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<h2>HL Paper 1</h2><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>4</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mn>2</mn>
<mi>n</mi>
</msup>
</mrow>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> </math></span> is an odd or even function, justifying your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using mathematical induction, prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mfrac> <mrow> <mi>m</mi> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{Z}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </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>Hence or otherwise, find an expression for the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 1"> <mi>n</mi> <mo>></mo> <mn>1</mn> </math></span>, the equation of the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_n}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> at <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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y - \pi = 0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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>even function <strong><em>A1</em></strong></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos kx = \cos ( - kx)"> <mi>cos</mi> <mo></mo> <mi>k</mi> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <strong>and</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is a product of even functions <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p>even function <strong><em>A1</em></strong></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\cos 2x)(\cos 4x) \ldots = \left( {\cos ( - 2x)} \right)\left( {\cos ( - 4x)} \right) \ldots "> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>…</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>…</mo> </math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>A0R1</em></strong>.</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>consider the case <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin 4x}}{{2\sin 2x}} = \frac{{2\sin 2x\cos 2x}}{{2\sin 2x}} = \cos 2x"> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p>hence true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> <strong><em>R1</em></strong></p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>, <em>ie</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\cos 2x)(\cos 4x) \ldots (\cos {2^k}x) = \frac{{\sin {2^{k + 1}}x}}{{{2^k}\sin 2x}}"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>…</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>M1 </em></strong>for “let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>” or “assume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>” or equivalent.</p>
<p> </p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1"> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span>:</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{k + 1}}(x) = {f_k}(x)(\cos {2^{k + 1}}x)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sin {2^{k + 1}}x}}{{{2^k}\sin 2x}}\cos {2^{k + 1}}x"> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{2\sin {2^{k + 1}}x\cos {2^{k + 1}}x}}{{{2^{k + 1}}\sin 2x}}"> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sin {2^{k + 2}}x}}{{{2^{k + 1}}\sin 2x}}"> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> true and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span> true <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow n = k + 1"> <mo stretchy="false">⇒</mo> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span> true. Hence true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }"> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> To obtain the final <strong><em>R1</em></strong>, all the previous <strong><em>M </em></strong>marks must have been awarded.</p>
<p> </p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’ = \frac{{vu' - uv'}}{{{v^2}}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mi>v</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>u</mi> <msup> <mi>v</mi> <mo>′</mo> </msup> </mrow> <mrow> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> (or correct product rule) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}(x) = \frac{{({2^n}\sin 2x)({2^{n + 1}}\cos {2^{n + 1}}x) - (\sin {2^{n + 1}}x)({2^{n + 1}}\cos 2x)}}{{{{({2^n}\sin 2x)}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for correct numerator and <strong><em>A1 </em></strong>for correct denominator.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = \frac{{\left( {{2^n}\sin \frac{\pi }{2}} \right)\left( {{2^{n + 1}}\cos {2^{n + 1}}\frac{\pi }{4}} \right) - \left( {\sin {2^{n + 1}}\frac{\pi }{4}} \right)\left( {{2^{n + 1}}\cos \frac{\pi }{2}} \right)}}{{{{\left( {{2^n}\sin \frac{\pi }{2}} \right)}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = \frac{{({2^n})\left( {{2^{n + 1}}\cos {2^{n + 1}}\frac{\pi }{4}} \right)}}{{{{({2^n})}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\cos {2^{n + 1}}\frac{\pi }{4}{\text{ }}( = 2\cos {2^{n - 1}}\pi )"> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>π</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = 2"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}\left( {\frac{\pi }{4}} \right) = 0"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> This <strong><em>A </em></strong>mark is independent from the previous marks.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2\left( {x - \frac{\pi }{4}} \right)"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y - \pi = 0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^4} + a{z^3} + b{z^2} + cz + d = 0"> <mrow> <msup> <mi>z</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mi>a</mi> <mrow> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>b</mi> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>c</mi> <mi>z</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}"> <mi>d</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}"> <mi>z</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> </math></span>.</p>
<p>Two of the roots of the equation are log<sub>2</sub>6 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\sqrt 3 "> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </math></span> and the sum of all the roots is 3 + log<sub>2</sub>3.</p>
<p>Show that 6<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="d"> <mi>d</mi> </math></span> + 12 = 0.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - i\sqrt 3 "> <mo>−</mo> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </math></span> is a root <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 + {\text{lo}}{{\text{g}}_2}3 - {\text{lo}}{{\text{g}}_2}6\left( { = 3 + {\text{lo}}{{\text{g}}_2}\frac{1}{2} = 3 - 1 = 2} \right)"> <mn>3</mn> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo>−</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span> is a root <em><strong>(A1)</strong></em></p>
<p>sum of roots: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - a = 3 + {\text{lo}}{{\text{g}}_2}3 \Rightarrow a = - 3 - {\text{lo}}{{\text{g}}_2}3"> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award M1 for use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - a"> <mo>−</mo> <mi>a</mi> </math></span> is equal to the sum of the roots, do not award if minus is missing.</p>
