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<h2>SL Paper 1</h2><div class="specification">
<p>Let <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>, </mtext>
</mrow>
<mi>p</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>p</mi>
<mo>></mo>
<mn>1</mn>
</math></span>. The point <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>, </mtext>
</mrow>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> 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 <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>. 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 binomial expansion <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>7</mn></msup><mo>=</mo><msup><mi>x</mi><mrow><mn>7</mn></mrow></msup><mo>+</mo><mi>a</mi><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo><mo>+</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>21</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>The third term in the expansion is the mean of the second term and the fourth term in the expansion.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</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><strong>EITHER</strong></p>
<p>recognises the required term (or coefficient) in the expansion <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><msup><mi>x</mi><mn>5</mn></msup><mo>=</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>7</mn></mmultiscripts><msup><mi>x</mi><mn>5</mn></msup><msup><mn>1</mn><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>7</mn></mmultiscripts></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>5</mn><mprescripts></mprescripts><mn>7</mn></mmultiscripts></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mrow><mn>7</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo><mn>5</mn><mo>!</mo></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>7</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo><mfenced><mrow><mn>7</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac></mrow></mfenced></math></p>
<p>correct working <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>1</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mo>×</mo><mn>6</mn></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>42</mn><mn>2</mn></mfrac></math></p>
<p><br><strong>OR</strong></p>
<p>lists terms from row <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> of Pascal’s triangle <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mn>7</mn><mo>,</mo><mo> </mo><mn>21</mn><mo>,</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>21</mn></math> <em><strong>AG</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>7</mn></math> <em><strong>(A1)</strong></em></p>
<p>correct equation <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>21</mn><msup><mi>x</mi><mn>5</mn></msup><mo>=</mo><mfrac><mrow><mi>a</mi><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>21</mn><msup><mi>x</mi><mn>5</mn></msup><mo>=</mo><mfrac><mrow><mn>7</mn><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>2</mn></mfrac></math></p>
<p>correct quadratic equation <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>42</mn><mi>x</mi><mo>+</mo><mn>35</mn><mo>=</mo><mn>0</mn></math> OR <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>5</mn><mo>=</mo><mn>0</mn></math> (or equivalent)</p>
<p>valid attempt to solve <strong>their</strong> quadratic <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><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>6</mn><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mn>5</mn></mfenced></msqrt></mrow><mrow><mn>2</mn><mfenced><mn>1</mn></mfenced></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award final <em><strong>A0</strong> </em>for obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>5</mn></math>.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The majority of candidates answered part (a) correctly, either by using the <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts></math> formula or Pascal's Triangle. In part (b) of the question, most candidates were able to correctly find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>7</mn></math> and set up a correct equation showing the mean of the second and fourth terms. While some struggled to complete the required algebra to solve the equation, the majority of candidates who found a correct quadratic equation were able to solve it correctly. A few candidates included <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> in their final answer, thus not earning the final mark.</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 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> </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>′</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>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℝ</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>></mo><mn>0</mn></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>></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>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>d</mi><mo>-</mo><mn>4</mn><mo><</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><</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><</mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo><</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo><</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>></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>></mo><mn>0</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>></mo><mn>0</mn></math> <em><strong>(R1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><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>></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>></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>Consider the functions <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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mn>4</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</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> </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>, 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>∈</mo><mi mathvariant="normal">ℤ</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>∈</mo><mi mathvariant="normal">ℤ</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 diagram shows the graph of a function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>, with domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 \leqslant x \leqslant 4">
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>4</mn>
</math></span>.</p>
<p><img src="images/Schermafbeelding_2018-02-11_om_09.13.25.png" alt="N17/5/MATME/SP1/ENG/TZ0/03"></p>
<p>The points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 2,{\text{ }}0)">
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(4,{\text{ }}7)">
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>7</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="f">
<mi>f</mi>
</math></span>.</p>
</div>
<div class="question">
<p>On the grid, sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}"> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><img src="images/Schermafbeelding_2018-02-11_om_10.32.42.png" alt="N17/5/MATME/SP1/ENG/TZ0/03.c/M"> <strong><em>A1A1A1 N3</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Award <strong><em>A1 </em></strong>for both end points within circles,</p>
<p><strong><em>A1 </em></strong>for images of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(2,{\text{ }}3)"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}2)"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </math></span> within circles,</p>
<p><strong><em>A1 </em></strong>for approximately correct reflection in <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>, concave up then concave down shape (do not accept line segments).</p>
<p> </p>
<p><strong><em>[3 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>Olava’s Pizza Company supplies and delivers large cheese pizzas.</p>
<p>The total cost to the customer, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>, in Papua New Guinean Kina (<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>PGK</mtext></math>), is modelled by the function</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>34</mn><mo>.</mo><mn>50</mn><mi>n</mi><mo>+</mo><mn>8</mn><mo>.</mo><mn>50</mn><mo> </mo><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo> </mo><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>,</mo></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, is the number of large cheese pizzas ordered. This total cost includes a fixed cost for delivery.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State, in the context of the question, what the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>34</mn><mo>.</mo><mn>50</mn></math> represents.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State, in the context of the question, what the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>50</mn></math> represents.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the minimum number of pizzas that can be ordered.</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>Kaelani has <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>450</mn><mo> </mo><mtext>PGK</mtext></math>.</p>
<p>Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.</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>the cost of <strong>each</strong> (large cheese) pizza / <strong>a</strong> pizza / <strong>one</strong> pizza / <strong>per</strong> pizza <em><strong>(A1) (C1)</strong></em><br><br><strong>Note:</strong> Award <em><strong>(A0)</strong></em> for “the cost of (large cheese) pizzas”. Do not accept “the <strong>minimum</strong> cost of a pizza”.</p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the (fixed) delivery cost <em><strong>(A1) (C1)</strong></em><br><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> <em><strong>(A1) (C1)</strong></em><br><em><strong><br>[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"><mn>450</mn><mo>=</mo><mn>34</mn><mo>.</mo><mn>50</mn><mi>n</mi><mo>+</mo><mn>8</mn><mo>.</mo><mn>50</mn></math> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for equating the cost equation to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>450</mn></math> (may be stated as an inequality).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>.</mo><mn>8</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>12</mn><mo>.</mo><mn>7971</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C3)</strong></em></p>
<p><strong><br>Note:</strong> The final answer must be an integer.<br>The final <em><strong>(A1)</strong></em><strong>(ft)</strong> is awarded for rounding their answer <strong>down</strong> to a whole number, provided their unrounded answer is seen.<br><em><strong><br><br>[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.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 graph of the quadratic function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = c + bx - {x^2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>c</mi>
<mo>+</mo>
<mi>b</mi>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> intersects the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}( - 1,{\text{ }}0)">
<mrow>
<mtext>A</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span> and has its vertex at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}(3,{\text{ }}16)">
<mrow>
<mtext>B</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>16</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-03-06_om_09.57.03.png" alt="N16/5/MATSD/SP1/ENG/TZ0/09"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the axis of symmetry for this graph.</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="b">
<mi>b</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>Write down the range 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>
<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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3">
<mi>x</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <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> constant, <strong><em>(A1) </em></strong>for the constant being 3.</p>
<p>The answer must be an equation.</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{{ - b}}{{2( - 1)}} = 3">
<mfrac>
<mrow>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mrow>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into axis of symmetry formula.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b - 2x = 0">
<mi>b</mi>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<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 correctly differentiating and equating to zero.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c + b( - 1) - {( - 1)^2} = 0">
<mi>c</mi>
<mo>+</mo>
<mi>b</mi>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> (or equivalent)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c + b(3) - {(3)^2} = 16">
<mi>c</mi>
<mo>+</mo>
<mi>b</mi>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>16</mn>
</math></span> (or equivalent) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 1,{\text{ }}0)">
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(3,{\text{ }}16)">
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>16</mn>
<mo stretchy="false">)</mo>
</math></span> in the original quadratic function.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(b = ){\text{ }}6">
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
</math></span> <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="( - \infty ,{\text{ 16]}}">
<mo stretchy="false">(</mo>
<mo>−</mo>
<mi mathvariant="normal">∞</mi>
<mo>,</mo>
<mrow>
<mtext> 16]</mtext>
</mrow>
</math></span><strong> OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="] - \infty ,{\text{ }}16]">
<mo stretchy="false">]</mo>
<mo>−</mo>
<mi mathvariant="normal">∞</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>16</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 two correct interval endpoints, <strong><em>(A1) </em></strong>for left endpoint excluded <strong>and </strong>right endpoint included.</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>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 > 0"> <mi>tan</mi> <mo></mo> <mi>θ</mi> <mo>></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 < x < \frac{\pi }{2}"> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</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 > 0"> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>></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>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</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 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>∈</mo><mi mathvariant="normal">ℝ</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>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a vertical asymptote and a horizontal asymptote.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the vertical asymptote.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the horizontal asymptote.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the set of axes below, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<p>On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAuYAAAGyCAYAAACobgEvAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAB9hSURBVHhe7d0NlJT1fejxH5RjohCO3RjvFtcJNVICJir4dqNeg6yo1crdSG40RmGJpMGX0mBpfEGtwReMsdCgTbiAsoh6clIhe+gNNwkuIW3ABiK+tNoYjgmZ4nabCjEEo+WS4c7MPtRdQYJNZvjP7Odzzhz+z3/GPGFe/s93Z58Z+u0uCgAA4KDqn/0JAAAcRMIcAAAS0OtUllyuKRsBAADVkM9vKf+5V5jvuQIAAKisnv3tVBYAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwB0hRV3tMyTWV/0W4EVPbo7NQmizE9jW3xojz58XGHeUJAOqIMAdIUWNLLMo/HQ9dMTRe/c7G+GE5xPvHoPd/OP5XvBh5YQ5Qd4Q5QLIOj/efcmLEq9vilV92h3j/I5vimGEnx3FHDihvA1A/hDlAsgbEEbljoiE2xY87Xy9u74zOFY/Ejyb8YQyzegPUHUs7QK3Y8Uw8tvGUmHrWEdkEAPVEmAMkbMCQY+Lk+Nd4Lv+j2PjgmshNPS+GWLkB6pLlHSB5/xFb/s+CaM9dEuOHHJLNAVBvhDlAyo7IxXENxT+HfCSuHp+zaAPUMWs8QMp+uT1ezl0Td3327Gi0YgPUNcs8QKoKnbGm7YcxbsHUGD3Icg1Q76z0AEl5JTbO/UiMuP6BWH7Hwth83qUxptF55QB9gTAHSMqu2P7yv8erz/wk+n98erQOH5zNA1Dv+u0uysaRyzVFPr8l2wIAACqpZ397xxwAABIgzAESt23btvIFgPomzAES9tprr8WJJx4fM2fOzGYAqFfCHCBhy5YtK//5/PPPRXt7e3kMQH0S5gCJ2rRpU9x00w3l8bx598XcuXOc0gJQx4Q5QKLmzJkTd911d3l8wgknxAUXXBj33ntveRuA+iPMARJUOm1l69atMWHChGwm4lOf+lSsXfvdWLduXTYDQD0R5gCJKZ2uMm3ateUPfB566KHZbERDQ0PccsstceONN5Q/FApAfRHmAInZuPHJ8ikspdNX3uycc8bFpZd+PJ566qlsBoB64V/+BEictRmgfvmXPwEAIDHCHAAAEiDMAQAgAcIcAAASIMwBACABwhwAABIgzAEAIAHCHAAAEiDMAaphx6ZYs3xOTBnRFLnc8TGl/V+yKwCgmzAHqKhCscnbY+ZH/ygmLtgaZ96zItZvfjYWtRydXQ8A3YQ5QAUVOlfEjPHXxvKmm2PVY7dH60Wjo9HKC8A+ODwAVEohHytm3RUrY0LMnnVJDB9kyQXgrTlKAFRI4cXV8eDKzhh4xnti061nRi7XFLkRV8a8DV1RyG4DAHv0212UjcsHjXx+S7YF9Wt1R0c2gkrZGS+tmhszH/1ljLn8shh/9vA4/JXnY8VDC6L9md+Ly+6YHucedUh22ze0Tp6UjXprW7wkG0FtGNvcnI2A/enV36Uw3+Poo4/KRlDfOh5/PBtVR7X3V2KflXNg+/zF7ifnnLv76PffsvvbP/9VNver3b948ou7zzv6D3ZftPgHxa0DczDWZo9l5VR7n33lfoVa1XONdyoLQNX0j0FDctEUr0b+Z686nQWAXoQ5QEUcFseOOi0Gvro6Vm3cls3tMTQuHJWLAdkWAJQIc4CK6B+Dz5oSsy/YGUvvnh9runaWvjsx1nzpgVh77qdj4skN2e0AoJswB6iU/rlouXdpzDvpezHx1GMiN/TsuPtXl8Yjd/jqRAD25sgAUEmDhkfLnX9b/sR9Pv9CfOPOT8Toxr2/jQUAhDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAvrtLsrGkcs1RT6/JduC+rW6oyMbQVpaJ0/KRr21LV6SjaA2jG1uzkbA/vTsb2FOn1QK82oeNKq9vxL7rJxq7/NgrM0ey8qp9j77yv0KtarnGu9UFgAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHOAqtgZXWvuiPNz58XcjTuyOQB4gzAHqIJC58q47ar58Xy2DQBvJswBKq2QjxWz7ovN7x2aTQDA3oQ5QEVtjxceuiceHH573Df1xGwOAPYmzAEqphA7NrbFn3/3tPjCp06Od2WzALAv/XYXZePI5Zoin9+SbUH9Wt3RkY2ggl5/IZbPfzreN3FCnNDQL7atmx/XLeyM8TNviYuPfWd2o95aJ0/KRr21LV6SjaA2jG1uzkbA/vTq71KY73H00UdlI6hvHY8/no2qo9r7K7HPyjmgff7qpd3fvmXa7jlP/iyb+H+7//Vr1xTX2XOLc7/I5g7MwVibPZaVU+199pX7FWpVzzXeqSwAlfDT9fFw27KY2/KB8rshudzQOHVae/GK54pz74/clPbo6r4lAJQJc4BKaGyJRfkt5V9Pdl82x/p5LcUrjovp7T+I/KKWaOy+JQCUCXMAAEiAMAcAgAQIc4CqGBCNLfdHPv/NmD56UDYHAG8Q5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJ6Le7KBtHLtcU+fyWbAvq1+qOjmwEaWmdPCkb9da2eEk2gtowtrk5GwH707O/hTl9UinMq3nQqPb+Suyzcqq9z4OxNnssK6fa++wr9yvUqp5rvFNZAAAgAcIcAAASIMwBACABwhwAABIgzAEAIAHCHAAAEiDMAQAgAcIcAAASIMwBACABwhwAABIgzAEAIAHCHAAAEiDMAQAgAcIcAAASIMwBACABwhwAABIgzAEAIAHCHAAAEiDMAQAgAcIcAAASIMwBACABwhygYnZG14b5MWVEU+RypctFMXPphugqZFcDQA/CHKAiCrFj44K4ub0xrt+Qj/zm78eyGUfG8pmXR+sXN8SO7FYAsIcwB6iIbfH97xwaV984PoYNKi61/RvjlGs/G9efGPH8kr+PH+7KbgYAGWEOUBFHxJjpV8boUpTv0f+wOPzId8TAC0fHsQOyOQDICHOAatn+YmxYOzguHveBGJxNAcAewhygKnZG5+qvxdc/PCOuOeuIbG5v3R8S7X0BoG/ot7soG5cPAPn8lmwL6tfqjo5sBNVR2Lou7r/vp/GHN4yPYe98e++JtE6eFG2Ll2RbUBvGNjdnI2B/eva3MKdPKoV5NQ8a1d5fiX1Wztve547nom32N2Pon1wbYxoPySYP3MFYmz2WlVPtffaV+xVqVc813qksAJVU6Iw1cx6LmDjlvxTlAPQdwhygUkpRfsfC2HzJ9Ggdvufjnjuja828mLvm5WwbALoJc4BKKEX5bZ+OiYsWxq3jRpZ/Vdl9OSZOverlGDW6IbshAHQT5gCV0H9IjJn1t+XzBve6/POsGDPY8gtAb44MAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAJyeWa9roA0Df0212UjcsHgHx+S7YF9Wt1R0c2gvS1Tp4UbYuXZFtQG8Y2N2cjYH969rcwp08qhXk1DxrV3l+JfVZOtfd5MNZmj2XlVHuffeV+hVrVc413KgsAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAta2rPabkmiKXOzWmtuejEDuja8P8mDKiOHfinNi4K7sdJE6YAwC1rbElFuX/KdqnHxUrH1wVT29oi/ue+x/xV/+8JfJPXxejB2S3g8QJcwCgDhweoz8+Kc59+i/iEw8cGddMPC4GZddArRDm1KTHH18VmzZtyrYAqBfPPPNMrFu3Ltt6m44cEWeeOCTOOP+kGKJwqEGettSUUox/7GMfiwULFsZhhx2WzQJQL0pr+4033hBXXXVVvPTSS9ns2/HzWLvhxdiebUEtEebUhNdeey0efvjhaG4+O8aPHx9LliyJo446KrsWgHoxbNiw+MY3vhnHH398fOhDp0V7e3v5GPDr7YzOFV+LTR/8UMTy1bFxeyGbh9ohzEle6dea559/XqxduzaeeOJ7cfnll8ehhx6aXQtAvSmt8aV3zDs6vh2PPvpoTJo06decvliIHS/8Tfx1vjluvL41Lo4XY/O/7YjO9nmxdNPr2W0gff12F2XjyOWaIp/fkm3BwVd6l/ymm27ItgDoy77yla/G6aefnm11K2xqi5bm2bHzis/HfTeOj2GDtsWamS0xcfnwmPHQnXHtKY3ehSRpPftbmJO00vmF06dPL4/vvPPO8q84f1Ol53nb4iUxtrk5m6m8g/Hass/KqfY+3a+V0xf2ubqjI1onT6rqPn+bf8fSb02nTfuTGDnyuLj55pudxkjd6fl68UMkSSstwKXzyS+77LLy+eVf/vKXD/BcQwBq2bZt2+Lzn/98OcqnT7+uvP6LcuqdMCd5pXMNW1payueXP/vss+XzzUvvoABQn0pfifuRj7TEz3/+8/ja19rLxwDoC4Q5NaP0TknpHZNbbrml/A5KaeEGoL6UvoXl9ttvj9mz74677rorGhoasmug/glzas4554wrv4MyevRJ2QwA9eKss84qf13imz/kCX2BMKcmld5B8S4KQP0pre2+Epe+SpjT51T7GxhKSt8CU232WTkHY5/V5rGsnIOxz2qve33hNQKVIMwBACABwhwAABIgzAEAIAHCHAAAEiDMAQAgAcKcPmh7vPTM38XcKadGLtcUuSnt0ZVdU3cKnbHm1osid+Kc2Lgrm6sTha51Ma/4GLZOnlR8HIfH+TMfiY1dO7NrE1Loig3zrowRpeda7qKYuXRDdBWy6yirmcfyt6WwtW5fl//p9ZdizfI5MWVE6Xl/fExp/5fsCmB/hDl9y44Xon3mJ2LmX3XEy2feFu3rfxT5RS3RmF1dX3ZF54rPx1VtT2XbdWTHhvjizasid/3j0bb4gVi/bHo0Lb8+Wlrnx8YdKVXvK7Hxi1fFhJW/H18uPtc2r/9M/M4j0+O2FfnQ5pmaeSx/W3bG1n/4an2+LssKsWNTe7Td8bmYuGBrnHnPili/+dlY1HJ0dj2wP8KcvqOQj/YZV8S05UfFZbdfH3e2XhCjGw/Jrqw/ha3rY9Zt/xTv/YOB2Uy9KMT27z8Zv3P1n0bLsMHF7QHReEprXH/9hyOe/3p854e/7L5ZAgrFQLlt7r/HFTdMjTHF51r/xjPiikvfGytvXBR/t12a19Jj+dtS6FwZjzz6kzp8XXYrdK6IGeOvjSeO+Fiseuz2aL1odDQqDThgXi70ETujc8Vfxo0rIy6YfVOc03RYNl+ndjwXjy9dF8MfnBtTh74jm6wX/WPwmKkxbfTh2XbJgHjX4b8bMfC0GHVsKo/t6/Hi2lXxdJwYp7x/z//Xd8b7zhgXJ766OlZt3JbN9WW18lj+lhRflw/d2h5Nn/l0Hb4uiwr5WDHrrlgZE2LSFWNi+CCJAW+XVw19Q+HH8a0HvxGvDvxgvG/T5+Kq8rmsp8aUeevq8HzfV2Ljwi/Ec8d9ND41uhg4fcIr8YMNT8fAi8fG6MGpLGs74qVNP4loOCZyRwzI5oqL7rsOjyPj3+MfN291Oss+pfhY/jZ0vy6/e+afxx8eOyibqy+FF1fHgys7Y+AZ74mXls4orrFNkRtxZczb0OW5DgdImNMnFF58IpY/HTGyeLBvvuLL8b8fmBvLZnww1t47Of74oRfq6KBRiB0bH44v/eyKmNQ8NOrz8L+3Qud34ytfPylmX3N6lE6ISMNr8cq/bc/Gb/Zq5H/2qljZhzQfy9/UG6/LOyYeF4dms/Vlz2+IRsXFHz4vmqf9VWxe/9WYccY/xr0T/iwe2vR6djtgf4Q5NWxXdLVf2/2uzFteur8NoPCLbZGP98RJ487