbsp;to obtain&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>=</mo><mn>2</mn><msup><mfenced><mn>1</mn></mfenced><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mi>C</mi></math>&nbsp;and so&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>1</mn></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>M1</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> into their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>(M1)</strong></p>
<p>so&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>4</mn></mfenced><mo>=</mo><mn>17</mn></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>A1</strong></p>
<p>&nbsp;</p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>1</mn><mn>4</mn></munderover><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo> </mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><munderover><mo>∫</mo><mn>1</mn><mn>4</mn></munderover><mn>3</mn><msqrt><mi>x</mi></msqrt><mo> </mo><mi mathvariant="normal">d</mi><mi>x</mi></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>(A1)</strong></p>
<p>attempts to integrate both sides&nbsp; &nbsp; &nbsp; &nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced><mn>1</mn><mn>4</mn></msubsup><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mn>2</mn><msup><mi>x</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></mrow></mfenced><mn>1</mn><mn>4</mn></msubsup></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>4</mn></mfenced><mo>-</mo><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>16</mn><mo>-</mo><mn>2</mn></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>M1</strong></p>
<p>uses&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>3</mn></math>&nbsp;to find their value of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>4</mn></mfenced></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>4</mn></mfenced><mo>-</mo><mn>3</mn><mo>=</mo><mn>16</mn><mo>-</mo><mn>2</mn></math></p>
<p>so&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>4</mn></mfenced><mo>=</mo><mn>17</mn></math>&nbsp; &nbsp; &nbsp; &nbsp;<strong>A1</strong></p>
<p>&nbsp;</p>
<p><strong>[6 marks]</strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{8x}}{{\sqrt {2{x^2} + 1} }}">
  <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>
      <msqrt>
        <mn>2</mn>
        <mrow>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </msqrt>
    </mrow>
  </mfrac>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = 5">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mn>0</mn>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>5</mn>
</math></span>, find <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>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to integrate     <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 2{x^2} + 1 \Rightarrow \frac{{{\text{d}}u}}{{{\text{d}}x}} = 4x">
  <mi>u</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
  </mrow>
  <mo>+</mo>
  <mn>1</mn>
  <mo stretchy="false">⇒</mo>
  <mfrac>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>u</mi>
    </mrow>
    <mrow>
      <mrow>
        <mtext>d</mtext>
      </mrow>
      <mi>x</mi>
    </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>4</mn>
  <mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{8x}}{{\sqrt {2{x^2} + 1} }}} {\text{d}}x = \int {\frac{2}{{\sqrt u }}} {\text{d}}u">
  <mo>∫</mo>
  <mrow>
    <mfrac>
      <mrow>
        <mn>8</mn>
        <mi>x</mi>
      </mrow>
      <mrow>
        <msqrt>
          <mn>2</mn>
          <mrow>
            <msup>
              <mi>x</mi>
              <mn>2</mn>
            </msup>
          </mrow>
          <mo>+</mo>
          <mn>1</mn>
        </msqrt>
      </mrow>
    </mfrac>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>x</mi>
  <mo>=</mo>
  <mo>∫</mo>
  <mrow>
    <mfrac>
      <mn>2</mn>
      <mrow>
        <msqrt>
          <mi>u</mi>
        </msqrt>
      </mrow>
    </mfrac>
  </mrow>
  <mrow>
    <mtext>d</mtext>
  </mrow>
  <mi>u</mi>
</math></span>      <em><strong>(A1)</strong></em></p>
<p><em><strong>EITHER</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\sqrt u \left( { + C} \right)">
  <mo>=</mo>
  <mn>4</mn>
  <msqrt>
    <mi>u</mi>
  </msqrt>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mo>+</mo>
      <mi>C</mi>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>      <em><strong>A1</strong></em></p>
<p><em><strong>OR</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\sqrt {2{x^2} + 1} \left( { + C} \right)">
  <mo>=</mo>
  <mn>4</mn>
  <msqrt>
    <mn>2</mn>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mn>1</mn>
  </msqrt>
  <mrow>
    <mo>(</mo>
    <mrow>
      <mo>+</mo>
      <mi>C</mi>
    </mrow>
    <mo>)</mo>
  </mrow>
</math></span>      <em><strong>A1</strong></em></p>
<p><em><strong>THEN</strong></em></p>
<p>correct substitution into <strong>their</strong> integrated function (must have <em>C</em>)       <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5 = 4 + C \Rightarrow C = 1">
  <mn>5</mn>
  <mo>=</mo>
  <mn>4</mn>
  <mo>+</mo>
  <mi>C</mi>
  <mo stretchy="false">⇒</mo>
  <mi>C</mi>
  <mo>=</mo>
  <mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 4\sqrt {2{x^2} + 1}  + 1">
  <mi>f</mi>
  <mrow>
    <mo>(</mo>
    <mi>x</mi>
    <mo>)</mo>
  </mrow>
  <mo>=</mo>
  <mn>4</mn>
  <msqrt>
    <mn>2</mn>
    <mrow>
      <msup>
        <mi>x</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <mo>+</mo>
    <mn>1</mn>
  </msqrt>
  <mo>+</mo>
  <mn>1</mn>
</math></span>      <em><strong>A1</strong></em></p>
<p><em><strong>[5 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 curve <em>y</em> = 5<em>x</em><sup>3 </sup>−&nbsp;3<em>x</em>.</p>
</div>

<div class="specification">
<p>The curve has a tangent at the point P(−1, −2).</p>
</div>

<div class="question">
<p>Find the equation of this tangent. Give your answer in the form <em>y</em> = <em>mx</em> + <em>c</em>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>(<em>y</em> − (−2)) = 12 (<em>x</em> − (−1))   <em><strong>  (M1)</strong></em></p>
<p><strong>OR</strong></p>
<p>−2 = 12(−1) + c   <em><strong>  (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong> (M1)</strong></em> for point <strong>and</strong> their gradient substituted into the equation of a line.</p>
<p> </p>
<p><em>y</em> = 12<em>x</em> + 10     <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (b).</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>