<p><strong>Note:</strong> If expanding the factored form of the equation, award <em><strong>M1</strong> </em>for equating <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> to the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^3}"> <mrow> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> </math></span>.</p>
<p> </p>
<p>product of roots: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( { - 1} \right)^4}d"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mn>4</mn> </msup> </mrow> <mi>d</mi> </math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\left( {{\text{lo}}{{\text{g}}_2}6} \right)\left( {i\sqrt 3 } \right)\left( { - i\sqrt 3 } \right)"> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6\,{\text{lo}}{{\text{g}}_2}6"> <mo>=</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - 6\,{\text{lo}}{{\text{g}}_2}6"> <mi>d</mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> </math></span></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6a + d + 12 = - 18 - 6\,{\text{lo}}{{\text{g}}_2}3 + 6\,{\text{lo}}{{\text{g}}_2}6 + 12"> <mn>6</mn> <mi>a</mi> <mo>+</mo> <mi>d</mi> <mo>+</mo> <mn>12</mn> <mo>=</mo> <mo>−</mo> <mn>18</mn> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> <mo>+</mo> <mn>12</mn> </math></span></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 6 + 6\,{\text{lo}}{{\text{g}}_2}2 = 0"> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>2</mn> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1A1AG</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong> </em>is for a correct use of one of the log laws.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 6 - 6\,{\text{lo}}{{\text{g}}_2}3 + 6\,{\text{lo}}{{\text{g}}_2}3 + 6\,{\text{lo}}{{\text{g}}_2}2 = 0"> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>2</mn> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1A1AG</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong> </em>is for a correct use of one of the log laws.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>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,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"></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>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the inequality <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mfrac><mrow><mn>2</mn><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>-</mo><mn>1</mn></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mn>1</mn></mrow></mfrac><mo><</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><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>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>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><strong><br>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>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempts to sketch <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>-</mo><mn>1</mn></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mn>1</mn></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p style="text-align:left;"><strong>Note:</strong> Award the <em><strong>(M1)</strong></em> if <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>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> are identified.</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><mfrac><mn>1</mn><mn>2</mn></mfrac></math> or <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 the use of a comma. Condone the use of ‘and’. Accept correct alternative notation.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>In the following Argand diagram, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>2</mtext></msub></math> are the vertices of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>1</mtext></msub><msub><mtext>OZ</mtext><mtext>2</mtext></msub></math> described anticlockwise.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub></math> represents the complex number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>α</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></math>. The point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>2</mn></msub></math> represents the complex number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>θ</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></math>.</p>
<p>Angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>θ</mi></math> are measured anticlockwise from the positive direction of the real axis such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>,</mo><mo> </mo><mi>θ</mi><mo><</mo><mn>2</mn><mi>π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math>.</p>
</div>
<div class="specification">
<p>In parts (c), (d) and (e), consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>1</mtext></msub><msub><mtext>OZ</mtext><mtext>2</mtext></msub></math> is an equilateral triangle.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math> be the distinct roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>z</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> and <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>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup></math> is the complex conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub><msub><mtext>OZ</mtext><mn>2</mn></msub></math> is a right-angled triangle.</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>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the result from part (c)(ii) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>b</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>z</mi><mo>+</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math>, deduce that only one equilateral triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub><msub><mtext>OZ</mtext><mn>2</mn></msub></math> can be formed from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the roots of this equation.</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"><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>-i</mtext><mi>θ</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>α</mi></mrow></msup><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>-i</mtext><mi>θ</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced></mrow></msup></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Accept working in modulus-argument form</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"><mtext>Re</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup></mrow></mfenced><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub><mo> </mo><mi>cos</mi><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>θ</mi><mo>=</mo><mtext>arcos</mtext><mo> </mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>,</mo><msub><mi>r</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi mathvariant="normal">π</mi></math>) <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub><msub><mtext>OZ</mtext><mn>2</mn></msub></math> is a right-angled triangle <em><strong>AG</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><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mfrac><mfenced><mrow><mo>=</mo><mfrac><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub></mfrac><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced></mrow></msup></mrow></mfenced><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub></math>) <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mi>e</mi><mrow><mi>i</mi><mfenced><mrow><mi>θ</mi><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></mfenced></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>θ</mi></mrow></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept working in either modulus-argument form to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></mfenced></math> or in Cartesian form to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mtext>i</mtext></mrow></mfenced></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup></math> <em><strong>M</strong><strong>1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mtext>i</mtext><mo>+</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><msub><mi>z</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> For candidates who work on the LHS and RHS separately to show equality, award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math>, <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math>. Accept working in either modulus-argument form or in Cartesian form.