rPq+8f0Oc8smp8ccjI55e/kS8mHohdbXHlH3+/XpcprRHZ9e3454vRVz92bOjoRZf3Qf49+z1LTqFn8aKWe0x/MHPRcuQWvnMwMDI/e5AC/Cb1eRj+esVerwu6/d8613xi5+9XHxqj4px//Ok8vrTv/G/xyevvjxGxvdj+dqf+EEUDoDjAjVsQDS23B/5/Jb9XPbzbQCDGuP3m94Rkd8Wv0j9iNHYEov2+ffrcVn0R9H/H74Wbd+6O1pG5rKvnvtQTPvWtohtc6LlmBr4yrID+nv2/Bad7bGl41vx7KV3xZ/2Ok85Be+J4878YMR/vBLbf/nGE2xX54+KmfKe+ODQd1uAe0n5sfxN7Iqf9nhdln64bJ08o7Zel/9l/WPQkFw0+Q0RHDDHBfqEAceOjgsHbo7lq/6pePjvbeCFo+PYN04BrmG9f1BpW7yk+OcTMe/choiG66L9R/X2lWU7o2vNong8zozrxgxJcDHb1wc9X48fP/tkbBs4NsaNLj4uZFJ/LH8Te7+B0Lb43jp8XR4Wx446LQbu84PNQ+PCUbniPQH8OsKcvmHw6XHN7AkRS/8y7l3TGYXSaTBrlsT8tafH9RNH19G5rH1FKeTuj/s2nxeXjHvjXPpCV0fMmvt3e/3wdbD0f9/Y+OQFO2P5w/83XtixK3Zs+ka0feUnccHsKXFWXX2w8TdRG48lv07/GHzWlJhdfL4vvXt+PLNtV/FB7Iw1X3og1p776Zh4sh9E4UA4MtBHHBJDWm6PFfNGxfqJp8YnJ18Zp979Wnzikdti4nBZXltKIXdPtE6cE0tvPS/7hp3u88+HnvoX8R+jRqbzg1b/XLTcuzRm/7evxriRQ2Nk8wMRUx+Ie1tyFt+yGnos+fWy5/u8k74Xc//sysgNPTvu/tWl8cgdl/jqRDhA/XYXZePyYlj6NRvUu9UdHTG2uTnbqrxq76/EPiun2vs8GGuzx7Jyqr3PvnK/Qq3qucb7ERYAABIgzAEAIAHCHAAAEiDMAQAgAT78SZ9U+mASpKj0D0PtS+l76aGW+PAnHJie/S3M6ZN8K0Jl2Gdl+FaWyukL++wr9yvUKt/KAgAAiRHmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ5QMTuja8P8mDKiKXK50uWimLl0Q3QVsqsBoAdhDlARhdixcUHc3N4Y12/IR37z92PZjCNj+czLo/WLG2JHdisA2EOYA1TEtvj+dw6Nq28cH8MGFZfa/o1xyrWfjetPjHh+yd/HD3dlNwOAjDAHqIgjYsz0K2N0Kcr36H9YHH7kO2LghaPj2AHZHABkhDlAtWx/MTasHRwXj/tADM6mAGCPfruLsnH5w0n5/JZsC+rX6o6ObATVsiu2rlsYtz55Qsy65vR491u8LdI6eVI26q1t8ZJsBLVhbHNzNgL2p2d/C3P6pFKYV/OgUe39ldhnpeyKx27/WFy3cH22vS8Nce68r8eilqOz7YhCZ3tcPTUff/zotb1PbzkAB2Nt9vypnGrvs6/cr1Creq7xTmUBeFsGRMPp15QX0be+PNsrymPHc/HQX/8oLl0w9W1HOQB9hyMEQCUVOmPNnMciJk6JMY2HZJMAsDdhDlAppSi/Y2FsvmR6tA7f83HPndG1Zl7MXfNytg0A3YQ5QCWUovy2T8fERQvj1nEjy+cQdl+OiVOvejlGjW7IbggA3YQ5QCX0HxJjZv3tPs4/L17+eVaMGWz5BaA3RwYAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABPTbXZSNI5drinx+S7YF9Wt1R0c2grS0Tp6UjXprW7wkG0FtGNvcnI2A/enZ38KcPqkU5tU8aFR7fyX2WTnV3ufBWJs9lpVT7X32lfsValXPNd6pLAAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAAJAAYQ4AAAkQ5gAAkABhDgAACRDmAACQAGEOAAAJEOYAVbEzutbcEefnzou5G3dkcwDwBmEOUAWFzpVx21Xz4/lsGwDeTJgDVFohHytm3Reb3zs0mwCAvQlzgIraHi88dE88OPz2uG/qidkcAOxNmANUTCF2bGyLP//uafGFT50c78pmAWBf+u0uysaRyzVFPr8l24L6tbqjIxtBBb3+Qiyf/3S8b+KEOKGhX2xbNz+uW9gZ42feEhcf+87sRr21Tp6UjXprW7wkG0FtGNvcnI2A/enZ38KcPqkU5tU8aFR7fyX2WSm74rHbP1YM7PXZ9r40xLlffDAuf2ppPNXyuZg++vDi3K7oav9MnDptU0xvX1acG9R90wNwMNZmz5/KqfY++8r9CrWq5xrvVBaAt2VANJx+TXkRfevLs7HoQ1vi4bZlMbflA+VFN5cbWozy9uJ//1xx7v2Rm9IeXd3/gwBQJswBKqGxJRb1ivXNsX5eS/GK42J6+w8iv6glGrtvCQBlwhwAABIgzAEAIAHCHKAqBkRjy/2Rz3/zbX3wE4C+Q5gDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJKDf7qJsHLlcU+TzW7ItqF+rOzqyEaSldfKkbNRb2+Il2Qhqw9jm5mwE7E/P/hbm9EmlMK/mQaPa+yuxz8qp9j4Pxtrssaycau+zr9yvUKt6rvFOZQEAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHAIAECHMAAEiAMAcAgAQIcwAASIAwBwCABAhzAABIgDAHqIYdm2LN8jkxZURT5HLHx5T2f8muAIBuwhygogrFJm+PmR/9o5i4YGucec+KWL/52VjUcnR2PQB0E+YAFVToXBEzxl8by5tujlWP3R6tF42ORisvAPvg8ABQKYV8rJh1V6yMCTF71iUxfJAlF4C35igBUCGFF1fHgys7Y+AZ74lNt54ZuVxT5EZcGfM2dEUhuw0A7NFvd1E2Lh808vkt2RbUr9UdHdkIKmVnvLRqbsx89Jcx5vLLYvzZw+PwV56PFQ8tiPZnfi8uu2N6nHvUIdlt39A6eVI26q1t8ZJsBLVhbHNzNgL2p2d/C3OAt2VXdLV/Jk6d1p5t70tDnDvvb+LqH0+LlgWnxUPfuy3GDC79grIQOzbeHx9t+es4ZNaKaG8dfkC/trQ2A9Svnmu8U1kA3pYB0dhyf3kRfetL6VtXjspu31P/GDQkF03xauR/9qrTWQDoRZgDVMRhceyo02Lgq6tj1cZt2dweQ+PCUbli4gPAG4Q5QEX0j8FnTYnZF+yMpXfPjzVdO0vfnRhrvvRArD330zHx5IbsdgDQTZgDVEr/XLTcuzTmnfS9mHjqMZEbenbc/atL45E7fHUiAHvz4U+AxFmbAeqXD38CAEBihDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDlAYl577bXYtu3N/4z/G1566aXybQCoL8IcIDFr1343pk6dus/4LgX7ZZd9PJ566qlsBoB6IcwBEnPOOePi3e9+dyxbtiybecPChQvjjDPOjNNPPz2bAaBeCHOABF133XVx0003xKZNm7KZiGeeeSZWrvx6zJgxI5sBoJ4Ic4AEDRs2LO666+6YM2dOebt0Wsu0aX8S06dfFw0NDeU5AOqLMAdI1IQJE2Lr1q3lcem0lpEjj4uWlpbyNgD1p9/uomwcuVxT5PNbsi0ADrbS6SsXXXRhefzEE9+Lo446qjwGoD707G/vmAMk7IQTTogbb5wZ8+bdL8oB6px3zAEA4CDxjjkAACRGmAMAQAKEOQAAJECYAwBAAoQ5AAAkQJgDAEAChDkAACRAmAMAQAKEOQAAJECYAwBAAoQ5AAAkoN/uomxc/rf6AQCA6snnt5T/7BXmAADAweFUFgAAOOgi/j8yXyBPSYWm2wAAAABJRU5ErkJggg=="></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, solve the inequality <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo><</mo><mn>2</mn></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</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">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p> </p>
<p>rational function shape with two branches in opposite quadrants, with two correctly positioned asymptotes and asymptotic behaviour shown <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The equations of the asymptotes are not required on the graph provided there is a clear indication of asymptotic behaviour at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn></math> (or at their FT asymptotes from part (a)).</p>
<p> </p>
<p>axes intercepts clearly shown at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept correct alternative correct notation, such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo>∞</mo></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>]</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>,</mo><mo>∞</mo><mo>[</mo></math>.</p>
<p> </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;">
<p>It is pleasing to note that many candidates were familiar with the shape of the graph of a rational function of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mrow><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></mrow></mfrac></math>, and a large number of them were able to sketch an appropriate graph. Part (c) was a struggle for the majority of candidates, with only a few answering correctly. Despite the word "hence" and the single mark available in this part, most candidates who attempted part (c) did so by trying to solve the inequality algebraically, rather than seeing the connection to the values in their graph.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p> </p>
</div>
<div class="specification">
<p>For the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></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 the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercepts.</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>find the coordinates of the vertex.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mi>k</mi></math>.</p>
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> and 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">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> <strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math>) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p>substituting their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> <strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to complete the square <strong><em>(M1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mfenced><mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>8</mn></math> <em><strong>A1A1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>></mo><mn>1</mn></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> contains the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>4</mn></mrow></mfenced></math>.</p>
</div>
<div class="specification">
<p>Consider the arithmetic sequence <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo> </mo><mo>,</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><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>Write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><msqrt><mn>32</mn></msqrt></mfenced></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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn></math> are four consecutive terms in a geometric sequence.</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and 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">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mfrac><mn>2</mn><mn>3</mn></mfrac></mfenced><mo>=</mo><mn>4</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo>=</mo><mn>4</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msup><mn>4</mn><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msup><mfenced><msup><mn>2</mn><mn>2</mn></msup></mfenced><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>64</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mi>a</mi><mn>3</mn></mroot><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>8</mn></math> <em><strong>AG</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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msub><mi>log</mi><mi>a</mi></msub><mo> </mo><mi>x</mi></math>.<br> Accept any equivalent expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo> </mo><mn>8</mn></mrow></mfrac></math>.</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>correct substitution <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><msqrt><mn>32</mn></msqrt></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>8</mn><mi>x</mi></msup><mo>=</mo><msup><mn>32</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></math></p>
<p>correct working involving log/index law <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>32</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>2</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><msup><mn>2</mn><mfrac><mn>5</mn><mn>2</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>8</mn><mo>=</mo><mn>3</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>ln</mi><mo> </mo><msup><mn>2</mn><mstyle displaystyle="true"><mfrac><mn>5</mn><mn>2</mn></mfrac></mstyle></msup></mrow><mrow><mi>ln</mi><mo> </mo><msup><mn>2</mn><mn>3</mn></msup></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow><mn>3</mn><mi>x</mi></mrow></msup><mo>=</mo><msup><mn>2</mn><mfrac><mn>5</mn><mn>2</mn></mfrac></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><msqrt><mn>32</mn></msqrt></mfenced><mo>=</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></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">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>equating a pair of differences<strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>-</mo><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><msub><mi>u</mi><mn>4</mn></msub><mo>-</mo><msub><mi>u</mi><mn>3</mn></msub><mfenced><mrow><mo>=</mo><msub><mi>u</mi><mn>3</mn></msub><mo>-</mo><msub><mi>u</mi><mn>2</mn></msub></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi><mo>-</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo>-</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo>-</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo>-</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mi>p</mi><mn>27</mn></mfrac></mfenced><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>125</mn><mi>q</mi></mfrac></mfenced><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>125</mn><mi>q</mi></mfrac></mfenced><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mi>q</mi><mi>p</mi></mfrac></mfenced></math> <strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>p</mi><mn>27</mn></mfrac><mo>=</mo><mfrac><mn>125</mn><mi>q</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>125</mn><mi>q</mi></mfrac><mo>=</mo><mfrac><mi>q</mi><mi>p</mi></mfrac></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn></math> are in geometric sequence <strong><em>AG</em></strong></p>
<p><strong><br>Note:</strong> If candidate assumes the sequence is geometric, award no marks for part (i). If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math> has been found, this will be awarded marks in part (ii).</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>expressing a pair of consecutive terms, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><msup><mn>8</mn><mi>d</mi></msup><mo>×</mo><mn>27</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><msup><mn>8</mn><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><msup><mn>8</mn><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn><mo>=</mo><msup><mn>8</mn><mrow><mn>3</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></math></p>
<p>two correct pairs of consecutive terms, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mn>8</mn><mi>d</mi></msup><mo>×</mo><mn>27</mn></mrow><mn>27</mn></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><msup><mn>8</mn><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></mstyle><mrow><msup><mn>8</mn><mi>d</mi></msup><mo>×</mo><mn>27</mn></mrow></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><msup><mn>8</mn><mrow><mn>3</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></mstyle><mrow><msup><mn>8</mn><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></mrow></mfrac></math> (must include 3 ratios)<strong> <em>A1</em></strong></p>
<p>all simplify to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>8</mn><mi>d</mi></msup></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn></math> are in geometric sequence <strong><em>AG</em></strong></p>
<p> </p>
<p><em><strong>[4</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (geometric, finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi></math>)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>4</mn></msub><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><msup><mi>r</mi><mn>3</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn><mo>=</mo><mn>27</mn><msup><mfenced><mi>r</mi></mfenced><mn>3</mn></msup></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math> (seen anywhere)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>27</mn><mi>r</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>125</mn><mi>q</mi></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>45</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>75</mn></math> <strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD 2 (arithmetic)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>4</mn></msub><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><mi>d</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo>+</mo><mn>3</mn><mi>d</mi></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>5</mn><mn>3</mn></mfrac></mfenced></math> (seen anywhere)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo>+</mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>5</mn><mn>3</mn></mfrac></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo>+</mo><mn>2</mn><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>5</mn><mn>3</mn></mfrac></mfenced></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>45</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>75</mn></math> <strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD 3 (geometric using proportion)</strong></p>
<p>recognizing proportion<strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo>=</mo><mn>125</mn><mo>×</mo><mn>27</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>125</mn><mi>p</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mn>27</mn><mi>q</mi></math></p>
<p>two correct proportion equations<strong> <em>A1</em></strong></p>
<p>attempt to eliminate either <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>125</mn><mo>×</mo><mfrac><mrow><mn>125</mn><mo>×</mo><mn>27</mn></mrow><mi>q</mi></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mn>27</mn><mo>×</mo><mfrac><mrow><mn>125</mn><mo>×</mo><mn>27</mn></mrow><mi>p</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>45</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>75</mn></math> <strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[5</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows the graph of the quadratic function <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><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> , with vertex <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>10</mn></mrow></mfenced></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 equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>k</mi></math> has two solutions. One of these solutions is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the other solution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>k</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>Complete the table below placing a tick (✔) to show whether the unknown parameters <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> are positive, zero or negative. The row for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> has been completed as an example.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></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>State the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is decreasing.</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"><mfenced><mrow><mi>x</mi><mo>=</mo></mrow></mfenced><mo> </mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mn>4</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo></mrow></mfenced><mo> </mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>2</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for correct calculation of the left symmetrical point.</p>
<p><em><strong><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo>-</mo><mn>6</mn></math> (A1) (C2)</strong></em></p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>(A1)(A1) (C2)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(A1)</strong></em> for each correct row.</p>
<p><em><strong><br></strong></em><em><strong>[2 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>x</mi><mo>></mo><mo>-</mo><mn>2</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mn>2</mn></math> <em><strong>(A1)(A1) (C2)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> seen as part of an inequality, <em><strong>(A1)</strong></em> for completely correct notation. Award <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em> for correct equivalent statement in words, for example “decreasing when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is greater than negative <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>”.</p>
<p><em><strong><br></strong></em><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> </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>The following diagram shows part of the graph of <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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>k</mi><mo>></mo><mn>0</mn></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>p</mi><mo>,</mo><mo> </mo><mfrac><mi>k</mi><mi>p</mi></mfrac></mrow></mfenced></math> 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> 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> </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>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> </mo><mi mathvariant="normal">m</mi><mo> </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, 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>≤</mo><mi>t</mi><mo>≤</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> </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>></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="question">
<p>The following table shows the probability distribution of a discrete random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>4</mn></math>.</p>
<p style="text-align:center;"><img src="data:image/png;base64,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"></p>
<p style="text-align:left;">Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>, justifying your answer.</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>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</mo><mi mathvariant="normal">P</mi><mfenced><mrow><mi>X</mi><mo>=</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>1</mn></mrow></mfenced></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><mn>7</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mi>k</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>k</mi><mo>+</mo><mn>1</mn><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempts to factorize their quadratic <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts use of the quadratic formula on their equation <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>5</mn><mo>±</mo><msqrt><msup><mn>5</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>4</mn></mfenced><mfenced><mn>1</mn></mfenced></msqrt></mrow><mn>8</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mo>-</mo><mn>5</mn><mo>±</mo><mn>3</mn></mrow><mn>8</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math> <strong>A1</strong></p>
<p>rejects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> as this value leads to invalid probabilities, for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mrow><mi>X</mi><mo>=</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mo>-</mo><mn>5</mn><mo><</mo><mn>0</mn></math> <strong>R1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>R0A1</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math> is stated without a valid reason given for rejecting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>.</p>
<p> </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="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>m</mi><mi>x</mi></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>m</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>-</mo><mn>9</mn></math> meets the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> at exactly one point.</p>
</div>
<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>p</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>q</mi></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> can also be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mi>k</mi></math>, where <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="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>4</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and 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.</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>h</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</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 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 both negative and increasing.</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><strong>METHOD 1 (discriminant)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>m</mi><mi>x</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>-</mo><mn>9</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>m</mi><mi>x</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></p>
<p>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo>=</mo><mn>0</mn></math> (seen anywhere) <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn><mi>m</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mi>m</mi></mfenced><mfenced><mn>9</mn></mfenced></math> (do not accept only in quadratic formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>) <em><strong> A1</strong></em></p>
<p>valid approach to solve quadratic for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mi>m</mi><mfenced><mrow><mi>m</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mn>36</mn><mo>±</mo><msqrt><msup><mn>36</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>0</mn></msqrt></mrow><mrow><mn>2</mn><mo>×</mo><mn>9</mn></mrow></mfrac></math></p>
<p>both solutions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>4</mn></math> <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>≠</mo><mn>0</mn></math> with a valid reason <em><strong> R1</strong></em></p>
<p>the two graphs would not intersect OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≠</mo><mo>-</mo><mn>9</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>4</mn></math> <em><strong> AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2 (equating slopes)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>m</mi><mi>x</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>-</mo><mn>9</mn></math> (seen anywhere) <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>2</mn><mi>m</mi><mi>x</mi><mo>-</mo><mn>2</mn><mi>m</mi></math> <em><strong> A1</strong></em></p>
<p>equating slopes, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi></math> (seen anywhere) <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>m</mi><mi>x</mi><mo>-</mo><mn>2</mn><mi>m</mi><mo>=</mo><mi>m</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong> A1</strong></em></p>
<p>substituting their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> value <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mn>3</mn><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mi>m</mi><mo>-</mo><mn>2</mn><mi>m</mi><mo>×</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>=</mo><mi>m</mi><mo>×</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>-</mo><mn>9</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>9</mn><mn>4</mn></mfrac><mi>m</mi><mo>-</mo><mfrac><mn>12</mn><mn>4</mn></mfrac><mi>m</mi><mo>=</mo><mfrac><mn>6</mn><mn>4</mn></mfrac><mi>m</mi><mo>-</mo><mn>9</mn></math> <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>9</mn><mi>m</mi></mrow><mn>4</mn></mfrac><mo>=</mo><mo>-</mo><mn>9</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>4</mn></math> <em><strong> AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3 (using <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></math>)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>m</mi><mi>x</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>-</mo><mn>9</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>m</mi><mi>x</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coord of vertex using <strong><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></math> <em>(M1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mfenced><mrow><mo>-</mo><mn>3</mn><mi>m</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>m</mi></mrow></mfrac></math> <em> A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong> A1</strong></em></p>
<p>substituting their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> value <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mn>3</mn><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mi>m</mi><mo>-</mo><mn>3</mn><mi>m</mi><mo>×</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>9</mn><mn>4</mn></mfrac><mi>m</mi><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mi>m</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math> <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mi>m</mi><mo>=</mo><mo>-</mo><mn>36</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>4</mn></math> <em><strong> AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>x</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>2</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>0</mn></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use valid approach <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>0</mn><mo>+</mo><mn>2</mn></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mo>-</mo><mn>8</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mo>×</mo><mn>4</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>,</mo><mo> </mo><mn>8</mn><mi>x</mi><mo>-</mo><mn>8</mn><mo>=</mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>k</mi><mo>=</mo><mo>-</mo><mn>4</mn></math> <em><strong> A1A1</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><strong>EITHER</strong></p>
<p>recognition <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>h</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> (may be seen on sketch)<em><strong> (M1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>recognition that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</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>x</mi></mfenced><mo>></mo><mn>0</mn></math><em><strong> (M1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo><</mo><mi>x</mi><mo><</mo><mn>2</mn></math> <em><strong> A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for two correct values, <em><strong>A1</strong></em> for correct inequality signs.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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>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>′</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>∈</mo><mi mathvariant="normal">ℝ</mi></math>. The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</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><mo>′</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>′</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><</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><</mo><mn>0</mn></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</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><</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="specification">
<p>Jean-Pierre jumps out of an airplane that is flying at constant altitude. Before opening his parachute, he goes through a period of freefall.</p>
<p>Jean-Pierre’s vertical speed during the time of freefall, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math>, in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, is modelled by the following function.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>K</mi><mo>-</mo><mn>60</mn><mfenced><mrow><mn>1</mn><mo>.</mo><msup><mn>2</mn><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mo> </mo><mo>,</mo><mo> </mo><mi>t</mi><mo>≥</mo><mn>0</mn></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, is the number of seconds after he jumps out of the airplane, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi></math> is a constant. A sketch of Jean-Pierre’s vertical speed against time is shown below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Jean-Pierre’s initial vertical speed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></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>K</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>In the context of the model, state what the horizontal asymptote represents.</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 Jean-Pierre’s vertical speed after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> seconds. Give your answer in <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>km</mtext><mo> </mo><msup><mtext>h</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></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 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>0</mn><mo>=</mo><mi>K</mi><mo>-</mo><mn>60</mn><mfenced><mrow><mn>1</mn><mo>.</mo><msup><mn>2</mn><mn>0</mn></msup></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for correctly substituted function equated to zero.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>K</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>60</mn></math> <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong><br></strong></em><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the (vertical) speed that Jean-Pierre is approaching (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> increases) <em><strong>(A1) (C1)<br></strong></em><strong>OR<br></strong>the limit of the (vertical) speed of Jean-Pierre <em><strong>(A1) (C1)</strong></em></p>
<p><em><strong><br></strong></em><strong>Note: </strong>Accept “maximum speed” or “terminal speed”.</p>
<p><em><strong><br></strong></em><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>S</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>60</mn><mo>-</mo><mn>60</mn><mfenced><mrow><mn>1</mn><mo>.</mo><msup><mn>2</mn><mrow><mo>-</mo><mn>10</mn></mrow></msup></mrow></mfenced></math> <em><strong>(M1)<br></strong></em></p>
<p><strong><br></strong><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correctly substituted function.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>S</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>50</mn><mo>.</mo><mn>3096</mn><mo>…</mo><mo> </mo><mfenced><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math> <em><strong>(A1)</strong></em><strong>(ft)</strong></p>
<p><strong><br>Note: </strong>Follow through from part (a).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>181</mn><mo> </mo><mfenced><msup><mtext>km h</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>181</mn><mo>.</mo><mn>114</mn><mo>…</mo><mo> </mo><mfenced><msup><mtext>km h</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em><strong>(ft) <em> (C3)</em></strong></p>
<p><br><strong>Note:</strong> Award the final <em><strong>(A1)</strong></em><strong>(ft)</strong> for correct conversion of their speed to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>km</mtext><mo> </mo><msup><mtext>h</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>.</p>
<p><em><strong><br></strong></em><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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 1 + {{\text{e}}^{ - x}}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = 2x + b">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>b</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>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> is a constant.</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="(g \circ f)(x)"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3"> <munder> <mrow> <mo form="prefix">lim</mo> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>3</mn> </math></span>, find 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">[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>attempt to form composite <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(1 + {{\text{e}}^{ - x}})"> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </math></span></p>
<p>correct function <strong><em>A1 N2</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 \circ f)(x) = 2 + b + 2{{\text{e}}^{ - x}},{\text{ }}2(1 + {{\text{e}}^{ - x}}) + b"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> </math></span></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>evidence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {\lim }\limits_{x \to \infty } (2 + b + 2{{\text{e}}^{ - x}}) = 2 + b + \mathop {\lim }\limits_{x \to \infty } (2{{\text{e}}^{ - x}})"> <munder> <mrow> <mo form="prefix">lim</mo> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mi>b</mi> <mo>+</mo> <munder> <mrow> <mo form="prefix">lim</mo> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <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="2 + b + 2{{\text{e}}^{ - \infty }}"> <mn>2</mn> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mrow> </math></span>, graph with horizontal asymptote when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \infty "> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></span></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>if candidate clearly has incorrect limit, such as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to 0,{\text{ }}{{\text{e}}^\infty },{\text{ }}2{{\text{e}}^0}"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi mathvariant="normal">∞</mi> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>0</mn> </msup> </mrow> </math></span>.</p>
<p> </p>
<p>evidence that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{ - x}} \to 0"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </math></span> (seen anywhere) <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="\mathop {\lim }\limits_{x \to \infty } ({{\text{e}}^{ - x}}) = 0,{\text{ }}1 + {{\text{e}}^{ - x}} \to 1,{\text{ }}2(1) + b = - 3,{\text{ }}{{\text{e}}^{{\text{large negative number}}}} \to 0"> <munder> <mrow> <mo form="prefix">lim</mo> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mo stretchy="false">→</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mrow> <mtext>large negative number</mtext> </mrow> </mrow> </msup> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </math></span>, graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {{\text{e}}^{ - x}}"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </math></span> or</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2{{\text{e}}^{ - x}}"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </math></span> with asymptote <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span>, graph of composite function with asymptote <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 3"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>3</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="2 + b = - 3"> <mn>2</mn> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = - 5"> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><msqrt><mi>x</mi><mo>+</mo><mn>3</mn></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mn>3</mn></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><msqrt><mi>x</mi><mo>+</mo><mn>3</mn></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe a sequence of transformations that transforms the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mn>0</mn></math> to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><msqrt><mi>x</mi><mo>+</mo><mn>3</mn></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mn>3</mn></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>State the range of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced></math>, stating its domain.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point(s) where the graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced></math> intersect.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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>for example,</p>
<p>a reflection in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis (in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math>) <strong>A1</strong></p>
<p>a horizontal translation (shift) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> units to the left <strong>A1</strong></p>
<p>a vertical translation (shift) down by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>A1</strong> for each correct transformation applied in a correct position in the sequence. Do not accept use of the “move” for a translation.</p>
<p><strong>Note:</strong> Award <strong>A1A1A1</strong> for a correct alternative sequence of transformations. For example,</p>
<p>a vertical translation (shift) down by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit, followed by a horizontal translation (shift) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> units to the left and then a reflection in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>.</p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>range is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≤</mo><mo>-</mo><mn>1</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Correct alternative notations include <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>]</mo><mo>-</mo><mo>∞</mo><mo>,</mo><mo>-</mo><mn>1</mn><mo>]</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mo>∞</mo><mo>,</mo><mo>-</mo><mn>1</mn><mo>]</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>≤</mo><mo>-</mo><mn>1</mn></math>.</p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>-</mo><msqrt><mi>y</mi><mo>+</mo><mn>3</mn></msqrt><mo>=</mo><mi>x</mi></math> <strong>M1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for interchanging <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> (can be done at a later stage).</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>y</mi><mo>+</mo><mn>3</mn></msqrt><mo>=</mo><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn><mfenced><mrow><mo>=</mo><mo>-</mo><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mn>3</mn><mo>=</mo><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <strong>A1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo> </mo><mfenced><mrow><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p>domain is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≤</mo><mo>-</mo><mn>1</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Correct alternative notations include <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>]</mo><mo>-</mo><mo>∞</mo><mo>,</mo><mo>-</mo><mn>1</mn><mo>]</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mo>∞</mo><mo>,</mo><mo>-</mo><mn>1</mn><mo>]</mo></math>.</p>
<p> </p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the point of intersection lies on the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>=</mo><mi>x</mi></math> <strong>M1</strong> </p>
<p>attempts to solve their quadratic equation <strong>M1</strong></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</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><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></msqrt></mrow><mn>2</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><mn>3</mn></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>-</mo><msqrt><mi>x</mi><mo>+</mo><mn>3</mn></msqrt><mo>=</mo><mi>x</mi></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn><mo>-</mo><msqrt><mi>x</mi><mo>+</mo><mn>3</mn></msqrt></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⇒</mo><mn>2</mn><msqrt><mi>x</mi><mo>+</mo><mn>3</mn></msqrt><mo>+</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mi>x</mi><mo>+</mo><mn>3</mn></msqrt><mo>=</mo><mo>-</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math></p>
<p>attempts to solve their quadratic equation <strong>M1</strong></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</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><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></msqrt></mrow><mn>2</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><mn>3</mn></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>1</mn></math> <strong>A1</strong></p>
<p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≤</mo><mo>-</mo><mn>1</mn></math>, the only solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> <strong>R1</strong></p>
<p>so the coordinates of the point of intersection are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>R0A1</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></math> is stated without a valid reason given for rejecting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced></math>.</p>
<p> </p>
<p><strong>[5 marks]</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>Line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span> intersects the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at point A and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis at point B, as shown on the diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_17.18.01.png" alt="M17/5/MATSD/SP1/ENG/TZ2/04"></p>
<p>The length of line segment OB is three times the length of line segment OA, where O is the origin.</p>
</div>
<div class="specification">
<p>Point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{(2, 6)}}">
<mrow>
<mtext>(2, 6)</mtext>
</mrow>
</math></span> lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>.</p>
</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="L">
<mi>L</mi>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx + c">
<mi>y</mi>
<mo>=</mo>
<mi>m</mi>
<mi>x</mi>
<mo>+</mo>
<mi>c</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 the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of point A.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 = - 3(2) + c">
<mn>6</mn>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>c</mi>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(y - 6) = - 3(x - 2)">
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>−</mo>
<mn>6</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for substitution of their gradient from part (a) into a correct equation with the coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(2,{\text{ }}6)">
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span> correctly substituted.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 3x + 12">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
</math></span> <strong><em>(A1)(</em>ft)</strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Award <strong><em>(A1)(</em>ft) </strong>for their correct equation. Follow through from part (a).</p>
<p>If no method seen, award <strong><em>(A1)(A0) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 3x">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</math></span>.</p>
<p>Award <strong><em>(A1)(A0) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3x + 12">
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
</math></span>.</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="0 = - 3x + 12">
<mn>0</mn>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> in their equation from part (b).</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x = ){\text{ }}4">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>4</mn>
</math></span> <strong><em>(A1)</em>(ft)</strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Follow through from their equation from part (b). Do not follow through if no method seen. Do not award the final <strong><em>(A1) </em></strong>if the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is negative or zero.</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">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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {x^4} - 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {x^3} - 4{x^2} + 2x + 6">
<mi>g</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>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</math></span></p>
<p>The functions intersect at points P and Q. Part of 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>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> are shown on the diagram.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the range of <em>f</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>Write down the <em>x</em>-coordinate of P and the <em>x</em>-coordinate of Q.</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 the values of <em>x</em> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) > g\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>></mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ { - 2,\,\,\infty } \right[{\text{ or }}\left[ { - 2,\,\,\infty } \right)"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∞</mi> </mrow> <mo>[</mo> </mrow> <mrow> <mtext> or </mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∞</mi> </mrow> <mo>)</mo> </mrow> </math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) \geqslant - 2{\text{ or }}y \geqslant - 2"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>⩾</mo> <mo>−</mo> <mn>2</mn> <mrow> <mtext> or </mtext> </mrow> <mi>y</mi> <mo>⩾</mo> <mo>−</mo> <mn>2</mn> </math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 \leqslant f\left( x \right) < \infty "> <mo>−</mo> <mn>2</mn> <mo>⩽</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </math></span> <em><strong>(A1)(A1) (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for −2 and <em><strong>(A1)</strong></em> for completely correct mathematical notation, including weak inequalities. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f \geqslant - 2"> <mi>f</mi> <mo>⩾</mo> <mo>−</mo> <mn>2</mn> </math></span>.</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>–1 and 1.52 (1.51839…) <em><strong>(A1)(A1) (C2)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for −1 and <em><strong>(A1)</strong></em> for 1.52 (1.51839).</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="x < - 1,\,\,\,x > 1.52"> <mi>x</mi> <mo><</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>></mo> <mn>1.52</mn> </math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \infty ,\,\, - 1} \right) \cup \left( {1.52,\,\,\infty } \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>∪</mo> <mrow> <mo>(</mo> <mrow> <mn>1.52</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∞</mi> </mrow> <mo>)</mo> </mrow> </math></span>. <strong><em>(A1)</em>(ft)</strong><strong><em>(A1)</em>(ft)</strong> <em><strong>(C2)</strong></em></p>
<p><strong>Note:</strong> Award <strong><em>(A1)</em>(ft)</strong> for <strong>both</strong> critical values in inequality or range statements such as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < - 1,\,\,\left( { - \infty ,\,\, - 1} \right),\,\,x > 1.52\,{\text{ or }}\left( {1.52,\,\,\infty } \right)"> <mi>x</mi> <mo><</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>></mo> <mn>1.52</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext> or </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1.52</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∞</mi> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p>Award the second <strong><em>(A1)</em>(ft)</strong> for correct strict inequality statements used with their critical values. If an incorrect use of strict and weak inequalities has already been penalized in (a), condone weak inequalities for this second mark and award <strong><em>(A1)</em>(ft)</strong>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {x^2} + bx + 11">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mi>x</mi>
<mo>+</mo>
<mn>11</mn>
</math></span>. The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 1{\text{, }}8} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow>
<mtext>, </mtext>
</mrow>
<mn>8</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> 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>
</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="b"> <mi>b</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>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {x^2}"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span> is transformed to obtain the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span>.</p>
<p>Describe this transformation.</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>valid attempt to substitute coordinates <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( { - 1} \right) = 8"> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>8</mn> </math></span></p>
<p>correct substitution <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( { - 1} \right)^2} + b\left( { - 1} \right) + 11 = 8"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>b</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>11</mn> <mo>=</mo> <mn>8</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - b + 11 = 8"> <mn>1</mn> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mn>11</mn> <mo>=</mo> <mn>8</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 4"> <mi>b</mi> <mo>=</mo> <mn>4</mn> </math></span> <em><strong>A1 N2</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>valid attempt to solve <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{x^2} + 4x + 4} \right) + 7"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>7</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = \frac{{ - 4}}{2}"> <mi>h</mi> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mn>2</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = g\left( { - 2} \right)"> <mi>k</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </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="{\left( {x + 2} \right)^2} + 7"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>7</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = - 2"> <mi>h</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 7"> <mi>k</mi> <mo>=</mo> <mn>7</mn> </math></span></p>
<p>translation or shift (do not accept move) of vector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 2} \\ 7 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> (accept left by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2"> <mn>2</mn> </math></span> and up by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="7"> <mn>7</mn> </math></span>) <em><strong>A1A1 N2</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <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> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Consider the function defined by <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> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> and its graph <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 <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 <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>, 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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>></mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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 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> (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> OR <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> <em><strong> A1</strong></em></p>
<p>correct working <em><strong> 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> OR cancelling <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math> OR <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> <em><strong> AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><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> (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> (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> <em><strong> 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> (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> <em><strong> 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> </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> (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> </mo><mo> </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> <em><strong> A1</strong></em></p>
<p>which is negative <em><strong> R1</strong></em></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is a local maximum <em><strong> AG</strong></em></p>
<p> </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> </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>></mo><mn>0</mn></math><em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>1</mn></math> <em><strong> A1</strong></em></p>
<p> </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 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"> <em><strong> A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p> </p>
<p><strong>Note: </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> 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> 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> </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="specification">
<p>The following table shows values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for different values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<p>Both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> are one-to-one functions.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mn>0</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"><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo><mo>(</mo><mn>0</mn><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of using composite function <em><strong>(M1)</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mn>0</mn></mfenced></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> <em><strong>A2</strong></em></p>
<p> </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;">
<p>This question was completed successfully by most of the candidates. In part (b) of the question, a few candidates did not recognize the notation for a composite function and instead incorrectly thought they were supposed to multiply values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mn>0</mn><mo>)</mo></math>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>Write down the equation of</p>
</div>
<div class="specification">
<p>Find the coordinates where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> crosses</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the vertical asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the horizontal asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</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>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> on the axes below.</p>
<p><img src="data:image/png;base64,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"></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><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><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><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">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><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">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></mfenced></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></math>) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for completely correct shape: two branches in correct quadrants with asymptotic behaviour.</p>
<p> </p>
<p><em><strong>[</strong></em><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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mo>…</mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>p</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Consider the case where the series is geometric.</p>
</div>
<div class="specification">
<p>Now consider the case where the series is arithmetic with common difference <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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of the series is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;">attempt to use a ratio from consecutive terms <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><msup><mi>r</mi><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mfenced><mfrac><mn>1</mn><mrow><mn>3</mn><mi>p</mi></mrow></mfrac></mfenced></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Candidates may use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>p</mi></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>…</mo></math> and consider the powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in geometric sequence</p>
<p style="text-align:left;">Award <em><strong>M1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>p</mi><mn>1</mn></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>p</mi></mfrac></math>.</p>
<p style="text-align:left;"><strong><br>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</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>p</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>M0A0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> with no other working seen.</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"><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mstyle></mrow></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mo>-</mo><mfrac><mn>3</mn><msqrt><mn>3</mn></msqrt></mfrac><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><msqrt><mn>3</mn></msqrt></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>1</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mtext>e</mtext><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">attempt to find a difference from consecutive terms or from <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;">correct equation <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mfenced><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;"><strong><br>Note:</strong> Candidates may use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>p</mi></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mo>…</mo></math> and consider the powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in arithmetic sequence.</p>
<p style="text-align:left;">Award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>-</mo><mn>1</mn><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>-</mo><mi>p</mi></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>p</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></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>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">attempt to use arithmetic mean <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>ln</mi><mo> </mo><mi>x</mi></mrow><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"><mn>2</mn><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>p</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></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>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 3</strong></p>
<p style="text-align:left;">attempt to find difference using <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>3</mn></msub></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>d</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>d</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow></mfenced></math></p>
<p style="text-align:left;">attempt to substitute into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> and equate to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</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"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math></p>
<p style="text-align:left;">correct working with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>6</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mfrac><mrow><mn>4</mn><mo>-</mo><mi>n</mi></mrow><mn>3</mn></mfrac></mfenced><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;">correct equation without <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>7</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>6</mn></mfrac><mo>=</mo><mo>-</mo><mn>3</mn></math> or equivalent</p>
<p style="text-align:left;"><strong><br>Note:</strong> Award as above if the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mi>p</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mo>…</mo></math> is considered leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>7</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn></math>.</p>
<p style="text-align:left;"><br>attempt to form a quadratic <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>7</mn><mi>n</mi><mo>-</mo><mn>18</mn><mo>=</mo><mn>0</mn></math></p>
<p style="text-align:left;">attempt to solve their quadratic <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>-</mo><mn>9</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>9</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">listing the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> terms of the sequence <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>0</mn><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mo>…</mo></math></p>
<p style="text-align:left;">recognizing first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> terms sum to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math><sup>th</sup> term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math><sup>th</sup> term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">sum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math><sup>th</sup> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math><sup>th</sup> term <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>9</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many candidates were able to identify the key relationship between consecutive terms for both geometric and arithmetic sequences. Substitution into the infinity sum formula was good with solving involving the natural logarithm done quite well. The complexity of the equation formed using 𝑆𝑛 was a stumbling block for some candidates. Those who factored out and cancelled the ln𝑥 expression were typically successful in solving the resulting quadratic.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="question">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mi>k</mi></math>.</p>
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> has no real roots.</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 – (discriminant)</strong></p>
<p>correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfenced><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mo>+</mo><mi>k</mi><mo> </mo><mo>,</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>5</mn><mo>+</mo><mi>k</mi><mo>=</mo><mn>0</mn></math></p>
<p>evidence of discriminant <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <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><mo>,</mo><mo> </mo><mtext>Δ</mtext></math></p>
<p>correct substitution into discriminant of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mrow><mo>-</mo><mn>5</mn><mo>+</mo><mi>k</mi></mrow></mfenced><mo> </mo><mo>,</mo><mo> </mo><mn>16</mn><mo>-</mo><mn>4</mn><mfenced><mrow><mi>k</mi><mo>-</mo><mn>5</mn></mrow></mfenced></math></p>
<p>recognizing discriminant is negative <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo><</mo><mn>0</mn><mo> </mo><mo>,</mo><mo> </mo><msup><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mrow><mo>-</mo><mn>5</mn><mo>+</mo><mi>k</mi></mrow></mfenced><mo><</mo><mn>0</mn><mo> </mo><mo>,</mo><mo> </mo><mn>16</mn><mo><</mo><mn>4</mn><mfenced><mrow><mi>k</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo> </mo><mo>,</mo><mo> </mo><mn>16</mn><mo>-</mo><mn>4</mn><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mn>5</mn></mfenced><mo><</mo><mn>0</mn></math></p>
<p>correct working (must be correct inequality) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mi>k</mi><mo><</mo><mo>-</mo><mn>36</mn><mo> </mo><mo>,</mo><mo> </mo><mi>k</mi><mo>-</mo><mn>5</mn><mo>></mo><mn>4</mn><mo> </mo><mo>,</mo><mo> </mo><mn>16</mn><mo>+</mo><mn>20</mn><mo>-</mo><mn>4</mn><mi>k</mi><mo><</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>9</mn></math> <em><strong>A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 2 – (transformation of vertex of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">f</mi></math>)</strong></p>
<p>valid approach for finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> vertex <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mo> </mo><mo>,</mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math></p>
<p>correct vertex of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mo> </mo><mn>9</mn></mrow></mfenced></math></p>
<p>correct vertex of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>9</mn></mrow></mfenced></math></p>
<p>correct vertex of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable></mfenced><mo> </mo><mo>,</mo><mo> </mo><mfenced><mrow><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>9</mn><mo>+</mo><mi>k</mi></mrow></mfenced></math></p>
<p>recognizing when vertex is above <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mo>+</mo><mi>k</mi><mo>></mo><mn>0</mn></math>, sketch</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>9</mn></math> <em><strong>A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 3 – (transformation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">f</mi></math>)</strong></p>
<p>recognizing vertical reflection of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo> </mo><mo>,</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>5</mn></math> , sketch</p>
<p>correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>-</mo><mn>5</mn><mo>+</mo><mi>k</mi></math></p>
<p>valid approach for finding vertex of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mo> </mo><mo>,</mo><mo> </mo><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math></p>
<p>correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> coordinate of vertex of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>9</mn><mo>+</mo><mi>k</mi><mo> </mo><mo>,</mo><mo> </mo><mfenced><mrow><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>9</mn><mo>+</mo><mi>k</mi></mrow></mfenced></math></p>
<p>recognizing when vertex is above <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mo>+</mo><mi>k</mi><mo>></mo><mn>0</mn></math> , sketch</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>9</mn></math> <em><strong>A1 N3</strong></em></p>
<p> </p>
<p><em><strong>[6 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 functions <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 defined such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{x + 3}}{4}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 8x + 5">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mi>x</mi>
<mo>+</mo>
<mn>5</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {g \circ f} \right)\left( x \right) = 2x + 11">
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>11</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {g \circ f} \right)^{ - 1}}\left( a \right) = 4">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>a</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[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>attempt to form composition <em><strong>M1</strong></em></p>
<p>correct substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( {\frac{{x + 3}}{4}} \right) = 8\left( {\frac{{x + 3}}{4}} \right) + 5">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>5</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {g \circ f} \right)\left( x \right) = 2x + 11">
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>11</mn>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute 4 (seen anywhere) <em><strong> (M1)</strong></em></p>
<p>correct equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 2 \times 4 + 11">
<mi>a</mi>
<mo>=</mo>
<mn>2</mn>
<mo>×</mo>
<mn>4</mn>
<mo>+</mo>
<mn>11</mn>
</math></span> <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 19 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>4</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>0</mn></math>.</p>
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>13</mn><mo>,</mo><mo> </mo><mn>7</mn></mrow></mfenced></math> lies on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</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>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>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>.</p>
<p>On the following grid, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAxQAAAHRCAYAAADzOf+/AAAgAElEQVR4Ae2972td2Znn2/+JA2KKthlD5+aNmaarGegGd70QRXEJ3WMa0jhO7GF6KJpANy2PQ4e8SOeCjUWYYmpykSc1BX3TSdvXlRBIKoh6UzQOgryoF0ZwB4pEiEsRCiFIYcLhGdZZ2tbRo72O9j57fc/Zj/QR2LLW2fu7v/uznvVdfrT14/eMNwhAAAIQgAAEIAABCEAAAgsS+L0Fz+M0CEAAAhCAAAQgAAEIQAACRkNBEUAAAhCAAAQgAAEIQAACCxOgoVgYHSdCAAIQgAAEIAABCEAAAjQU1AAEIAABCEAAAhCAAAQgsDABGoqF0XEiBCAAAQhAAAIQgAAEIHCqoVi79DnjDwyoAWqAGqAGqAFqgBqgBqgBaqBUA7NtVGtDMXsA/4YABCAAAQhAAAIQgAAEINAQSE3G7BsNxSwN/g0BCEAAAhCAAAQgAAEIzCVAQzEXDy9CAAIQgAAEIAABCEAAAvMI0FDMo8NrEIAABCAAAQhAAAIQgMBcAjQUc/HwIgQgAAEIQAACEIAABCAwjwANxTw6vAYBCEAAAhCAAAQgAAEIzCVAQzEXDy9CAAIQgAAEIAABCEAAAvMI0FDMo8NrEIAABCAAAQhAAAIQgMBcAjQUc/HwIgQgAAEIQAACEIAABCAwjwANxTw6vAYBCEAAAhCAAAQgAAEIzCVAQzEXDy9CAAIQgAAEIAABCEAAAvMI0FDMo8NrEIAABCAAAQhAAAIQgMBcAjQUc/HwIgQgAAEIQAACEIAABCAwjwANxTw6vAYBCEAAAhCAAAQgAAEIzCVAQzEXDy9CAAL9CfzO9h//taVwWbt0xV67/8wOpyITO9i+Z1cvv2lP9l70l+UMCEAAAhCAAARGSYCGYpTTgikInAMCk+f2aP2Kra1v2e4k389k/337+vUv2aPdz87BDXILEIAABCAAAQgkAjQU1AEEICAicGg79//M1v7gvu387ugSqcm4/R3bOTzqMERXRhYCEIAABCAAgeURoKFYHmuuBIELRuAz2926YWuX/tqe7KeO4oXtPd6w//z4Y6OduGClwO1CAAIQgMC5JkBDca6nl5uDwCoJNN9LcdRQHD6zzXtPbY9uYpWTwrUhAAEIQAAC1QnQUFRHiiAEIJAJHH0T9qU/swfPntv2P3zr1Ddj//rXv7YPP/wQYBCAAAQgAAEIBCZAQxF48rAOgbET+N3OffvCpWv2Vze/Zt/Y3jv1pU7/8c6d6Tdy/fKXvxz7reAPAhCAAAQgAIECARqKAhiGIQCB4QQmu1v2+qVrdmvro6MfHXus+eTx46MfLfs5+/d//Mf229/+9vhF/gUBCEAAAhCAQBgCNBRhpgqjEIhHYLL7jt25977tu++b+M1vfjNtJr79j//4sqn4n++8E+8GcQwBCEAAAhCAwHQvn8Xwe7MfpH/7jsO/zscQgAAEWgmkb8LeeNeet/yI2I2///uXTyVSxjRPK3Z3d1ulGIQABCAAAQhAYLwEfL9AQzHeucIZBMZP4PCZPbj+J/Z3Wz+0zY23W3/fxM9+9rPpJyqab8ZOIZS+3OnPv/jF6Z/x3yQOIQABCEAAAhCYJUBDMUuDf0MAAsMIHGzb3ctX7LWNd21n/8UprdQ4pO+XSE8omrcmhNLTifTv9LSCNwhAAAIQgAAE4hBo9vLGMU8oGhK8hwAEqhNI3zORGor0PRTN22wIvfVf35o2FenHyfIGAQhAAAIQgEAMArN7eXJMQxFj3nAJgXAE0o+GbXsCMRtCbU8wwt0ohiEAAQhAAAIXjMDsXp5unYbighUAtwuBZRFIX8o0+6VOzXV9CKXGgx8j29DhPQQgAAEIQGD8BPxeTkMx/jnDIQTOFQEfQufq5rgZCEAAAhCAwAUg4PdyGooLMOncIgTGRMCH0Ji84QUCEIAABCAAgbMJ+L2chuJsZhwBAQhUJOBDqKI0UhCAAAQgAAEILIGA38tpKJYAnUtAAALHBHwIHb/CvyAAAQhAAAIQiEDA7+U0FBFmDY8QOEcEfAido1vjViAAAQhAAAIXgoDfy2koLsS0c5MQGA8BH0LjcYYTCEAAAhCAAAS6EPB7OQ1FF2ocAwEIVCPgQ6iaMEIQgAAEIAABCCyFgN/LaSiWgp2LQAACDQEfQs047yEAAQhAAAIQiEHA7+U0FDHmDZcQODcEfAidmxvjRiAAAQhAAAIXhIDfy2koLsjEc5sQGAsBH0Jj8YUPCEAAAhCAAAS6EfB7OQ1FN24cNWIC7/3oJ5b+NG/Nx6WxZnz2uGYsabSNt435Y9uuX9Kdd+zYrtXcw6K+Uuj4P42mZ9iML3qt5nyve5F4w+A4D7rUUcOry7Hz6oiaa89Oz2WsvGv56lJHbddqaov3EIhCgIYiykzhszOBJpw7n9DhQIVmumwkXZVXH0IdpuPMQ1ReI+lG8qpaCzCIlTHUQY42Vd2eGZwcAIEBBPxezhOKATA5FQIQ6E/Ah1B/Bc6AAAQgAAEIQGCVBPxeTkOxytng2hC4gAR8CF1ABNwyBCAAAQhAIDQBv5fTUISeTswnAorHxQpNlVeVroqBD6EaVazyGkk3ktdoNRuJbSSv1EFOP9Wc1chWNCBQIuD3chqKEinGwxBQhLFCMwGNpKvy6kOoRqGpvEbSjeRVtRZgECtjqIOcfqq6rZGtaECgRMDv5TQUJVKMhyGgCGOFZgIaSVfl1YdQjUJTeY2kG8mrai3AIFbGUAc5/VR1WyNb0YBAiYDfy2koSqQYD0NAEcYKzQQ0kq7Kqw+hGoWm8hpJN5JX1VqAQayMoQ5y+qnqtka2ogGBEgG/l9NQlEgxDgEISAj4EJJcBFEIQAACEIAABGQE/F5OQyFDjfCyCCg+u6PQTDwi6aq8+hCqUScqr5F0I3lVrQUYxMoY6iCnn6pua2QrGhAoEfB7OQ1FiRTjYQgowlihmYBG0lV59SFUo9BUXiPpRvKqWgswiJUx1EFOP1Xd1shWNCBQIuD3chqKEinGwxBQhLFCMwGNpKvy6kOoRqGpvEbSjeRVtRZgECtjqIOcfqq6rZGtaECgRMDv5TQUJVKMhyGgCGOFZgIaSVfl1YdQjUJTeY2kG8mrai3AIFbGUAc5/VR1WyNb0YBAiYDfy2koSqQYhwAEJAR8CEkugigEIAABCEAAAjICfi+noZChRhgCEGgj4EOo7RjGIAABCEAAAhAYLwG/l9NQjHeucNaRgOJxsUIz3U4k3WOvEzvc/ak9uHnNUoCsXbpmt+7/1HYPJx1n6ORhPoROvrrYR8deFzu/dFYk3UheE2+FX4WmyqtKFwaa2lLNl1I3afMGARUBv5fTUKhIo7s0AooNVKGZgETSbbxO9h7bncvX7NbWR3aYbmKyZ9v31u3qxrYdLDDLPoQWkDh1SuP11AsDByLpRvKapkXhV6Gp8qrShYGmtlTzpdRN2rxBQEXA7+U0FCrS6C6NgGIDVWgmIJF0s9fPbHfrhq1dvmfbB8dPJCa7W/a6G+s64T6Eup4377hIXFV1AINY64s6yCtaUbcKTdV8KXUzYf6GgIaA38tpKDScUYXAOSEwsYPte3b10hv2YOfTo3vKY18Y0ROKcwKb24AABCAAAQiEIEBDEWKaMAmBERE4fGYPrl+xtcu37dHzA5vsv2/f2HjbfrH/YiGTPoQWEuEkCEAAAhCAAARWRsDv5TyhWNlUcOFaBBSPtxWa6X4j6c56nex/aJvNN2Wvb9nu8Vc/9Z5GH0K9BVpOmPXa8vLCQ5F0I3lNE6Lwq9BUeVXpwkBTW6r5Uuombd4goCLg93IaChVpdJdGIG2gs5to83FprBmfPa4ZS6bbxtvG/LHNDQ89duj5fXx1uVY6xg6f23/5ym376l/8ka1desX+3Y1v2f5RUzGrka6dQuasP1PNI2Cz5zfjbWN97qvPsWO7FgyO1/PQuelyPrzh3dSJz42mNvx4+ji9Nec170tjpfHmvNnrTIX5CwIBCNBQBJgkLPYjoAhjhWa6q0i6L70efmSPbm/Yk730JU4vbH/7m/bapSv22v1n+ac+9ZuuabPR85QzD3/p9cwj+x0QSTeS1zQLCr8KTZVXlS4MNLWlmi+lbtLmDQIqAjQUKrLoroyAYgNVaCZAkXSz16Of8nTiy5wmdrizaa9dumGPdj/rPe8+hHoLtJwQiauqDmAQa31RB3khK+pWoamaL6VuJszfENAQ8Hs5X/Kk4YwqBM4JgU9se+NVWzvRUJjZwbbdvTyehuKcwOY2IAABCEAAAiEI0FCEmCZMQmAsBJqnETO/2M4O7PnWbfvC7ce2t8A3Z/sQGsud4gMCEIAABCAAgW4E/F7OE4pu3DhqxAQUj7cVmglhJN1jry9sf+ddu5t+dOz0G66v2a37P7XdwwW6iaNv2q5dTsde6ypH0o3kVbUWYBArY6iDnFequq2bhqhB4CQBGoqTPPjoHBBQhLFCM6GOpKvy6kOoRgmqvEbSjeRVtRZgECtjqIOcfqq6rZGtaECgRMDv5TyhKJFiPAwBRRgrNBPQSLoqrz6EahSaymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFOAQgICHgQ0hyEUQhAAEIQAACEJAR8Hs5DYUMNcIQgEAbAR9CbccwBgEIQAACEIDAeAn4vZyGYrxzhbOOBBSPixWa6XYi6aq8+hDqOM1zD1N5jaQbyatqLcAgVsZQBznWVHU7NzR5EQIDCfi9nIZiIFBOXz0BRRgrNBOpSLoqrz6EalSQymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSK8TAEFGGs0ExAI+mqvPoQqlFoKq+RdCN5Va0FGMTKGOogp5+qbmtkKxoQKBHwezkNRYkU4xCAgISADyHJRRCFAAQgAAEIQEBGwO/lNBQy1Agvi4DiszsKzcQjkq7Kqw+hGnWi8hpJN5JX1VqAQayMoQ5y+qnqtka2ogGBEgG/l9NQlEgxHoaAIowVmgloJF2VVx9CNQpN5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAwBRRgrNBPQSLoqrz6EahSaymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSKcQhAQELAh5DkIohCAAIQgAAEICAj4PdyGgoZaoQhAIE2Aj6E2o5hDAIQgAAEIACB8RLwezkNxXjnCmcdCSgeFys00+1E0lV59SHUcZrnHqbyGkk3klfVWoBBrIyhDnKsqep2bmjyIgQGEvB7OQ3FQKCcvnoCKYxnA7n5uDTWjM8e14ylu2kbbxvzxzYkhh479Pw+vrpcKx3jNZsxPz490DFMoeP/lM5vxtP72T9tum1jzfldfDXHzl6nGetzfp9ju1yr8dDlWBjk2vRz0Idhn2PhfX5516qDLuu27VpNbfEeAlEI0FBEmSl8dibQhHPnEzocqNBMl42kq/LqQ6jDdJx5iMprJN1IXlVrAQaxMoY6yNGmqtszg5MDIDCAgN/LeUIxACanQgAC/Qn4EOqvwBkQgAAEIAABCKySgN/LaShWORtcGwIXkIAPoQuIgFuGAAQgAAEIhCbg93IaitDTiflEQPG4WKGp8qrSVTHwIVSjilVeI+lG8hqtZiOxjeSVOsjpp5qzGtmKBgRKBPxeTkNRIsV4GAKKMFZoJqCRdFVefQjVKDSV10i6kbyq1gIMYmUMdZDTT1W3NbIVDQiUCPi9nIaiRIrxMAQUYazQTEAj6aq8+hCqUWgqr5F0I3lVrQUYxMoY6iCnn6pua2QrGhAoEfB7OQ1FiRTjYQgowlihmYBG0lV59SFUo9BUXiPpRvKqWgswiJUx1EFOP1Xd1shWNCBQIuD3chqKEinGIQABCQEfQpKLIAoBCEAAAhCAgIyA38tpKGSoEYYABNoI+BBqO4YxCEAAAhCAAATGS8Dv5TQU450rnHUkoHhcrNBMtxNJV+XVh1DHaZ57mMprJN1IXlVrAQaxMoY6yLGmqtu5ocmLEBhIwO/lNBQDgXL66gkowlihmUhF0lV59SFUo4JUXiPpRvKqWgswiJUx1EFOP1Xd1shWNCBQIuD3chqKEinGwxBQhLFCMwGNpKvy6kOoRqGpvEbSjeRVtRZgECtjqIOcfqq6rZGtaECgRMDv5TQUJVKMQwACEgI+hCQXQRQCEIAABCAAARkBv5fTUMhQIwwBCLQR8CHUdgxjEIAABCAAAQiMl4Dfy2koxjtXOOtIQPG4WKGZbieSbqvXyb7tPP0ne3Dzmq1detXubn/ScZaOD/MhdPzK4v9q9bq43MszI+lG8poAK/wqNFVeVbow0NSWar6UukmbNwioCPi9nIZCRRrdpRFQbKAKzQQkkq73Otn/0DZvXrOrN+/bv2zv2uGCM+xDaEGZE6d5rydeHPBBJN1IXtOUKPwqNFVeVbow0NSWar6UukmbNwioCPi9nIZCRRrdpRFQbKAKzQQkku4Jr4fP7MH1V+3Www9tfzJsan0IDVPLZ5/wWkPwSCOSbiSvCa/Cr0JT5VWlCwNNbanmS6mbtHmDgIqA38tpKFSk0V0aAcUGqtBMQCLpHnv91Hbuv2FXbz+2vYHNRGLgQ6hGoRx7raF2rBFJN5LXRFjhV6Gp8qrShYGmtlTzpdRN2rxBQEXA7+U0FCrS6ELgnBCY7G7Z65fW7e7Wu0ffO/E5u3pz036+e7DQHfoQWkiEkyAAAQhAAAIQWBkBv5fTUKxsKrhwLQKKz8gpNNP9RtLNXj+z3a0btnZ9w97Z2bfpA4rDj+xR+qbs65u2c9j/kYUPoRp1EImrqg5gEGt9UQd55SvqVqGpmi+lbibM3xDQEPB7OQ2FhjOqSySg2DwUmglJJN3s9RPb3njVrm5s2+zziPzU4oq9vvU8Nxk95tuHUI9Ti4dG4qqqAxjEWl/UQV7OirpVaKrmS6mbCfM3BDQE/F5OQ6HhjOoSCaTNY3YDaT4ujTXjs8c1Y8l223jbmD+2ueWhxw49v4+vM6/19Lv2N3/4yrSh+P4R5+k5T79rj9avnBpP104hc9afpNG8tXloG+tzX32OHdu1GjZj8zVvvmrzrsWgC8Na16rNAN7zM6I271p1sGjNNfPNewhEIUBDEWWm8NmZQLMRdD6hw4EKzXTZSLrZa35Csba+ZbuzX900eW6P1j/PE4oOtdR2iKIOFJoxa7aN+LCxSGwjeVXVFwyG1TtnQ6ALARqKLpQ4BgIQOCIwsYPte3b18pv2ZO/FMZWDbbt7+YY92v3seKzjv3wIdTyNwyAAAQhAAAIQGAkBv5fzJU8jmRhsQGC0BCYf25Pb6RfavWPP0zdhT/btFw9v2+v3ny30y+18CI32vjEGAQhAAAIQgEArAb+X01C0YmIwEgHF422FZmIaSfeE18Nd+/n9L9vV6fdHrNvd7z1b+Bfc+RCqUWsnvNYQPNKIpBvJq2otwCBWxlAHOWhUdXsUY7yDgISA38tpKCSYEV0mAUUYKzQTk0i6Kq8+hGrUisprJN1IXlVrAQaxMoY6yOmnqtsa2YoGBEoE/F5OQ1EixXgYAoowVmgmoJF0VV59CNUoNJXXSLqRvKrWAgxiZQx1kNNPVbc1shUNCJQI+L2chqJEivEwBBRhrNBMQCPpqrz6EKpRaCqvkXQjeVWtBRjEyhjqIKefqm5rZCsaECgR8Hs5DUWJFOMQgICEgA8hyUUQhQAEIAABCEBARsDv5TQUMtQIL4uA4rM7Cs3EI5KuyqsPoRp1ovIaSTeSV9VagEGsjKEOcvqp6rZGtqIBgRIBv5fTUJRIMR6GgCKMFZoJaCRdlVcfQjUKTeU1km4kr6q1AINYGUMd5PRT1W2NbEUDAiUCfi+noSiRYjwMAUUYKzQT0Ei6Kq8+hGoUmsprJN1IXlVrAQaxMoY6yOmnqtsa2YoGBEoE/F5OQ1EixXgYAoowVmgmoJF0VV59CNUoNJXXSLqRvKrWAgxiZQx1kNNPVbc1shUNCJQI+L2chqJEinEIQEBCwIeQ5CKIQgACEIAABCAgI+D3choKGWqEIQCBNgI+hNqOYQwCEIAABCAAgfES8Hs5DcV45wpnHQkoHhcrNNPtRNJVefUh1HGa5x6m8hpJN5JX1VqAQayMoQ5yrKnqdm5o8iIEBhLwezkNxUCgnL56AoowVmgmUpF0VV59CNWoIJXXSLqRvKrWAgxiZQx1kNNPVbc1shUNCJQI+L2chqJEivEwBBRhrNBMQCPpqrz6EKpRaCqvkXQjeVWtBRjEyhjqIKefqm5rZCsaECgR8Hs5DUWJFOMQgICEgA8hyUUQhQAEIAABCEBARsDv5TQUMtQIQwACbQR8CLUdwxgEIAABCEAAAuMl4PdyGorxzhXOOhJQPC5WaKbbiaSr8upDqOM0zz1M5TWSbiSvqrUAg1gZQx3kWFPV7dzQ5EUIDCTg93IaioFAOX31BFIYzwZy83FprBmfPa4ZS3fTNt425o9tSAw9duj5fXx1uVY6xms2Y358eqBjmELH/ymd34yn97N/2nTbxprzu/hqjp29TjPW5/w+x3a5VuOhy7EwyLXp56APwz7Hwvv88q5VB13Wbdu1mtriPQSiEKChiDJT+OxMoAnnzid0OFChmS4bSVfl1YdQh+k48xCV10i6kbyq1gIMYmUMdZCjTVW3ZwYnB0BgAAG/l/OEYgBMTh0HAUUYKzQTrUi6Kq8+hGpUkcprJN1IXlVrAQaxMoY6yOmnqtsa2YoGBEoE/F5OQ1EixTgEICAh4ENIchFEIQABCEAAAhCQEfB7OQ2FDDXCyyKg+OyOQjPxiKSr8upDqEadqLxG0o3kVbUWYBArY6iDnH6quq2RrWhAoETA7+U0FCVSjIchoAhjhWYCGklX5dWHUI1CU3mNpBvJq2otwCBWxlAHOf1UdVsjW9GAQImA38tpKEqkGA9DQBHGCs0ENJKuyqsPoRqFpvIaSTeSV9VagEGsjKEOcvqp6rZGtqIBgRIBv5fTUJRIMR6GgCKMFZoJaCRdlVcfQjUKTeU1km4kr6q1AINYGUMd5PRT1W2NbEUDAiUCfi+noSiRYhwCEJAQ8CEkuQiiEIAABCAAAQjICPi9nIZChhphCECgjYAPobZjGIMABCAAAQhAYLwE/F5OQzHeucJZRwKKx8UKzXQ7kXRVXn0IdZzmuYepvEbSjeRVtRZgECtjqIMca6q6nRuavAiBgQT8Xk5DMRAop6+egCKMFZqJVCRdlVcfQjUqSOU1km4kr6q1AINYGUMd5PRT1W2NbEUDAiUCfi+noSiRYjwMAUUYKzQT0Ei6Kq8+hGoUmsprJN1IXlVrAQaxMoY6yOmnqtsa2YoGBEoE/F5OQ1EixXgYAoowVmgmoJF0VV59CNUoNJXXSLqRvKrWAgxiZQx1kNNPVbc1shUNCJQI+L2chqJEinEIQEBCwIeQ5CKIQgACEIAABCAgI+D3choKGWqEl0VA8dkdhWbiEUm35HWy99juXL5hj3Y/W2iKfQgtJOJOKnl1h/X+MJJuJK9pIhR+FZoqrypdGGhqSzVfSt2kzRsEVAT8Xk5DoSKN7tIIKDZQhWYCEkm31evkY3ty+5qtXaKhGFLgrWyHCAarLdVaUHBVeVXpwiBWzqrqIOnyBgElARoKJV20V0JAsYEqNBOcSLqnvX5qO/fftL/b+JJdpaEYVOun2Q6Sm56s0Ixfs8O5wiAzjFRfkbwq66tO9aMCgXYCNBTtXBgNTECxeSg0E+JIuie9Tuxw5zt26/6/2t72PRqKgevlJNuBYkenKzRj12wdrjDIHCPVVySvyvqqtwJQgsBpAjQUp5kwAgEInEXg8Jk9uPkd2zn8nR2MsKE4yz6vQwACEIAABCBQjwANRT2WKEHgghBIX+r0NXuw86mZTWgoLsisc5sQgAAEIACBEgEaihIZxsMSUDzeVmgmwJF0s9f0pU5v29cff2yTaYWMs6GIxFVVBzCItb6og2mgSDKRtZDZ8jcElARoKJR00V4JgbR5zG4gzcelsWZ89rhmLN1A23jbmD+2ufmhxw49v4+vM6/1g027c+MfbG/ScPmxff9bf2m/f+lPX/7Y2FmNdO0UMmf9Sec0b7PnN+NtY33uq8+xY7sWDI7X89C56XI+vOHd1InPjaY2/Hj6OL015zXvS2Ol8ea82etMhfkLAgEI0FAEmCQs9iOgCGOFZrqrSLrv/ejHR1/eNKdBWN+y3fzoovOk+RDqfOKcAyNxVdUBDGKtL+ogL2hF3So0VfOl1M2E+RsCGgJ+L+f3UGg4owqBc0pgnF/ydE5hc1sQgAAEIACBURKgoRjltGAKAlEI0FBEmSl8QgACEIAABFQEaChUZNFdGQHF422FZgIUSbfd6zgbinavw0sykm4kr6q1AINYGUMd5IxS1e3wBEQBAmUCNBRlNrwSlIAijBWaCW8kXZVXH0I1yk7lNZJuJK+qtQCDWBlDHeT0U9VtjWxFAwIlAn4v53soSqQYD0NAEcYKzQQ0kq7Kqw+hGoWm8hpJN5JX1VqAQayMoQ5y+qnqtka2ogGBEgG/l9NQlEgxHoaAIowVmgloJF2VVx9CNQpN5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiHAIQkBDwISS5CKIQgAAEIAABCMgI+L2chkKGGmEIQKCNgA+htmMYgwAEIAABCEBgvAT8Xk5DMd65wllHAorHxQrNdDuRdFVefQh1nOa5h6m8RtKN5FW1FmAQK2OogxxrqrqdG5q8CIGBBPxeTkMxECinr56AIowVmolUJF2VVx9CNSpI5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAwBRVMYIc4AACAASURBVBgrNBPQSLoqrz6EahSaymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFOAQgICHgQ0hyEUQhAAEIQAACEJAR8Hs5DYUMNcIQgEAbAR9CbccwBgEIQAACEIDAeAn4vZyGYrxzhbOOBBSPixWa6XYi6aq8+hDqOM1zD1N5jaQbyatqLcAgVsZQBznWVHU7NzR5EQIDCfi9nIZiIFBOXz0BRRgrNBOpSLoqrz6EalSQymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSK8TAEFGGs0ExAI+mqvPoQqlFoKq+RdCN5Va0FGMTKGOogp5+qbmtkKxoQKBHwezkNRYkU4xCAgISADyHJRRCFAAQgAAEIQEBGwO/lNBQy1Agvi4DiszsKzcQjkq7Kqw+hGnWi8hpJN5JX1VqAQayMoQ5y+qnqtka2ogGBEgG/l9NQlEgxHoaAIowVmgloJF2VVx9CNQpN5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAyBFMazgdx8XBprxmePa8bSTbeNt435YxtgQ48den4fX12ulY7xms2YH58e6Bim0PF/Suc34+n97J823bax5vwuvppjZ6/TjPU5v8+xXa7VeOhyLAxybfo56MOwz7HwPr+8a9VBl3Xbdq2mtngPgSgEaCiizBQ+OxNowrnzCR0OVGimy0bSVXn1IdRhOs48ROU1km4kr6q1AINYGUMd5GhT1e2ZwckBEBhAwO/lPKEYAJNTIQCB/gR8CPVX4AwIQAACEIAABFZJwO/lNBSrnA2uDYELSMCH0AVEwC1DAAIQgAAEQhPwezkNRejpxHwioHhcrNBUeVXpqhj4EKpRxSqvkXQjeY1Ws5HYRvJKHeT0U81ZjWxFAwIlAn4vp6EokWI8DAFFGCs0E9BIuiqvPoRqFJrKayTdSF5VawEGsTKGOsjpp6rbGtmKBgRKBPxeTkNRIsV4GAKKMFZoJqCRdFVefQjVKDSV10i6kbyq1gIMYmUMdZDTT1W3NbIVDQiUCPi9nIaiRIrxMAQUYazQTEAj6aq8+hCqUWgqr5F0I3lVrQUYxMoY6iCnn6pua2QrGhAoEfB7OQ1FiRTjEICAhIAPIclFEIUABCAAAQhAQEbA7+U0FDLUCC+LgOKzOwrNxCOSrsqrD6EadaLyGkk3klfVWoBBrIyhDnL6qeq2RraiAYESAb+X01CUSDEehoAijBWaCWgkXZVXH0I1Ck3lNZJuJK+qtQCDWBlDHeT0U9VtjWxFAwIlAn4vp6EokWI8DAFFGCs0E9BIuiqvPoRqFJrKayTdSF5VawEGsTKGOsjpp6rbGtmKBgRKBPxeTkNRIsV4GAKKMFZoJqCRdFVefQjVKDSV10i6kbyq1gIMYmUMdZDTT1W3NbIVDQiUCPi9nIaiRIpxCEBAQsCHkOQiiEIAAhCAAAQgICPg93IaChlqhCEAgTYCPoTajmEMAhCAAAQgAIHxEvB7OQ3FeOcKZx0JKB4XKzTT7UTSPfZ6YLvvb9qty5+zFCBr1zfsnZ19m3ScH3+YDyH/+iIfH3td5OzyOZF0I3lNxBV+FZoqrypdGGhqSzVfSt2kzRsEVAT8Xk5DoSKN7tIIKDZQhWYCEkk3e31he0/fss33d+0w3cBk337x8Mt29dIb9mDn04Xm2IfQQiLupEhck3WFX4WmyqtKFwaa2lLNl0qXOkhkeYOAloDfy2kotLxRXwIBxeah0EwoIulOvU7+l33wwa9OPo2YPLdH61fs6sa2HSwwvz6EFpA4dUokrqo6gEGs9UUd5GWsqFuFpmq+lLqZMH9DQEPA7+U0FBrOqELgHBP4xLY3Xh1VQ3GOYXNrEIAABCAAgdERoKEY3ZRgCALRCKSGYt3uPP745JOLjrfhQ6jjaRwGAQhAAAIQgMBICPi9nCcUI5kYbCxOQPF4W6GZ7jCSbtHrwbbdXd+0ncPFvi3bh9DiM398ZtHr8SEL/SuSbiSvaTIUfhWaKq8qXRhoaks1X0rdpM0bBFQE/F5OQ6Eije7SCKQNdHYTbT4ujTXjs8c1Y8l023jbmD+2ueGhxw49v4+vLtdKxxxr/sAefuWL9tWHP2hu9wSvNJhC5qw/jeaxbp7DZryLr8bA0GOHnu/vYZ6vLtdKx3jNZsyPz7tWn2O7+FrmtZr7Heqry/m1rgXv9uz0XMbKu5avRWuuWV+8h0AUAjQUUWYKn50JNBtB5xM6HKjQTJeNpHva68QOd962u1sf5Z/41IFj2yE+hNqO6Tt22mtfhfbjI+lG8ppoK/wqNFVeVbow0NSWar6UukmbNwioCPi9nCcUKtLoLo2AYgNVaCYgkXS918neT2zze8OaicTAh1CNQvFea2hGny8Y1CIQe93WohBpjUXyqsyZWnOPDgTaCPi9nIaijRJjEIDACQKT/W3bfLht+y+/bWJih89/aI8++OTEcV0+8CHU5RyOgQAEIAABCEBgPAT8Xk5DMZ65wcmCBBSfjVJoptuLpJu9Tuxw97HdvX6l5fsibtij3c96z5oPod4CLSdE4qqqAxjEWl/UQV7IirpVaKrmS6mbCfM3BDQE/F5OQ6HhjOoSCSg2D4VmQhJJN3md7D22O5cL32S9vmW7L59YdJ9wH0LdzywfGYmrqg5gEGt9UQd5PSvqVqGpmi+lbibM3xDQEPB7OQ2FhjOqSySg2DwUmglJJF2VVx9CNUpF5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAwBRRgrNBPQSLoqrz6EahSaymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFOAQgICHgQ0hyEUQhAAEIQAACEJAR8Hs5DYUMNcIQgEAbAR9CbccwBgEIQAACEIDAeAn4vZyGYrxzhbOOBBSPixWa6XYi6aq8+hDqOM1zD1N5jaQbyatqLcAgVsZQBznWVHU7NzR5EQIDCfi9nIZiIFBOXz0BRRgrNBOpSLoqrz6EalSQymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSK8TAEFGGs0ExAI+mqvPoQqlFoKq+RdCN5Va0FGMTKGOogp5+qbmtkKxoQKBHwezkNRYkU4xCAgISADyHJRRCFAAQgAAEIQEBGwO/lNBQy1Agvi4DiszsKzcQjkq7Kqw+hGnWi8hpJN5JX1VqAQayMoQ5y+qnqtka2ogGBEgG/l9NQlEgxHoaAIowVmgloJF2VVx9CNQpN5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAwBRRgrNBPQSLoqrz6EahSaymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSKcQhAQELAh5DkIohCAAIQgAAEICAj4PdyGgoZaoQhAIE2Aj6E2o5hDAIQgAAEIACB8RLwezkNxXjnCmcdCSgeFys00+1E0lV59SHUcZrnHqbyGkk3klfVWoBBrIyhDnKsqep2bmjyIgQGEvB7OQ3FQKCcvnoCKYxnA7n5uDTWjM8e14ylu2kbbxvzxzYkhh479Pw+vrpcKx3jNZsxPz490DFMoeP/lM5vxtP72T9tum1jzfldfDXHzl6nGetzfp9ju1yr8dDlWBjk2vRz0Idhn2PhfX5516qDLuu27VpNbfEeAlEI0FBEmSl8dibQhHPnEzocqNBMl42kq/LqQ6jDdJx5iMprJN1IXlVrAQaxMoY6yNGmqtszg5MDIDCAgN/LeUIxACanQgAC/Qn4EOqvwBkQgAAEIAABCKySgN/LaShWORtcGwIXkIAPoQuIgFuGAAQgAAEIhCbg93IaitDTiflEQPG4WKGp8qrSVTHwIVSjilVeI+lG8hqtZiOxjeSVOsjpp5qzGtmKBgRKBPxeTkNRIsV4GAKKMFZoJqCRdFVefQjVKDSV10i6kbyq1gIMYmUMdZDTT1W3NbIVDQiUCPi9nIaiRIrxMAQUYazQTEAj6aq8+hCqUWgqr5F0I3lVrQUYxMoY6iCnn6pua2QrGhAoEfB7OQ1FiRTjYQgowlihmYBG0lV59SFUo9BUXiPpRvKqWgswiJUx1EFOP1Xd1shWNCBQIuD3chqKEinGIQABCQEfQpKLIAoBCEAAAhCAgIyA38tpKGSoEYYABNoI+BBqO4YxCEAAAhCAAATGS8Dv5TQU450rnHUkoHhcrNBMtxNJV+XVh1DHaZ57mMprJN1IXlVrAQaxMoY6yLGmqtu5ocmLEBhIwO/lNBQDgXL66gkowlihmUhF0lV59SFUo4JUXiPpRvKqWgswiJUx1EFOP1Xd1shWNCBQIuD3chqKEinGwxBQhLFCMwGNpKvy6kOoRqGpvEbSjeRVtRZgECtjqIOcfqq6rZGtaECgRMDv5TQUJVKMQwACEgI+hCQXQRQCEIAABCAAARkBv5fTUMhQIwwBCLQR8CHUdgxjEIAABCAAAQiMl4Dfy2koxjtXOOtIQPG4WKGZbieSrsqrD6GO0zz3MJXXSLqRvKrWAgxiZQx1kGNNVbdzQ5MXITCQgN/LaSgGAuX01RNQhLFCM5GKpKvy6kOoRgWpvEbSjeRVtRZgECtjqIOcfmfV7ZPHj+3DDz+sEZVoQKAaAb+X01BUQ4vQqgicFcaL+FJoJh+RdFVefQgtMj/+HJXXSLqRvKrWAgxiZQx1kJPsrLp967++ZSk3N/7+7+03v/mNjz8+hsBKCPi9nIZiJdPARWsSOCuMF7mWQjP5iKSr8upDaJH58eeovEbSjeRVtRZgECtjqIOcZF3q9mc/+5n9+z/+4+mf9MSCNwismoDfy2koVj0jXB8CF4yAD6ELdvvcLgQgAIGFCPz2t7+1b//jP06fVvzHO3fs17/+9UI6nASBGgT8Xk5DUYMqGisl0OWzO30NKjSTh0i6Kq8+hPrOTdvxKq+RdCN5Va0FGMTKGOogp1nfuv3lL39pf/7FL04bi//5zjuWGg3eILBsAn4vp6FY9gxwveoE+oZxFwMKzXTdSLoqrz6EuszHWceovEbSjeRVtRZgECtjqIOcbIvUbWoiUjOR8jQ1FzytOGuX4PXaBPxeTkNRmzB6SyWQCpo/MKAGqAFqgBq4yDWQvseCNwgsk0Bab7NvNBSzNPh3SAK+qGvchEIz+VLoJs1FPsN1FieFVyWDs+5nkddhoKlZ6iBXo6K+FJqq+VLpnncG6elE+sbsdJ/pG7XTl0DxBoFlE/DrjIZi2TPA9aoT8EVd4wIKzeRLoavQVHlV6cJAU1uq+VLpUgfUgaq2xqK7u7vL90+kyeBt5QR83tJQrHxKMDCUgC/qoXrpfIWmSjeSVxjk6lTMmUJTNV8qXRjEyi7qoHsepKcSze+jSN8zkRoLs1/Zk5ufn+5XV28/tr2J2WT/Q9u8ec3WLv2ZPdg5zBfgbwgICPi8paEQQEZyuQR8Ude4ukIz+VLoJs3Vf8nTC9vfedfuXr9ia5eu2a2HH9r+pH0mVAzarzZsVOFVWQfD7rb9bBho1i11kOtNUV8KTdV8ddFNX9KUvrQp3dfp30ExscOdTXvt0g17tPOBbW68a88PC+GbkfM3BKoQ8OuMhqIKVkRWSWD1/5nufvcqrwrd7ppHG9r1b9r2/guzyZ5t33vDXrv/zNo+P+ZDqDu98pHdvZY12l5ReE3XUfiN5DUxUPhVcFV5pQ4SAeqgSx2k3zmRfkt2+Sc5HT2puPymPdl7kcHyNwTEBHyG01CIgSOvJ6D4T4RfKLXuQuVVodtZc/LcHq3/id3d/uQY08G23b18wx7tfnY8dvQvBdvOXk+5mT+g8JquqPAbyWtioPCr4KrySh0kAtRBnTr4zHa3btjazce2n7HyNwTkBHyG01DIkXOBiAT8QhnzPaza62R3y15Pj9tPNA+f2PbGn9jrW8/NP3xftd8+c4nXPrT6HQvbfry6Hh2Ja7qnSH7H6/Woobh8z7YPfOJ2rRyOg0A/An490FD048fRF4SAXyhjvu3Vei1tZKmheNXW1rds1+1vq/Xbbybx2o9Xn6Nh24dW92MjcU13FcnvWL1O9p7a1zf+1m5dfvXkk+LuZcOREOhNwK8HGoreCDlhbAQUX+ag0EzcIul281pqHErjmv9AdPPav3Ij6UbyqloLMIiVMdRBzqRBdXv4kT3aeNt2Dv//l0+Ff7f31L7xvdNPh/snIGdAoEyAhqLMhleCEhgUxoV7VmimS0XS7ea11DiUxmkoVHXQbb4KBT9nOJJuJK/UQS46xZwpNFXztbDu9HvXrtja9Xv2ZPfAzCZ2sH3Prp7xU/Yydf6GwHACNBTDGaIwMgKKzUOhmbBF0u3mdRxf8tTNa//CjaQbyatqLcAgVsZQBzmTVHXbP/E4AwLdCdBQdGfFkUEIKMJYoZlwRtLt6nX6Tdn+mwGnnz37/NK+Kbur174lHUk3klfVWoBBrIyhDnIiqeq2b95xPAT6EKCh6EOLYyHgCEz2Htudwo9DdYeu4MMD231/025d/tz0Gx3Xrm/YOzv7p37KUnVjI/ixsVXv6XDXfn7/y3b1UuJ4xV7beNd20u/XiPA2+die3H61tZEbl/30ixB/Yv9yxPnqxralL9oY1duJOvicXb25aT+ffmnJGFxO7HD3A3vyw/t26w9KP9mn+y+blN/R4a5tP/4ne3Bzvf2bhk+wXvWaO7Dd7afT2vxCl7oMs+bks8wFLhgBGooLNuHcbkUC043jmq2d+hGpFa+xsNQL23v6lm2+v5t/mdxk337xMP2n+A17sPPpwqrdTlz9L7br5rPDUZOP7b2Hb7/8j+Nk/0PbvHnN1q5v2s7of/vsC9t7/KZdvXRl3A1FU5uXv2wPfviB7Y6R69Fav3rznaPfOvzC9re/aa/5J3EdSkpxSHoq+H9+6cv2V+mTB62e+q1JhceXmukTDm/csFtfStnZ8lOIRrXm0pdw3ra/uvnn008onN3oBllzLyeDf0CgHgEainosURoJAcXj4tOan9rO/Tft7za+ZFcHNBSndetAfO/pI/vgg1+dfBpx9E17Z2+K7R76eX1h+8/eOno6sm53v/fM9t2Pi22u4kOoGR/yvp/X8pUm/9+H9sHMb5pNuvn3bLT8R6gsc+YrtfweX2hi//zwP9mtv00/OrJuQ1HXa1pHb9jVm2/Zg3feO7Zf6V+1vPo5n+q2PYkb6HuY3/bvXyp6nfPLJrvcxjCvVlxHT7/74MSaS148/y7+Zo8Z6tUK2XlSNzVt36my5k7qzt4J/4bAeAn4vZwfGzveucJZRwKKMD6pebRx3P9X25v+FA3/S9w6Gl3691Dkn7S0nIaiOwMfQt3PLB95cr7Kx/V9Zao7/Y/YyBuKw2f21df+k+3svW93R9tQHH3W/PKb9mTvheT7iarVwXTOr9hr959Nn/i9rIPilxf1rax8/DC/5YYi/4fc51TKg/ZfNtnF/TCv5SahVXfgmmvV7HKTzTFdGorDZ/bg5neqrLnBfhvfvIfAEgn4vZyGYonwuZSGgCKMT2g2G8fh745+LJ/fqLvf1wnd7qedeWS7bvoPxLrdefzxyScXZ6rlA9o1O5485zAfQnMO7fySyutUN/3n5g/yf4I7GzrjwLp+02f9v2ZfffgDs6P/CLf9hvIzLBVfruZ1+p+0z9trG//39OvTfz99j0r6sqfmy/SKDrq/UM2r5Scpa+lHcG59ZP/89F3b/od7tvms7vckDfNbaiie2u7WjZYvhSr/KOcuhId5XaChGLDmhno9+wlFXnPTLyetsOYG++0ygRwDgcoE/F5OQ1EZMHLnjcDMxvHy53wv3lAslU7a6NbH97X/PoSWyqT3xdLPdv+GvX70merep8tPaJ6e5c+kKxqKWreQP2s+++VwB/Z86/aSvs9ngbtovtdj+s35Y1zz7Q2FWalxKI0vwGaBU7p/GdMI1lzhCUW+7ThrboFp4hQIdCbg93Iais7oOPDiEUgbx9v29Zef4W9+cdAY/3PhZ2e2EfKvrfZjH0KrdXPG1V8+nSp8Q8gZp8tfPnxmm/ee2l5jr8JnSzWej9aO/wbio/+4ra1v2W5zDxoDC6ge2O7j79iD+39rr6Wm4vo3bXtUP+3rnDYUY1hz8xqKMGtugZLnFAj0IOD3chqKHvA4dJwE+j0u/pU9ufn5/GNVp595PPoRq+7f//Yr37Hf+Y2jwhOKfl5/Z/uP//pMr2t/cN8e/r8/mZmc3Ajd3foo/8SnmVf6/LOf1+7KPoS6n1k+UuP1U3t456/s0fP6P9C0jt9Pbefh/zX9foREZqopaCjqeD39n9+se3q8PMtnv1LHa7pOenryNfvPR59MePrOt+zr19NvJa77xG+Y33Z27/0o8pc8pZr+5uA1N4xr+q7w5/Zo/Yr57z9770c/OLHmphVZYc0N9nv20uAICFQn4PdyGorqiBFcNoEUxrOB3HxcGmvGZ49rxpL3PP5j+/63/tKmX+ftmo20iKZ/1rfs7af52qfPnz/eMGrz0DZ27Kt8r42Hyd5P7M7X/pv98xwuzbFdrtXnWH9fL1k1zI7eN5r+vprxLr78tZpz2sbbxuZf64X9j6+/aV/7bz8s1lZzvr+H/tdqau5kzTT307w/pfv9b9vd5neOOL6Z+w37m//+9KiejxvORq957/2n8eatOaY0Vho/fX5eT+knpM16evmfX7eWGt3Z6zdj3u/pax2vkS7nN7qzxz7971+z149+mtvL8R9s2oPr+Zc1Pj1aW81rbR7axtqu1Yz1ua987FN7+2/+1Nb+zV/a9kF+vNP4Sf7/6Gi8GXvv6Xftb/7w+CeAvRyfyYm2sT6+5p0/9XTp/7D/8K3/Z4rm9LHpR1+//XLNpdebt9PH5ldmx+cdO3vcWbrTY6esXrHf/4tvv/z9KNPx73/b/sO/af8kVNua63StGf7NPfAeAhEIpJqffaOhmKXBv0MSmA3tWjfQrjn8S57adYe7bnQn+9u2+XB75ke2Tuzw+Q/t0Qef9L5Io9n7xDNO8CF0xuGdXq7rNf3Ogbdtc3vP0n8c89uBPf/eu/bB0X/cOpmac1Bdv/lCU80Kny31tqt5nXq79vKHBGTd9HX9i//kIY3X0+tc4TV5H8a29ITiJ0efYf+Tk79Ebsp/8S/XHOb1rG/KPl5zx1/5tviaG+q1/ISiyYOZyquw5gb7nbHDPyGwLAJ+L6ehWBZ5riMjoAjjds3T/9Hoe1Ptun1VTh//3o9+bIe7j+1u+rKMU5+tXuw/ESqvPoRO303/kXpe09fN38tfM+85Vvw6/3p+j1lNNSv85+ZYMf+rntejXwJ2+fb0S1re+9F7+XeXVPzBAdW8pq/jv37Fml9sN11fz9+xWwN+8pDnmj4e5ndOQ2Hpyx437bXm+z4me7Z9742XPwa3zctZY8O8zmsoflh9zQ31SkNxVjXwOgRs+n+NWQ40FLM0+DcE5hIY3lDMlR/w4mTvsd0pfflLxf8ID7D48lRFQ/FSfNA/mt962/YlDcdfKjLoEuqTBQ1FXcsHtvv+5tEvQLxir228azuj+kbn47ud7D+zdzbWXzboV29uvvwN6sdHrehf03merdO2+uz+yya1d3H006VmG/SXmTS2Nddk/CzbMz4hM/o1p51d1C8uAb+X01Bc3Fo4N3c++LNRLSQUmukykXRVXn0IteDvPaTyGkk3klfVWoBBrIyhDnLUqeq2d5ByAgR6EPB7OQ1FD3gcOk4CijBWaCZ6kXRVXn0I1agqlddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSK8TAEFGGs0ExAI+mqvPoQqlFoKq+RdCN5Va0FGMTKGOogp5+qbmtkKxoQKBHwezkNRYkU42EIKMJYoZmARtJVefUhVKPQVF4j6UbyqloLMIiVMdRBTj9V3dbIVjQgUCLg93IaihIpxiEAAQkBH0KSiyAKAQhAAAIQgICMgN/LaShkqBGGAATaCPgQajuGMQhAAAIQgAAExkvA7+U0FOOdK5x1JKB4XKzQTLcTSVfl1YdQx2mee5jKayTdSF5VawEGsTKGOsixpqrbuaHJixAYSMDv5TQUA4Fy+uoJKMJYoZlIRdJVefUhVKOCVF4j6UbyqloLMIiVMdRBTj9V3dbIVjQgUCLg93IaihIpxsMQUISxQjMBjaSr8upDqEahqbxG0o3kVbUWYBArY6iDnH6quq2RrWhAoETA7+U0FCVSjIchoAhjhWYCGklX5dWHUI1CU3mNpBvJq2otwCBWxlAHOf1UdVsjW9GAQImA38tpKEqkGIcABCQEfAhJLoIoBCAAAQhAAAIyAn4vp6GQoUZ4WQQUn91RaCYekXRVXn0I1agTlddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSK8TAEFGGs0ExAI+keez2w3fc37dblz1kKkLXrG/bOzr5NFqwQH0ILypw47djrieHBH0TSjeQ1TYzCr0JT5VWlCwNNbanmS6mbtHmDgIqA38tpKFSk0V0aAcUGqtBMQCLpZq8vbO/pW7b5/q4dphuY7NsvHn7Zrl56wx7sfLrQHPsQWkjEnRSJa7Ku8KvQVHlV6cJAU1uq+VLpUgeJLG8Q0BLwezkNhZY36ksgoNg8FJoJRSTdqdfJ/7IPPvjVyacRk+f2aP2KXd3YtoMF5teH0AISp06JxFVVBzCItb6og7yMFXWr0FTNl1I3E+ZvCGgI+L2chkLDGVUInGMCn9j2xqujaijOMWxuDQIQgAAEIDA6AjQUo5sSDEEgObg8IQAAIABJREFUGoHUUKzbnccfn3xy0fE2fAh1PI3DIAABCEAAAhAYCQG/l/OEYiQTg43FCSgebys00x1G0i16Pdi2u+ubtnO42Ldl+xBafOaPzyx6PT5koX9F0o3kNU2Gwq9CU+VVpQsDTW2p5kupm7R5g4CKgN/LaShUpNFdGoG0gc5uos3HpbFmfPa4ZiyZbhtvG/PHNjc89Nih5/fx1eVa6ZhjzR/Yw6980b768AfN7Z7glQKmy59G81g3z2Ez3sVXY2DosUPP9/cwz1eXa6VjvGYz5sfnXavPsV18LfNazf0O9dXl/FrXgnd7dnouY+Vdy9eiNdesL95DIAoBGoooM4XPzgSajaDzCR0OVGimy45G9+gbq+f/5/8Ve33r+cyXNU3scOdtu7v1Uf6JTx04th3iQ6jtmL5jo+Ha0bjCr0Iz3U4k3UheVWxhEKtmVXXQMYo4DAILE/B7OU8oFkbJiRC4WAQmez+xze8NayYSMR9CF4sidwsBCEAAAhCIT8Dv5TQU8eeUO4CAnMBkf9s2H27b/stvm5jY4fMf2qMPPul9bR9CvQU4AQIQgAAEIACBlRLwezkNxUqng4vXIKB4xK/QTPcaSTd7ndjh7mO7e/1Ky/dG3LBHu5/1nkIfQr0FWk6IxFVVBzCItb6og7yQFXWr0FTNl1I3E+ZvCGgI+L2chkLDGdUlElBsHgrNhCSSbvI62Xtsdy4XvtF6fct2Xz6x6D7hPoS6n1k+MhJXVR3AINb6og7yelbUrUJTNV9K3UyYvyGgIeD3choKDWdUl0hAsXkoNBOSSLoqrz6EapSKymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSKcQhAQELAh5DkIohCAAIQgAAEICAj4PdyGgoZaoSXRUDx2R2FZuIRSVfl1YdQjTpReY2kG8mrai3AIFbGUAc5/VR1WyNb0YBAiYDfy2koSqQYD0NAEcYKzQQ0kq7Kqw+hGoWm8hpJN5JX1VqAQayMoQ5y+qnqtka2ogGBEgG/l9NQlEgxHoaAIowVmgloJF2VVx9CNQpN5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAwBRRgrNBPQSLoqrz6EahSaymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFOAQgICHgQ0hyEUQhAAEIQAACEJAR8Hs5DYUMNcIQgEAbAR9CbccwBgEIQAACEIDAeAn4vZyGYrxzhbOOBBSPixWa6XYi6aq8+hDqOM1zD1N5jaQbyatqLcAgVsZQBznWVHU7NzR5EQIDCfi9nIZiIFBOXz0BRRgrNBOpSLoqrz6EalSQymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSK8TAEFGGs0ExAI+mqvPoQqlFoKq+RdCN5Va0FGMTKGOogp5+qbmtkKxoQKBHwezkNRYkU4xCAgISADyHJRRCFAAQgAAEIQEBGwO/lNBQy1Agvi4DiszsKzcQjkq7Kqw+hGnWi8hpJN5JX1VqAQayMoQ5y+qnqtka2ogGBEgG/l9NQlEgxHoaAIowVmgloJF2VVx9CNQpN5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAyBFMazgdx8XBprxmePa8bSTbeNt435YxtgQ48den4fX12ulY7xms2YH58eOMMwBU7bn9L5zXh6P/vH6zbH+euXxtvOb46dvU4z5nXnnd/n2C7Xajx0OXaZvpZ5rVoMujCsda3adQDvnDueazNffnwer2XWwaLXavzzHgJRCNBQRJkpfHYmMLvBdD7pjAMVmumSkXRVXn0InTEVnV5WeY2kG8mrai3AIFbGUAc53lR12yk8OQgCCxLwezlPKBYEyWkQgMBiBHwILabCWRCAAAQgAAEIrIqA38tpKFY1E1wXAheUgA+hC4qB24YABCAAAQiEJeD3chqKsFOJ8YaA4nGxQjP5jaSr8upDqJnHIe9VXiPpRvKqWgswiJUx1EFOPVXdDslUzoXAWQT8Xk5DcRYxXh89AUUYKzQTyEi6Kq8+hGoUmMprJN1IXlVrAQaxMoY6yOmnqtsa2YoGBEoE/F5OQ1EixXgYAoowVmgmoJF0VV59CNUoNJXXSLqRvKrWAgxiZQx1kNNPVbc1shUNCJQI+L2chqJEinEIQEBCwIeQ5CKIQgACEIAABCAgI+D3choKGWqEIQCBNgI+hNqOYQwCEIAABCAAgfES8Hs5DcV45wpnHQkoHhcrNNPtRNJVefUh1HGa5x6m8hpJN5JX1VqAQayMoQ5yrKnqdm5o8iIEBhLwezkNxUCgnL56AoowVmgmUpF0VV59CNWoIJXXSLqRvKrWAgxiZQx1kNNPVbc1shUNCJQI+L2chqJEivEwBBRhrNBMQCPpqrz6EKpRaCqvkXQjeVWtBRjEyhjqIKefqm5rZCsaECgR8Hs5DUWJFONhCCjCWKGZgEbSVXn1IVSj0FReI+lG8qpaCzCIlTHUQU4/Vd3WyFY0IFAi4PdyGooSKcYhAAEJAR9CkosgCgEIQAACEICAjIDfy2koZKgRhgAE2gj4EGo7hjEIQAACEIAABMZLwO/lNBTjnSucdSSgeFys0Ey3E0m35HWy99juXL5hj3Y/6zhDJw/zIXTy1cU+KnldTO34rEi6kbwmwgq/Ck2VV5UuDDS1pZovpW7S5g0CKgJ+L6ehUJFGd2kEFBuoQjMBiaTb6nXysT25fc3WLtFQDCnwVrZDBIPVlmotKLiqvKp0YRArZ1V1kHR5g4CSAA2Fki7aKyGg2EAVmglOJN3TXj+1nftv2t9tfMmu0lAMqvXTbAfJTU9WaMav2eFcYZAZRqqvSF6V9VWn+lGBQDsBGop2LoxCAAJzCUzscOc7duv+v9re9r3RNRRzrfMiBCAAAQhAAAJVCdBQVMWJGAQuCIHDZ/bg5nds5/B3dkBDcUEmnduEAAQgAAEItBOgoWjnwmhgAorH2wrNhDiS7rHX9KVOX7MHO5+a2WSUDcWx17qFHEk3klfVWoBBrIyhDnJeqeq2bhqiBoGTBGgoTvLgo3NAIIXxbCA3H5fGmvHZ45qxhKNtvG3MH9ugHHrs0PP7+Dr7Wj+2f374pn398cf2dMr5x/b9b/2l/f6lP335U55mNVLAdPmTzmneZs9vxtvG+txXn2PHdi0YHK/noXPT5Xx4w7upE58bTW348fRxemvOa96XxkrjzXmz15kK8xcEAhCgoQgwSVjsR0ARxgrNdFej0Z08t0frV874z/8r9vr9t+3Bvae2N2nmhCcUDYkh7xV1oNAcVc12AA6DEWVMh/lS1Rd10BE+h0FgAAEaigHwOHWcBBSbh0Iz0Yuk+96Pfnz05U1znjqsb9nuy2ajW334EOp21vyjInFV1QEMYq0v6iCvaUXdKjRV86XUzYT5GwIaAn4v5/dQaDijCoFzSmCcTyjOKWxuCwIQgAAEIDBKAjQUo5wWTA0hoPhslEIz3WMk3Xav42wo2r0Oqap8biTdSF5VawEGsTKGOtDmTFbnbwhoCNBQaLiiukICiv9EKDQToki67V5pKGqUejvbYcoKzfNRs8O4wiDzi1Rfkbwq62t45aMAgTIBGooyG14JSkCxeSg0E95IuiqvPoRqlJ3KayTdSF5VawEGsTKGOsjpp6rbGtmKBgRKBPxezvdQlEgxHoaAIowVmgloJF2VVx9CNQpN5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiHAIQkBDwISS5CKIQgAAEIAABCMgI+L2chkKGGmEIQKCNgA+htmMYgwAEIAABCEBgvAT8Xk5DMd65wllHAorHxQrNdDuRdFVefQh1nOa5h6m8RtKN5FW1FmAQK2OogxxrqrqdG5q8CIGBBPxeTkMxECinr56AIowVmolUJF2VVx9CNSpI5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAwBRRgrNBPQSLoqrz6EahSaymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSKcQhAQELAh5DkIohCAAIQgAAEICAj4PdyGgoZaoSXRUDx2R2FZuIRSVfl1YdQjTpReY2kG8mrai3AIFbGUAc5/VR1WyNb0YBAiYDfy2koSqQYD0NAEcYKzQQ0kq7Kqw+hGoWm8hpJN5JX1VqAQayMoQ5y+qnqtka2ogGBEgG/l9NQlEgxHoaAIowVmgloJF2VVx9CNQpN5TWSbiSvqrUAg1gZQx3k9FPVbY1sRQMCJQJ+L6ehKJFiPAwBRRgrNBPQSLoqrz6EahSaymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFOAQgICHgQ0hyEUQhAAEIQAACEJAR8Hs5DYUMNcIQgEAbAR9CbccwBgEIQAACEIDAeAn4vZyGYrxzhbOOBBSPixWa6XYi6aq8+hDqOM1zD1N5jaQbyatqLcAgVsZQBznWVHU7NzR5EQIDCfi9nIZiIFBOXz2BFMazgdx8XBprxmePa8bS3bSNt435YxsSQ48den4fX12ulY7xms2YH58eOMMwBU7bn9L5zXh6P/vH6zbH+euXxtvOb46dvU4z5nXnnd/n2C7Xajx0OXaZvpZ5rVoMujCsda3adQDvnDueazNffnwer2XWwaLXavzzHgJRCNBQRJkpfHYmMLvBdD7pjAMVmumSkXRVXn0InTEVnV5WeY2kG8mrai3AIFbGUAc53lR12yk8OQgCCxLwezlPKBYEyWkQgMBiBHwILabCWRCAAAQgAAEIrIqA38tpKFY1E1wXAheUgA+hC4qB24YABCAAAQiEJeD3chqKsFOJ8YaA4nGxQjP5jaSr8upDqJnHIe9VXiPpRvKqWgswiJUx1EFOPVXdDslUzoXAWQT8Xk5DcRYxXh89AUUYKzQTyEi6Kq8+hGoUmMprJN1IXlVrAQaxMoY6yOmnqtsa2YoGBEoE/F5OQ1EixXgYAoowVmgmoJF0VV59CNUoNJXXSLqRvKrWAgxiZQx1kNNPVbc1shUNCJQI+L2chqJEivEwBBRhrNBMQCPpqrz6EKpRaCqvkXQjeVWtBRjEyhjqIKefqm5rZCsaECgR8Hs5DUWJFOMQgICEgA8hyUUQhQAEIAABCEBARsDv5TQUMtQIL4uA4rM7Cs3EI5KuyqsPoRp1ovIaSTeSV9VagEGsjKEOcvqp6rZGtqIBgRIBv5fTUJRIMR6GgCKMFZoJaCRdlVcfQjUKTeU1km4kr6q1AINYGUMd5PRT1W2NbEUDAiUCfi+noSiRYjwMAUUYKzQT0Ei6Kq8+hGoUmsprJN1IXlVrAQaxMoY6yOmnqtsa2YoGBEoE/F5OQ1EixXgYAoowVmgmoJF0VV59CNUoNJXXSLqRvKrWAgxiZQx1kNNPVbc1shUNCJQI+L2chqJEinEIQEBCwIeQ5CKIQgACEIAABCAgI+D3choKGWqEIQCBNgI+hNqOYQwCEIAABCAAgfES8Hs5DcV45wpnHQkoHhcrNNPtRNJt9TrZt52n/2QPbl6ztUuv2t3tTzrO0vFhPoSOX1n8X61eF5d7eWYk3UheE2CFX4WmyqtKFwaa2lLNl1I3afMGARUBv5fTUKhIo7s0AooNVKGZgETS9V4n+x/a5s1rdvXmffuX7V07XHCGfQgtKHPiNO/1xIsDPoikG8lrmhKFX4WmyqtKFwaa2lLNl1I3afMGARUBv5fTUKhIo7s0AooNVKGZgETSPeH18Jk9uP6q3Xr4oe1Phk2tD6FhavnsE15rCB5pRNKN5DXhVfhVaKq8qnRhoKkt1XwpdZM2bxBQEfB7OQ2FijS6SyOg2EAVmglIJN1jr5/azv037Ortx7Y3sJlIDHwI1SiUY6811I41IulG8poIK/wqNFVeVbow0NSWar6UukmbNwioCPi9nIZCRRpdCJwTApPdLXv90rrd3Xr36HsnPmdXb27az3cPFrpDH0ILiXASBCAAAQhAAAIrI+D3chqKlU0FF65FQPEZOYVmut9IutnrZ7a7dcPWrm/YOzv7Nn1AcfiRPUrflH1903YO+z+y8CFUow4icVXVAQxirS/qIK98Rd0qNFXzpdTNhPkbAhoCfi+nodBwRnWJBBSbh0IzIYmkm71+Ytsbr9rVjW2bfR6Rn1pcsde3nucmo8d8+xDqcWrx0EhcVXUAg1jrizrIy1lRtwpN1XwpdTNh/oaAhoDfy2koNJxRXSKBtHnMbiDNx6WxZnz2uGYs2W4bbxvzxza3PPTYoed38vX0u/Zo/cr0+xlSKLT/ecX+6M279uYfvjJtKL5/xHnq7+j81GjMjrfrnNZPGs1b2/22jXW6r4JupGs1bGDQvhb71EEXhvA+zs8uvIaupbHyruWrC8O2azVceQ+BKARoKKLMFD47E2jCufMJHQ5UaKbLRtLNXvMTirX1Ldud/eqmyXN7tP55nlB0qKW2QxR1oNCMWbNtxIeNRWIbyauqvmAwrN45GwJdCNBQdKHEMRCAwBGBiR1s37Orl9+0J3svjqkcbNvdyzfs0e5nx2Md/+VDqONpHAYBCEAAAhCAwEgI+L2cL3kaycRgAwKjJTD52J7cTr/Q7h17nr4Je7Jvv3h4216//2yhX27nQ2i0940xCEAAAhCAAARaCfi9nIaiFRODkQgoHm8rNBPTSLonvB7u2s/vf9muTr/fYt3ufu/Zwr/gzodQjVo74bWG4JFGJN1IXlVrAQaxMoY6yEGjqtujGOMdBCQE/F5OQyHBjOgyCSjCWKGZmETSVXn1IVSjVlReI+lG8qpaCzCIlTHUQU4/Vd3WyFY0IFAi4PdyGooSKcbDEFCEsUIzAY2kq/LqQ6hGoam8RtKN5FW1FmAQK2Oog5x+qrqtka1oQKBEwO/lNBQlUoxDAAISAj6EJBdBFAIQgAAEIAABGQG/l9NQyFAjDAEItBHwIdR2DGMQgAAEIAABCIyXgN/LaSjGO1c460hA8bhYoZluJ5KuyqsPoY7TPPcwlddIupG8qtYCDGJlDHWQY01Vt3NDkxchMJCA38tpKAYC5fTVE1CEsUIzkYqkq/LqQ6hGBam8RtKN5FW1FmAQK2Oog5x+qrqtka1oQKBEwO/lNBQlUoyHIaAIY4VmAhpJV+XVh1CNQlN5jaQbyatqLcAgVsZQBzn9VHVbI1vRgECJgN/LaShKpBgPQ0ARxgrNBDSSrsqrD6EahabyGkk3klfVWoBBrIyhDnL6qeq2RraiAYESAb+X01CUSDEOAQhICPgQklwEUQhAAAIQgAAEZAT8Xk5DIUONMAQg0EbAh1DbMYxBAAIQgAAEIDBeAn4vp6EY71zhrCMBxeNihWa6nUi6Kq8+hDpO89zDVF4j6UbyqloLMIiVMdRBjjVV3c4NTV6EwEACfi+noRgIlNNXT0ARxgrNRCqSrsqrD6EaFaTyGkk3klfVWoBBrIyhDnL6qeq2RraiAYESAb+X01CUSDEehoAijBWaCWgkXZVXH0I1Ck3lNZJuJK+qtQCDWBlDHeT0U9VtjWxFAwIlAn4vp6EokWIcAhCQEPAhJLkIohCAAAQgAAEIyAj4vZyGQoYaYQhAoI2AD6G2YxiDAAQgAAEIQGC8BPxeTkMx3rnCWUcCisfFCs10O5F0VV59CHWc5rmHqbxG0o3kVbUWYBArY6iDHGuqup0bmrwIgYEE/F5OQzEQKKevnkAK49lAbj4ujTXjs8c1Y+lu2sbbxvyxDYmhxw49v4+vLtdKx3jNZsyPTw+cYZgCp+1P6fxmPL2f/eN1m+P89Uvjbec3x85epxnzuvPO73Nsl2s1Hrocu0xfy7xWLQZdGNa6Vu06gHfOHc+1mS8/Po/XMutg0Ws1/nkPgSgEaCiizBQ+OxOY3WA6n3TGgQrNdMlIuiqvPoTOmIpOL6u8RtKN5FW1FmAQK2OogxxvqrrtFJ4cBIEFCfi9nCcUC4LktPEQUISxQjMRi6Sr8upDqEYlqbxG0o3kVbUWYBArY6iDnH6quq2RrWhAoETA7+U0FCVSjEMAAhICPoQkF0EUAhCAAAQgAAEZAb+X01DIUCO8LAKKz+4oNBOPSLoqrz6EatSJymsk3UheVWsBBrEyhjrI6aeq2xrZigYESgT8Xk5DUSLFeBgCijBWaCagkXRVXn0I1Sg0lddIupG8qtYCDGJlDHWQ009VtzWyFQ0IlAj4vZyGokSK8TAEFGGs0ExAI+mqvPoQqlFoKq+RdCN5Va0FGMTKGOogp5+qbmtkKxoQKBHwezkNRYkU42EIKMJYoZmARtJVefUhVKPQVF4j6UbyqloLMIiVMdRBTj9V3dbIVjQgUCLg93IaihIpxiEAAQkBH0KSiyAKAQhAAAIQgICMgN/LaShkqBGGAATaCPgQajuGMQhAAAIQgAAExkvA7+U0FOOdK5x1JKB4XKzQTLcTSVfl1YdQx2mee5jKayTdSF5VawEGsTKGOsixpqrbuaHJixAYSMDv5TQUA4Fy+uoJKMJYoZlIRdJVefUhVKOCVF4j6UbyqloLMIiVMdRBTj9V3dbIVjQgUCLg93IaihIpxsMQUISxQjMBjaSr8upDqEahqbxG0o3kVbUWYBArY6iDnH6quq2RrWhAoETA7+U0FCVSjEMAAhICPoQkF0EUAhCAAAQgAAEZAb+X01DIUCMMAQi0EfAh1HYMYxCAAAQgAAEIjJeA38tpKMY7VzjrSEDxuFihmW4nkq7Kqw+hjtM89zCV10i6kbyq1gIMYmUMdZBjTVW3c0OTFyEwkIDfy2koBgLl9NUTUISxQjORiqR77HVih7s/tQc3r1kKkLVL1+zW/Z/a7uFkocn3IbSQiDvp2Kt7YeCHkXQjeU3TovCr0FR5VenCQFNbqvlS6iZt3iCgIuD3choKFWl0l0ZAsYEqNBOQSLqN18neY7tz+Zrd2vrIDtNNTPZs+966Xd3YtoMFZtmH0AISp05pvJ56YeBAJN1IXtO0KPwqNFVeVbow0NSWar6UukmbNwioCPi9nIZCRRrdpRFQbKAKzQQkkm72+pntbt2wtcv3bPvg+InEZHfLXndjXSfch1DX8+YdF4mrqg5gEGt9UQd5RSvqVqGpmi+lbibM3xDQEPB7OQ2FhjOqEDgnBCZ2sH3Prl56wx7sfHp0T3nsCyN6QnFOYHMbEIAABCAAgRAEaChCTBMmITAiAofP7MH1K7Z2+bY9en5gk/337Rsbb9sv9l8sZNKH0EIinAQBCEAAAhCAwMoI+L2cJxQrmwouXIuA4vG2QjPdbyTdWa+T/Q9ts/mm7PUt2z3+6qfe0+hDqLdAywmzXlteXngokm4kr2lCFH4VmiqvKl0YaGpLNV9K3aTNGwRUBPxeTkOhIo3u0gikDXR2E20+Lo0147PHNWPJdNt425g/trnhoccOPb+Pry7XSsfY4XP7L1+5bV/9iz+ytUuv2L+78S3bP2oqZjVSwHT5M9U8AjZ7fjPeNtbnvvocO7ZrweB4PQ+dmy7nwxveTZ343Ghqw4+nj9Nbc17zvjRWGm/Om73OVJi/IBCAAA1FgEnCYj8CijBWaKa7Go3u5Lk9Wr9yxn/+X7HXt57b5PAje3R7w57spS9xemH729+01y5dsdfuP8s/9anfdE2v2fOUMw8fDdczneYDFH4VmsltJN1IXlVsYRCrZlV1kJOGvyGgI0BDoWOLMgTOIYGjn/J04sucJna4s2mvXbphj3Y/633PPoR6C3ACBCAAAQhAAAIrJeD3cr7kaaXTwcUhMHYCn9j2xqu2dqKhMLODbbt7mYZi7LOHPwhAAAIQgICCAA2FgiqaKyWgeMSv0EyQIulmr83TiJlfbGcH9nzrtn3h9mPbW+Cbs30I1SieSFxVdQCDWOuLOsgrX1G3Ck3VfCl1M2H+hoCGgN/LeUKh4YzqEgkoNg+FZkISSffY6wvb33nX7qYfHTv9putrduv+T233cIFuwozvoRDVwfF81V18kXQjeVXlAQxi5ayqDuqmAGoQOE2AhuI0E0aCE1BsoArNhDmSrsqrD6Ea5afyGkk3klfVWoBBrIyhDnL6qeq2RraiAYESAb+X84SiRIrxMAQUYazQTEAj6aq8+hCqUWgqr5F0I3lVrQUYxMoY6iCnn6pua2QrGhAoEfB7OQ1FiRTjEICAhIAPIclFEIUABCAAAQhAQEbA7+U0FDLUCC+LgOKzOwrNxCOSrsqrD6EadaLyGkk3klfVWoBBrIyhDnL6qeq2RraiAYESAb+X01CUSDEehoAijBWaCWgkXZVXH0I1Ck3lNZJuJK+qtQCDWBlDHeT0U9VtjWxFAwIlAn4vp6EokWI8DAFFGCs0E9BIuiqvPoRqFJrKayTdSF5VawEGsTKGOsjpp6rbGtmKBgRKBPxeTkNRIsV4GAKKMFZoJqCRdFVefQjVKDSV10i6kbyq1gIMYmUMdZDTT1W3NbIVDQiUCPi9nIaiRIpxCEBAQsCHkOQiiEIAAhCAAAQgICPg93IaChlqhCEAgTYCPoTajmEMAhCAAAQgAIHxEvB7OQ3FeOcKZx0JKB4XKzTT7UTSVXn1IdRxmucepvIaSTeSV9VagEGsjKEOcqyp6nZuaPIiBAYS8Hs5DcVAoJy+egKKMFZoJlKRdFVefQjVqCCV10i6kbyq1gIMYmUMdZDTT1W3NbIVDQiUCPi9nIaiRIrxMAQUYazQTEAj6aq8+hCqUWgqr5F0I3lVrQUYxMoY6iCnn6pua2QrGhAoEfCC1xa7AAADmElEQVR7OQ1FiRTjYQgowlihmYBG0lV59SFUo9BUXiPpRvKqWgswiJUx1EFOP1Xd1shWNCBQIuD3chqKEinGIQABCQEfQpKLIAoBCEAAAhCAgIyA38tpKGSoEV4WAcVndxSaiUckXZVXH0I16kTlNZJuJK+qtQCDWBlDHeT0U9VtjWxFAwIlAn4vp6EokWI8DAFFGCs0E9BIuiqvPoRqFJrKayTdSF5VawEGsTKGOsjpp6rbGtmKBgRKBPxeTkNRIsV4GAIpjGcDufm4NNaMzx7XjKWbbhtvG/PHNsCGHjv0/D6+ulwrHeM1mzE/Pj1whmEKnLY/pfOb8fR+9o/XbY7z1y+Nt53fHDt7nWbM6847v8+xXa7VeOhy7DJ9LfNatRh0YVjrWrXrAN45dzzXZr78+Dxey6yDRa/V+Oc9BKIQoKGIMlP4hAAEIAABCEAAAhCAwAgJ0FCMcFKwBAEIQAACEIAABCAAgSgEaCiizBQ+IQABCEAAAhCAAAQgMEICNBQjnBQsQQACEIAABCAAAQhAIAoBGoooM4VPCEAAAhCAAAQgAAEIjJAADcUIJwVLEIAABCAAAQhAAAIQiEKAhiLKTOETAhCAAAQgAAEIQAACIyRAQzHCScESBCAAAQhAAAIQgAAEohCgoYgyU/iEAAQgAAEIQAACEIDACAnQUIxwUrAEAQhAAAIQgAAEIACBKARoKKLMFD4hAAEIQAACEIAABCAwQgI0FCOcFCxBAAIQgAAEIAABCEAgCgEaiigzhU8IQAACEIAABCAAAQiMkAANxQgnBUsQgAAEIAABCEAAAhCIQoCGIspM4RMCEIAABCAAAQhAAAIjJEBDMcJJwRIEIAABCEAAAhCAAASiEKChiDJT+IQABCAAAQhAAAIQgMAICdBQjHBSsAQBCEAAAhCAAAQgAIEoBGgooswUPiEAAQhAAAIQgAAEIDBCAjQUI5wULEEAAhCAAAQgAAEIQCAKARqKKDOFTwhAAAIQgAAEIAABCIyQAA3FCCcFSxCAAAQgAAEIQAACEIhCgIYiykzhEwIQgAAEIAABCEAAAiMkQEMxwknBEgQgAAEIQAACEIAABKIQ6NRQpIP4AwNqgBqgBqgBaoAaoAaoAWqAGmirgdnm5/dmP+DfEIAABCAAAQhAAAIQgAAE+hCgoehDi2MhAAEIQAACEIAABCAAgRMEaChO4OADCEAAAhCAAAQgAAEIQKAPgf8NuMv7L2BebnQAAAAASUVORK5CYII="></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>attempt to substitute coordinates (in any order) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>13</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>7</mn><mo> </mo><mo>,</mo><mo> </mo><mi>a</mi><mo> </mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>7</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>13</mn><mo> </mo><mo>,</mo><mo> </mo><mi>a</mi><mo> </mo><mi>log</mi><mo> </mo><mn>9</mn><mo>=</mo><mn>7</mn></math></p>
<p>finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mn>9</mn><mo>=</mo><mn>2</mn></math> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mn>9</mn><mo>=</mo><mn>2</mn><mo> </mo><mo>,</mo><mo> </mo><mn>2</mn><mi>a</mi><mo>=</mo><mn>7</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></math> <em><strong>A1 N2</strong></em> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1A1A1 N3</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct shape of logarithmic function (must be increasing and concave down).<br><strong>Only</strong> if the shape is correct, award the following:<br><em><strong>A1</strong> </em>for being asymptotic to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math><br><em><strong>A1</strong> </em>for curve including both points in circles. </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the graph of the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mfrac><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
<p style="text-align: center;"><img 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KxniPbtJaCdduoBUESwFzsrOCKACBwls8dQdB5cO22dF0+5IIKUNCnLOgFEYD1DtG+WgASQY6edo8zZQPgRAhUJIIKK8Kl6OwFNi9Qc+dQLIkhNlPIsE0AElrND2/YSyDF1VJUigr3oWcERAUTgKJk9hqKLqlJPHRVHRNBjb+o3ZkTQb+5dRK6nZ6WeOiowiMBF9yCIhQQQwUJQrGaTgHbYqaeOKlJEYDPftCoPAUSQhyulFiCgu41qh536Pv1qOiIokECqMEMAEZhJBQ05lECuqaNqByI4NBus3zIBRNBy9jpvu8YGUt91NCJFBJEErz0QQAQ9ZNlpjJotlPquoxEVIogkeO2BACLoIctOY8xxs7mIChFEErz2QAAR9JBlpzFqZ53ygfVDTIhgSIP33gkgAu8ZdhpfnDGU+mZzERciiCR47YEAIughyw5jzDljSLgQgcNOQ0iTBBDBJBp+sEwg54whxY0ILGeftqUmgAhSE6W8IgQ0Y0j3Gcq1IIJcZCnXIgFEYDErtGkvgVz3GIoVI4JIgtceCCCCHrLsMEbtqHPcYyiiQgSRBK89EEAEPWTZWYy6t5B21Jo5lGtBBLnIUq5FAojAYlZo0ywBXTuQe0edu/zZAPkRAoUJIILCwKluO4Gvj4+z3Voitg4RRBK89kAAEfSQZWcx6tYSkkHOBRHkpEvZ1gggAmsZoT17CWgnnevWErFyRBBJ8NoDAUTQQ5YdxRgHinM8jGaICREMafDeOwFE4D3DzuIrMVAsZIjAWcchnFkCiGAWDz9aI1BioFgxIwJrmac9OQkggpx0KTs5gRIDxWo0IkieOgo0TAARGE4OTbtIQDvonFcUxxoRQSTBaw8EEEEPWXYSYxwoznlFcUSFCCIJXnsggAh6yLKTGE9PT8NbV64UiQYRFMFMJUYIIAIjiaAZ+wnkvvX0sAWIYEiD994JIALvGXYU33tXrwY9kKbEgghKUKYOKwQQgZVM0I5ZAk+fPj2fyZPrGcW7lSOCXSJ89kwAEXjOrqPYNFOo5M65ZF2O0kQojRJABI0mrrdml7qQLHJFBJEErz0QQAQ9ZNlBjLkfTbmLCBHsEuGzZwKIwHN2ncQWxwdKXEgWkSGCSILXHggggh6y3HiMcXxAQii1IIJSpKnHAgFEYCELtGGWgMYHdGqo5IIIStKmrtoEEEHtDFD/XgKSQO4nku02AhHsEuGzZwKIwHN2HcRWY3xA2BCBg85DCIsJIILFqFixBoEa4wOKExHUyDZ11iKACGqRp95FBEpfPxAbhQgiCV57IIAIeshywzGWvL/QEBMiGNLgvXcCiMB7hhuOT88d0A651P2FhqgQwZAG770TQATeM9xwfHpQfannD+xiQgS7RPjsmQAi8JzdxmP79MaNoH81FkRQgzp11iKACGqRp969BHQ0oKOCGgsiqEGdOmsRQAS1yFPvLAGNC2hnXOL5xGMNQQRjVPjOKwFE4DWzjcdV47YSQ2SIYEiD994JIALvGW40Pk0bLX1biSEqRDCkwXvvBBCB9ww3GF+cNvrw4cNqrUcE1dBTcQUCiKACdKqcJ3B6ehp0RFBzQQQ16VN3aQKIoDRx6ttLQHcbvfXFrb3r5VwBEeSkS9nWCCACaxnpvD217ja6ix0R7BLhs2cCiMBzdhuMrebVxENciGBIg/feCSAC7xluLD5dSVz7tJCQIYLGOo7z5uaeOIEInHeglsKzclpIzBBBSz3Hf1s1bqZ/uW7AiAj896FmIrRyWkjAEEEz3aaLhuqPJB0tq1/q+hp9TrkggpQ0KWsTASunhRQEItiUSjbOREBHBJparX96el+qBRGkIkk5mwhYOi2kQBDBpnSycUYC+r+iowL1Uf3xlOLooKgI1HD+wYA+QB+gD6TrAykmVxQVQUZJUnTjBCxcRDZEqB0VCwQsEmj+iMAiVNpUn0C8t1CuGRFrIkQEa6ixTW4CjBHkJkz51QjcPjmpfm+h3eARwS4RPtckoKMAZg3VzAB1ZydQ+5bTYwEigjEqfFeLANcR1CJPvUUI6FBXO91aTyKbChIRTJHh+xoEZv9/PD4JH8SJOG9/Hs5+eBaePb4bPv/jb8PlS78Ln539sLfJDBbvRcQKOQloxoP+2rG2IAJrGaE98wR+Co9OPg6vX3ozfHLyv+Hzo/8J9394Nr/J4FdEMIDB27IEdN6z5gPq56JFBHN0+M0kge/vhE9efSVcfvXjcPvRTwc1EREchIuVUxKIt5RIcUFMynapLESQmijl5Sfwbbhz9Ga4/M6X4cHyg4HzZiGC/NmhhgkC1q4dGDYTEQxp8L4JAs++Cbc/1LjA++GrBz8e1GREcBAuVk5FQLfV1c429+1117YXEawlx3Z1CGiM4ChcO/o4/F7jBHe+PagZiOAgXKycioDVQeIYHyKIJHhtgcCzRyfhz386CY9+vh++eue18PrR38Kjs/8Ofzn5Jiw5S4QIWsiyszZqTEA7Wo0RWF0QgdXM0K4hgZ/PboY3Lr0Wfn90Eh6czxL6LpzdfDdcvvRO+OTkftg/cfR5aYhgSJX3RQicnp6ezxayOEgcASCCSILXHggggh6ybCxGi1cS7yJCBLtE+OyZACLwnF2DselhGtrJzl4paaDdiMBAEmhCMQKIoBhqKhKBo+vXz2+eZZ0GIrCeIdqXkgAiSEmTsmYJxCmjlm43PdVgRDBFhu89EkAEHrNqNCbrU0aH2BDBkAbvvRNABN4zbCS++PAZy1NGh6gQwZAG770TQATeM2wkPj1sW7OFWlkQQSuZop0pCCCCFBQpY5aA5buMTjUcEUyR4XuPBBCBx6wai0mPotTtpi1fQLaLDBHsEuGzZwKIwHN2DcQWjwYkg5YWRNBStmjrVgKIYCtBtp8l0OLRgAJCBLNp5UdnBBCBs4RaCqfVowExRASWehJtyU0AEeQm3HH5rR4NKGWIoOOO22HoiKDDpJcIueWjAfFBBCV6CXVYIYAIrGTCWTtaPhpQKhCBsw5JOLMEEMEsHn5cQ6D1owHFjAjWZJ5tWiWACFrNnOF2t340ILSIwHAHo2nJCSCC5Ej7LrC1ewpNZQsRTJHhe48EEIHHrFaMSXcYbemeQlOoEMEUGb73SAAReMxqpZji8wb0FLLWF0TQegZp/yEEEMEhtFh3loCePqZ/HhZE4CGLxLCUACJYSor1ZgnEZxG38PSx2UB++RERLKHEOl4IIAIvmawch8YFND7gZUEEXjJJHEsIIIIllFhnlkCcLqoZQ14WROAlk8SxhAAiWEKJdSYJeLh4bCw4RDBGhe+8EkAEXjNbKK44XVRC8LQgAk/ZJJZ9BBDBPkL8PklAA8PaYXqYLrobJCLYJcJnzwQQgefsZo7to2vX3EwX3UWFCHaJ8NkzAUTgObsZY/M4QDzEhQiGNHjvnQAi8J7hDPFpdpAeRi8ZeF0QgdfMEtcYAUQwRoXvZgno6mEP9xOaCxIRzNHhN28EEIG3jGaOx9sVxFO4EMEUGb73SAAReMxqppjiNQNfHx9nqsFOsYjATi5oSX4CiCA/Yzc1fHrjxvkpIW/XDIwlCBGMUeE7rwQQgdfMJo4rnhLyeM3AGCpEMEaF77wSQAReM5swrnhKSEcEvSyIoJdME6cIIAL6wV4CEoCmi/ZwSijCQASRBK89EEAEPWR5Q4y9nRKKqBBBJMFrDwQQQQ9ZXhljPCXUwyyhXUSIYJcInz0TQASes7sxtp5mCe2iQgS7RPjsmQAi8JzdDbGdnp6e31nUy6MnD0WBCA4lxvotE0AELWcvU9sfPnzo/l5C+9Ahgn2E+N0TAUTgKZuJYtHtpfWv5wUR9Jz9/mJHBP3lfDZiDQxrqqin5w/PBjzxIyKYAMPXLgkgApdpXRdUnCp69+7ddQU42goROEomoewlgAj2IupjhfiMAT2DmCWcD5TDAQK9EEAEvWR6T5xxXKCnq4fnkHBEMEeH37wRQATeMroinjguoNlCLM8JIAJ6Qk8EEEFP2R6JlXGBESiBU0PjVPjWKwFE4DWzC+JiXGAaEkcE02z4xR8BROAvp4si0lgA4wLTqBDBNBt+8UcAEfjL6aKINDtI1wswLjCOCxGMc+FbnwQQgc+8zkal6wS0o+vlaWOzMCZ+RAQTYPjaJQFE4DKt00HpCEA7udsnJ9Mr8QvXEdAHuiKACDpKd3y+QE+PnFybXo4I1pJjuxYJIIIWs7ayzQwOLweHCJazYs32CSCC9nO4KAIGhxdh+nUlRPArCt50QAARdJDk+JAZBoeXJxsRLGfFmu0TQATt53A2gnjlsGTAspwAIljOijXbJ4AI2s/hZATxSWPcUXQS0eQPiGASDT84JIAIHCZVIWmG0HtXr55fPcwdRQ9PMiI4nBlbtEsAEbSbu9mWa4aQRIAEZjFN/ogIJtHwg0MCiMBhUpkhtD2piGA7Q0pohwAiaCdXi1qqK4a1E2OG0CJckyshgkk0/OCQACJwlNR4DyGeObw9qYhgO0NKaIcAImgnV7MtffDgwfmRADOEZjEt/hERLEbFig4IIAIHSWSaaPokIoL0TCnRLgFEYDc3i1oWbySnWULMEFqEbNFKiGARJlZyQgARNJxI7fi5kVyeBCKCPFwp1SYBRGAzL4taJQnwlLFFqA5eCREcjIwNGiaACBpNHtcK5E0cIsjLl9JtEUAEtvKxqDWSgHZUminEkocAIsjDlVJtEkAENvMy2ap4S2muFZhElOQHRJAEI4U0QgARNJIoNZMLxsolCxGUY01N9Qkggvo5WNSCKAEuGFuEa/NKiGAzQgpoiAAiaCBZXDBWPkmIoDxzaqxHABHUY7+o5iiBT2/cWLQ+K6UhgAjScKSUNgggAsN5ihLgquHySUIE5ZlTYz0CiKAe+9maddUwTxibRZT1R0SQFS+FGyOACIwlRM3h1hH1k4II6ueAFpQjgAjKsV5UU5SAbh3x5MmTRduwUnoCiCA9U0q0SwARGMrNUAIaH2CpRwAR1GNPzeUJIILyzCdrPLp+nZvITdIp+wMiKMub2uoSQAR1+f9aOzeR+xWFiTeIwEQaaEQhAoigEOi5auJN5DgdNEep7G+IoCxvaqtLABHU5R+iBLiJXOVE7FSPCHaA8NE1AURQMb1IoCL8PVUjgj2A+NkVAURQKZ1IoBL4hdUigoWgWM0FAURQIY3xTqKcDqoAf2GViGAhKFZzQQARFE4jEigMfGV1iGAlODZrkgAiKJi2KAE9ZYzFNgFEYDs/tC4tAUSQludkaVECGhtgsU8AEdjPES1MRwARpGM5WRISmERj9gdEYDY1NCwDAUSQAeqwSCQwpNHOe0TQTq5o6XYCiGA7w8kSkMAkGvM/IALzKaKBCQkggoQwh0UhgSGN9t4jgvZyRovXE0AE69lNbokEJtE08wMiaCZVNDQBAUSQAOKwiPicYWYHDam09x4RtJczWryeACJYz+7ClkjgApJmv0AEzaaOhq8ggAhWQBvbBAmMUWn3O0TQbu5o+eEEEMHhzC5sgQQuIGn+C0TQfAoJ4AACiOAAWGOrIoExKu1/hwjazyERLCeACJazurBmlMBH164FPXiexQ8BROAnl0SynwAi2M9odA0kMIrFzZeIwE0qCWQBAUSwANLuKkhgl4i/z4jAX06JaJoAIphmM/oLEhjF4u5LROAupQQ0QwARzMDZ/QkJ7BLx+xkR+M0tkV0kgAguMhn9BgmMYnH7JSJwm1oCGyGACEag7H6FBHaJ+P+MCPznmAhfEEAEL1iMvkMCo1jcf4kI3KeYAAcEEMEAxu5bJLBLpJ/PiKCfXBNpCIhgohcggQkwnXyNCDpJNGGeE0AEIx0BCYxA6ewrRNBZwjsPFxHsdAAksAOk04+IoNPEdxo2IhgkHgkMYHT+FhF03gE6Cx8R/JJwJNBZz98TLiLYA4ifXRFABCEEJOCqTycJBhEkwUghjRDoXgRIoJGeWriZiKAwcKqrSqBrESCBqn3PdOWIwHR6aFxiAt2KAAkk7knOikMEzhJKOLMEuhQBEpjtE/wYQr57wD4AAAMPSURBVEAEdIOeCHQnAiTQU/deHysiWM+OLdsj0JUIkEB7HbRWixFBLfLUW4NANyJAAjW6V7t1IoJ2c0fLDyfQhQiePn0a3rt6NXx07VrQexYI7COACPYR4ndPBNyLQDt+CQAJeOq2+WNBBPkZU4MdAq5FgATsdLTWWoIIWssY7d1CwK0IkMCWbsG2iIA+0BMBlyJAAj114TyxIoI8XCnVJgF3IkACNjtaa61CBK1ljPZuIeBKBEhgS1dg2yEBRDCkwXvvBNyIAAl476pl40MEZXlTW10CLkQQJfDWlSvnzxaoi5TaPRBABB6ySAxLCTQvAiSwNNWsdwgBRHAILdZtnUDTIkACrXc/u+1HBHZzQ8vSE2hWBEggfWegxBcEEMELFrzzT6BJESAB/x2zdoSIoHYGqL8kgeZEgARKdo9+60IE/ea+x8ibEgES6LGL1okZEdThTq11CDQjAiRQp4P0Wisi6DXzfcbdjAhufXErcJ1An520RtSIoAZ16qxFoAkRIIFa3aPfehFBv7nvMXLzIkACPXbL+jEjgvo5oAXlCJgWARIo1xGo6WUCiOBlHnzyTcCsCCQB/We8e/eu7wwQnUkCiMBkWmhUJgImRfD18TESyJRwil1GABEs48RaPgiYFMHDhw/DvXv3fBAmiiYJIIIm00ajVxIwKYKVsbAZBJIRQATJUFJQAwQQQQNJoonlCSCC8sypsR4BRFCPPTUbJoAIDCeHpiUngAiSI6VADwQQgYcsEsNSAohgKSnW64oAIugq3d0Hiwi67wIAGCOACMao8J1XAojAa2aJaxMBRLAJHxs3RgARNJYwmluGACIow5labBBABDbyQCuMEUAExhJCc7ISQARZ8VJ4qwQQQauZo91rCCCCNdTYxj0BROA+xQQ4IIAIBjB4CwEIQKBHAoigx6wTMwQgAIEBAUQwgMFbCEAAAj0SQAQ9Zp2YIQABCAwIIIIBDN5CAAIQ6JEAIugx68QMAQhAYEAAEQxg8BYCEIBAjwQQQY9ZJ2YIQAACAwKIYACDtxCAAAR6JPD/B+ajFCxAxsEAAAAASUVORK5CYII="></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the zero of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></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>Write down the coordinates of the local minimum point.</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>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><mo>-</mo><mi>x</mi></math>.</p>
<p>Solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mi>x</mi><mo>+</mo><mfrac><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for equating the function to zero.</p>
<p><em><strong><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>29</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>28942</mn><mo>…</mo></mrow></mfenced></math> (A1) (C2)</strong></em></p>
<p><em><strong><br></strong></em><strong>Note:</strong> Award <em><strong>(C1)</strong></em> for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-value given as part of a coordinate pair or alongside an explicitly stated <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-value.</p>
<p><em><strong><br></strong></em><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>.</mo><mn>88</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>33</mn></mrow></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfenced><mrow><mn>2</mn><mo>.</mo><mn>88449</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>32674</mn><mo>…</mo></mrow></mfenced></mfenced></math> <em><strong>(A1)(A1) (C2)</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>88</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>33</mn></math>.</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mi>x</mi><mo>=</mo><mi>x</mi><mo>+</mo><mfrac><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> (or equivalent) <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for equating the functions or for a sketch of the two functions.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>43</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>.</mo><mn>43080</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1) (C2)</strong></em></p>
<p><strong><br>Note: </strong>Do not award the final <em><strong>(</strong><strong>A1)</strong></em> if the answer is seen as part of a coordinate pair or a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-value is explicitly stated, unless already penalized in part (a).</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<br><hr><br><div class="specification">
<p>The points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> have position vectors <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 2} \\ 4 \\ { - 4} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 6 \\ 8 \\ 0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> respectively.</p>
<p>Point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}">
<mrow>
<mtext>C</mtext>
</mrow>
</math></span> has position vector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 1} \\ k \\ 0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{O}}">
<mrow>
<mtext>O</mtext>
</mrow>
</math></span> be the origin.</p>
</div>
<div class="specification">
<p>Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OA</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span>.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OB</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\widehat {\text{O}}{\text{C}} = {\text{B}}\widehat {\text{O}}{\text{C}}"> <mrow> <mtext>A</mtext> </mrow> <mrow> <mover> <mtext>O</mtext> <mo>^</mo> </mover> </mrow> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <mrow> <mtext>B</mtext> </mrow> <mrow> <mover> <mtext>O</mtext> <mo>^</mo> </mover> </mrow> <mrow> <mtext>C</mtext> </mrow> </math></span>, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 7"> <mi>k</mi> <mo>=</mo> <mn>7</mn> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the area of triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AOC}}"> <mrow> <mtext>AOC</mtext> </mrow> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OA</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span> or into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OB</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span> (in (ii)) <em><strong>(A1)</strong></em> </p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 \times \left( { - 1} \right) + 4 \times k"> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mi>k</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 \times \left( { - 1} \right) + 8 \times k"> <mn>6</mn> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>8</mn> <mo>×</mo> <mi>k</mi> </math></span></p>
<p>correct expression <em><strong>A1</strong></em><em><strong> N1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + 4k"> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 2"> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </math></span></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>correct expression <em><strong>A1</strong></em><em><strong> N1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8k - 6"> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6 + 8k"> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>8</mn> <mi>k</mi> </math></span></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding magnitudes (seen anywhere) <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{\left( { - 2} \right)}^2} + {{\left( 4 \right)}^2} + {{\left( { - 4} \right)}^2}} \,\,\left( { = 6} \right)"> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{\left( 6 \right)}^2} + {{\left( 8 \right)}^2} + {0^2}} \,\,\left( { = 10} \right)"> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>correct substitution of their values into formula for angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AOC}}"> <mrow> <mtext>AOC</mtext> </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{cos}}\,\theta = \frac{{2 + 4k}}{{\sqrt {{{\left( { - 2} \right)}^2} + {{\left( 4 \right)}^2} + {{\left( { - 4} \right)}^2}} \left| {\overrightarrow {{\text{OC}}} } \right|}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </mrow> <mrow> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </math></span></p>
<p>correct substitution of their values into formula for angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BOC}}"> <mrow> <mtext>BOC</mtext> </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{cos}}\,\theta = \frac{{8k - 6}}{{\sqrt {{{\left( 6 \right)}^2} + {{\left( 8 \right)}^2} + {0^2}} \left| {\overrightarrow {{\text{OC}}} } \right|}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </math></span></p>
<p>recognizing that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,{\text{A}}\widehat {\text{O}}{\text{C}} = {\text{cos}}\,{\text{B}}\widehat {\text{O}}{\text{C}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>A</mtext> </mrow> <mrow> <mover> <mtext>O</mtext> <mo>^</mo> </mover> </mrow> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>B</mtext> </mrow> <mrow> <mover> <mtext>O</mtext> <mo>^</mo> </mover> </mrow> <mrow> <mtext>C</mtext> </mrow> </math></span> (seen anywhere) <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 + 4k}}{{\left| {\overrightarrow {{\text{OC}}} } \right|\sqrt {{{\left( { - 2} \right)}^2} + {{\left( 4 \right)}^2} + {{\left( { - 4} \right)}^2}} }} = \frac{{8k - 6}}{{\left| {\overrightarrow {{\text{OC}}} } \right|\sqrt {{6^2} + {{\left( 8 \right)}^2} + {0^2}} }}"> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <msqrt> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2 + 4k}}{{6\sqrt {1 + {k^2}} }} = \frac{{8k - 6}}{{10\sqrt {1 + {k^2}} }}"> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </mrow> <mrow> <mn>6</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <mn>10</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span></p>
<p>correct working (without radicals) <em><strong>(A2)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10\left( {2 + 4k} \right) = 6\left( {8k - 6} \right)"> <mn>10</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>6</mn> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mi>k</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="11{k^2} - 79k + 14 = 0"> <mn>11</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>79</mn> <mi>k</mi> <mo>+</mo> <mn>14</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct working clearly leading to the required answer <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="20 + 36 = 48k"><mn>20</mn><mo>+</mo><mn>36</mn><mo>=</mo><mn>48</mn><mi>k</mi><mo>-</mo><mn>40</mn><mi>k</mi></math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="56 = 8k"> <mn>56</mn> <mo>=</mo> <mn>8</mn> <mi>k</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 7"> <mi>k</mi> <mo>=</mo> <mn>7</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{2}{{11}}"> <mi>k</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mrow> <mn>11</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {k - 7} \right)\left( {11k - 2} \right) = 0"> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>7</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>11</mn> <mi>k</mi> <mo>−</mo> <mn>2</mn> </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="k = 7"> <mi>k</mi> <mo>=</mo> <mn>7</mn> </math></span> <em><strong>AG</strong></em><em><strong> N0</strong></em></p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding magnitude of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span> (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="\sqrt {{{\left( { - 1} \right)}^2} + {7^2} + {0^2}} "> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>7</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {50} "> <msqrt> <mn>50</mn> </msqrt> </math></span></p>
<p>valid attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</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="{\text{cos}}\,\theta = \frac{{2 + 28}}{{6\sqrt {{{\left( { - 1} \right)}^2} + {7^2} + {0^2}} }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>28</mn> </mrow> <mrow> <mn>6</mn> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>7</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{{56 - 6}}{{10\sqrt {{{\left( { - 1} \right)}^2} + {7^2} + {0^2}} }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>56</mn> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <mn>10</mn> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>7</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\sqrt {26} } \right)^2} = {6^2} + {\left( {\sqrt {50} } \right)^2} - 2\left( 6 \right)\sqrt {50} \,{\text{cos}}\,\theta "> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>26</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>50</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> <msqrt> <mn>50</mn> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span></p>
<p>finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </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{cos}}\,\theta = \frac{5}{{\sqrt {50} }}\,\,\,\left( { = \frac{1}{{\sqrt 2 }}} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>5</mn> <mrow> <msqrt> <mn>50</mn> </msqrt> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{4}"> <mi>θ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = {\text{cos}}\,\theta "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \sqrt {1 - \frac{{25}}{{50}}} "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mn>25</mn> </mrow> <mrow> <mn>50</mn> </mrow> </mfrac> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \sqrt {1 - {\text{co}}{{\text{s}}^2}\,\theta } "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \frac{{\sqrt 2 }}{2}"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span></p>
<p>correct substitution of <strong>their</strong> values into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}ab\,{\text{sin}}\,{\text{C}}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>C</mtext> </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="\frac{1}{2} \times 6 \times \sqrt {50} "><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn><mo>×</mo><msqrt><mn>50</mn></msqrt><mo>×</mo><msqrt><mn>1</mn><mo>-</mo><mfrac><mn>25</mn><mn>50</mn></mfrac></msqrt></math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 6 \times \sqrt {50} \times \frac{5}{{\sqrt {50} }}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>6</mn> <mo>×</mo> <msqrt> <mn>50</mn> </msqrt> <mo>×</mo> <mfrac> <mn>5</mn> <mrow> <msqrt> <mn>50</mn> </msqrt> </mrow> </mfrac> </math></span></p>
<p>area is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15"> <mn>15</mn> </math></span> <em><strong>A1</strong></em><em><strong> 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 graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>6</mn></math> is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="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><mfenced><mn>2</mn></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mn>2</mn></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>1</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>6</mn></math>. On the axes above, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>.</p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>2</mn></mfenced><mo>=</mo><mn>6</mn></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mn>2</mn></mfenced><mo>=</mo><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">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:150px;"><img 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"> <em><strong> M1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for an attempt to apply any vertical stretch or vertical translation, <em><strong>A1</strong></em> for a correct horizontal line segment between <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> (located roughly at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math>),<br><em><strong>A1</strong></em> for a correct concave down parabola including max point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></math> and for correct end points at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>6</mn><mo>,</mo><mn>0</mn><mo>)</mo></math> (within circles). Points do not need to be labelled.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<br><hr><br><div class="specification">
<p>Consider the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>(</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>20</mn><mo>)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mo>(</mo><mo>-</mo><mn>14</mn><mo>,</mo><mo> </mo><mn>12</mn><mo>)</mo></math>. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[BC]</mtext></math>.</p>
</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">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>k</mi><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math>.</p>
<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">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"><msub><mi>m</mi><mtext>BC</mtext></msub><mo>=</mo><mfrac><mrow><mn>12</mn><mo>-</mo><mn>6</mn></mrow><mrow><mo>-</mo><mn>14</mn><mo>-</mo><mn>4</mn></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mi>L</mi></msub><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><msub><mi>m</mi><mtext>BC</mtext></msub></mfrac></math> using their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mtext>BC</mtext></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mi>L</mi></msub><mo>=</mo><mn>3</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>20</mn><mo>=</mo><mn>3</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mo> </mo><mo> </mo><mi>y</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>26</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>26</mn></math></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>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>k</mi><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math> into their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>-</mo><mn>20</mn><mo>=</mo><mn>3</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>=</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>26</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>-</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Finding the gradient of a line was well understood and many candidates also correctly found the perpendicular slope. Even with an error in their part (a), follow through marks in part (b) allowed many candidates to earn full marks for finding k despite their incorrect equation resulting 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="question">
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> are defined for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mn>2</mn></mfenced><mo>=</mo><mo>-</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>5</mn></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
</div>
<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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>-</mo><mn>2</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mn>2</mn></mfenced><mo>=</mo><mo>-</mo><mn>3</mn><mo>⇒</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>-</mo><mn>2</mn><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mi>b</mi></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>5</mn><mo>⇒</mo><mo>-</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>5</mn></math> <strong>A1</strong></p>
<p>a valid attempt to solve their two linear equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> <strong>M1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>3</mn></math> <strong>A1</strong></p>
<p> </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="specification">
<p>Consider the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mi>x</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>x</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo></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>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>π</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></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><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math></p>
<p>recognising to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><msqrt><mn>3</mn></msqrt></math> (values may be seen in right triangle) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mi>π</mi><mn>6</mn></mfrac></math> (seen anywhere) (accept degrees) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>6</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>7</mn><mi>π</mi></mrow><mn>6</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>12</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>7</mn><mi>π</mi></mrow><mn>12</mn></mfrac></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong></em> if any additional solutions are seen.<br>Award <em><strong>A1A0</strong> </em>for correct answers in degrees.<br>Award <em><strong>A0A0</strong> </em>for correct answers in degrees with additional values.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Determining the composite function was very well done. In part (b) very few candidates showed any recognition that tan (or cot) were required to solve this trigonometric equation. Many saw the 2<em>x</em> and simply employed one of the double angle rules but could not then progress to an answer.</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>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> </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>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = ax + b + \frac{c}{x}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mfrac>
<mi>c</mi>
<mi>x</mi>
</mfrac>
</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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> are positive integers.</p>
<p>Part 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> is shown on the axes below. The graph of the function has its local maximum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( - 2,{\text{ }} - 2)">
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span> and its local minimum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(2,{\text{ }}6)">
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_11.28.21.png" alt="M17/5/MATSD/SP1/ENG/TZ1/12"></p>
</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="y = - 6"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> </math></span> on the axes.</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>Write down the number of solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 6"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>6</mn> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the range of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = k"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </math></span> has no solution.</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><img src="images/Schermafbeelding_2017-08-15_om_15.32.51.png" alt="M17/5/MATSD/SP1/ENG/TZ1/21.b.i/M"> <strong><em>(A1)</em></strong> <strong><em>(C1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> The command term “Draw” states: “A ruler (straight edge) should be used for straight lines”; do not accept a freehand <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 6"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> </math></span> line.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2 <strong><em>(A1)</em>(ft)</strong> <strong><em>(C1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from part (b)(i).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 < k < 6"> <mo>−</mo> <mn>2</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mn>6</mn> </math></span> <strong><em>(A1)(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(A1) </em></strong>for both end points correct and <strong><em>(A1) </em></strong>for correct <strong>strict </strong>inequalities.</p>
<p>Award at most <strong><em>(A1)(A0) </em></strong>if the stated variable is different from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> for example <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 < x < 6"> <mo>−</mo> <mn>2</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>6</mn> </math></span> is <strong><em>(A1)(A0)</em></strong>.</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">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="question">
<p>Consider the vectors <em><strong>a</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> and <em><strong>b</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p>Find the possible values of <em>p</em> for which <strong><em>a</em></strong> and <strong><em>b</em></strong> are parallel.</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>(eliminating <em>k</em>)</p>
<p>recognizing parallel vectors are multiples of each other <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em><strong>a</strong></em> = <em>k<strong>b</strong></em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> = <em>k</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{p + 1}}{3} = \frac{8}{{2p}}"> <mfrac> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> <mo>=</mo> <mfrac> <mn>8</mn> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </math></span>, 3<em>k</em> = <em>p</em> + 1 and 2<em>kp</em> = 8</p>
<p>correct working (must be quadratic) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 2<em>p</em><sup>2</sup> + 2<em>p</em> = 24, <em>p</em><sup>2</sup> + <em>p</em> – 12, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 = \frac{{{p^2} + p}}{4}"> <mn>3</mn> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>p</mi> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p>valid attempt to solve <strong>their</strong> quadratic equation <em><strong>(M1)</strong></em></p>
<p><em>eg </em>factorizing, formula, completing the square</p>
<p>evidence of correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> (<em>p</em> + 4)(<em>p</em> – 3), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{ - 2 \pm \sqrt {4 - 4\left( 2 \right)\left( { - 24} \right)} }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mo>±</mo> <msqrt> <mn>4</mn> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>24</mn> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p><em>p</em> = –4, <em>p</em> = 3 <strong> <em>A1A1 N4</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong> (solving for <em>k</em>)</p>
<p>recognizing parallel vectors are multiples of each other <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em><strong>a</strong></em> = <em>k<strong>b</strong></em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> = <em>k</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, 3<em>k</em> = <em>p</em> + 1 and 2<em>kp</em> = 8</p>
<p>correct working (must be quadratic) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 3<em>k</em><sup>2</sup> – <em>k</em> = 4, 3<em>k</em><sup>2</sup> – <em>k</em> – 4, 4<em>k</em><sup>2</sup> = 3 – <em>k</em></p>
<p>one correct value for <em>k</em> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <em>k</em> = –1, <em>k</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{3}"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span>, <em>k</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{4}"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span></p>
<p>substituting <strong>their</strong> value(s) of <em>k</em> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right) = \frac{3}{4}\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\left( {\frac{4}{3}} \right) = p + 1"> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\left( {\frac{4}{3}} \right)p = 8"> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>p</mi> <mo>=</mo> <mn>8</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 1} \right)\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><em>p</em> = –4, <em>p</em> = 3 <em><strong>A1A1 N4</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong> (working with angles and cosine formula)</p>
<p>recognizing angle between parallel vectors is 0 and/or 180° <em><strong>M1</strong></em></p>
<p><em>eg</em> cos <em>θ</em> = ±1, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \bullet b = \left| a \right|\left| b \right|"> <mi>a</mi> <mo>∙</mo> <mi>b</mi> <mo>=</mo> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mi>b</mi> <mo>|</mo> </mrow> </math></span></p>
<p>correct substitution of scalar product and magnitudes into equation <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\left( {p + 1} \right) + 2p\left( 8 \right)}}{{\sqrt {{3^2} + {{\left( {2p} \right)}^2}} \sqrt {{{\left( {p + 1} \right)}^2} + {8^2}} }} = \pm 1"> <mfrac> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mrow> <msqrt> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>8</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>=</mo> <mo>±</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="19p + 3 = \sqrt {4{p^2} + 9} \sqrt {{p^2} + 2p + 65} "> <mn>19</mn> <mi>p</mi> <mo>+</mo> <mn>3</mn> <mo>=</mo> <msqrt> <mn>4</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>9</mn> </msqrt> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>65</mn> </msqrt> </math></span></p>
<p>correct working (must include both ± ) <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( {p + 1} \right) + 2p\left( 8 \right) = \pm \sqrt {{3^2} + {{\left( {2p} \right)}^2}} \sqrt {{{\left( {p + 1} \right)}^2} + {8^2}} "> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>±</mo> <msqrt> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>8</mn> <mn>2</mn> </msup> </mrow> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="19p + 3 = \pm \sqrt {4{p^2} + 9} \sqrt {{p^2} + 2p + 65} "> <mn>19</mn> <mi>p</mi> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mo>±</mo> <msqrt> <mn>4</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>9</mn> </msqrt> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>65</mn> </msqrt> </math></span></p>
<p>correct quartic equation <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="361\,{p^2} + 114p + 9 = 4{p^4} + 8{p^3} + 269{p^2} + 18p + 585"> <mn>361</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>114</mn> <mi>p</mi> <mo>+</mo> <mn>9</mn> <mo>=</mo> <mn>4</mn> <mrow> <msup> <mi>p</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mrow> <msup> <mi>p</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>269</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>18</mn> <mi>p</mi> <mo>+</mo> <mn>585</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4{p^4} + 8{p^3} - 92{p^2} - 96p + 576 = 0"> <mn>4</mn> <mrow> <msup> <mi>p</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mrow> <msup> <mi>p</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>92</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>96</mn> <mi>p</mi> <mo>+</mo> <mn>576</mn> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p^4} + 2{p^3} - 23{p^2} - 24p + 144 = 0"> <mrow> <msup> <mi>p</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mi>p</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>23</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>24</mn> <mi>p</mi> <mo>+</mo> <mn>144</mn> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {p + 4} \right)^2}{\left( {p - 3} \right)^2} = 0"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><em>p</em> = –4, <em>p</em> = 3 <em><strong>A2 N4</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {x^2} - 4x + 5">
<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>
<mn>4</mn>
<mi>x</mi>
<mo>+</mo>
<mn>5</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The function can also be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {(x - h)^2} + k">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>k</mi>
</math></span>.</p>
</div>
<div class="question">
<p>(i) Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>.</p>
<p>(ii) 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>(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 2">
<mi>h</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <strong><em>A1 N1</em></strong></p>
<p>(ii) <strong>METHOD 1</strong></p>
<p>valid attempt to find <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><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(2)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p>correct substitution into <strong>their </strong>function <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="{(2)^2} - 4(2) + 5">
<mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>5</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 1">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>valid attempt 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="{x^2} - 4x + 4">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</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="({x^2} - 4x + 4) - 4 + 5,{\text{ }}{(x - 2)^2} + 1">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>4</mn>
<mo>+</mo>
<mn>5</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 1">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>, with derivative <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 2{x^2} + 5kx + 3{k^2} + 2">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mi>k</mi>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x{\text{, }}k \in \mathbb{R}">
<mi>x</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the discriminant of <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> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - 16"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>16</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> is an increasing function, find all possible 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">[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>correct substitution into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{b^2} - 4ac"> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </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="{\left( {5k} \right)^2} - 4\left( 2 \right)\left( {3{k^2} + 2} \right)"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {5k} \right)^2} - 8\left( {3{k^2} + 2} \right)"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>8</mn> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>correct expansion of each term <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="25{k^2} - 24{k^2} - 16"> <mn>25</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>24</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>16</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="25{k^2} - \left( {24{k^2} + 16} \right)"> <mn>25</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>24</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>16</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - 16"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>16</mn> </math></span> <em><strong>AG</strong></em><em><strong> N0</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 <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) \geqslant 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>recognizing discriminant <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" < 0"> <mo><</mo> <mn>0</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \leqslant 0"> <mo>⩽</mo> <mn>0</mn> </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="D < 0"> <mi>D</mi> <mo><</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - 16 \leqslant 0"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>16</mn> <mo>⩽</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} < 16"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo><</mo> <mn>16</mn> </math></span></p>
<p>two correct values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>/endpoints (even if inequalities are incorrect) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \pm 4"> <mi>k</mi> <mo>=</mo> <mo>±</mo> <mn>4</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k < - 4"> <mi>k</mi> <mo><</mo> <mo>−</mo> <mn>4</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k > 4"> <mi>k</mi> <mo>></mo> <mn>4</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| k \right| < 4"> <mrow> <mo>|</mo> <mi>k</mi> <mo>|</mo> </mrow> <mo><</mo> <mn>4</mn> </math></span></p>
<p>correct interval <em><strong>A1</strong></em><em><strong> N2</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4 < k < 4"> <mo>−</mo> <mn>4</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mn>4</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4 \leqslant k \leqslant 4"> <mo>−</mo> <mn>4</mn> <mo>⩽</mo> <mi>k</mi> <mo>⩽</mo> <mn>4</mn> </math></span></p>
<p><strong>Note:</strong> Candidates may work with an equation, then write the intervals with inequalities at the end. If inequalities are not seen until the candidate’s final correct answer, <em><strong>M0M0A1A1</strong></em> may be awarded.<br>If candidate is working with incorrect inequalitie(s) at the beginning, then gets the correct final answer, award <em><strong>M0M0A1A0</strong></em> or <em><strong>M1M0A1A0</strong></em> or<em><strong> M0M1A1A0</strong></em> in line with the markscheme.</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>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>></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 coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math>, 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> <em><strong> (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> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>±</mo><msqrt><mn>17</mn></msqrt></math> <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> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to differentiate (must include <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi></math> and/or <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>) <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><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> <em><strong>A1</strong></em></p>
<p>setting their derivative <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> <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> OR <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> (or equivalent) <em><strong>A1</strong></em></p>
<p>valid attempt to solve their quadratic <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> if the candidate’s final answer includes additional solutions (such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>8</mn></math>).</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br>