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mi>b</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>+</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><mo>=</mo><mn>3</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> into their expression <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>b</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn><mi>b</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> is not clearly recognized, award maximum <em><strong>(A0)A1A1M1A0</strong></em>.</p>
<p> </p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>b</mi><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mi>b</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>2</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>+</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><mo>=</mo><mn>3</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> into their expression <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>b</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn><mi>b</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> is not clearly recognized, award maximum <em><strong>(A0)A1A1M1A0</strong></em>.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>b</mi><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>×</mo><mn>12</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>±</mo><mn>6</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><msup><mi>z</mi><mn>2</mn></msup><mo>±</mo><mn>6</mn><mi>z</mi><mo>+</mo><mn>12</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>6</mn><mo>:</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt><mtext>i</mtext><mo>,</mo><mo> </mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mn>3</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></math> which does not satisfy <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math> <em><strong>R1</strong></em></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>6</mn><mo>:</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mo>-</mo><mn>3</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mtext>i</mtext><mo>,</mo><mo> </mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>so (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math>), only one equilateral triangle can be formed from point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the two roots of this equation <em><strong>AG</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>The vast majority of candidates scored full marks in parts (a) and (b). If they did not, it was normally due to the lack of rigour in setting out of the answer to a "show that" question. Part (c) was, though, more often than not poorly done. Many candidates could not use the given condition (equilateral triangle) to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math>. Part (d) was well answered by a rather high number of candidates.</p>
<p>Only a handful of students made good progress in (e), not even finding the possible values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let the roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^3} = - 3 + \sqrt 3 {\text{i}}">
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>+</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>On an Argand diagram, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span> are represented by the points U, V and W respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + \sqrt 3 {\text{i}}"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> <mrow> <mtext>i</mtext> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{{\text{e}}^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r > 0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \pi < \theta \leqslant \pi "> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo>⩽</mo> <mi>π</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> expressing your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{{\text{e}}^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r > 0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \pi < \theta \leqslant \pi "> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo>⩽</mo> <mi>π</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>Find the area of triangle UVW.</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>By considering the sum of the roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span>, show that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{7\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}} = 0"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to find modulus <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 2\sqrt 3 \left( { = \sqrt {12} } \right)"> <mi>r</mi> <mo>=</mo> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <msqrt> <mn>12</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to find argument in the correct quadrant <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \pi + {\text{arctan}}\left( { - \frac{{\sqrt 3 }}{3}} \right)"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>+</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{5\pi }}{6}"> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + \sqrt 3 {\text{i}} = \sqrt {12} {{\text{e}}^{\frac{{5\pi {\text{i}}}}{6}}}\left( { = 2\sqrt 3 {{\text{e}}^{\frac{{5\pi {\text{i}}}}{6}}}} \right)"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <msqrt> <mn>12</mn> </msqrt> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></span></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find a root using de Moivre’s theorem <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}{{\text{e}}^{\frac{{5\pi {\text{i}}}}{{18}}}}"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to find further two roots by adding and subtracting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> to the argument <em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}{{\text{e}}^{ - \frac{{7\pi {\text{i}}}}{{18}}}}"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}{{\text{e}}^{\frac{{17\pi {\text{i}}}}{{18}}}}"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Ignore labels for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> at this stage.</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><em><strong>METHOD 1</strong></em><br>attempting to find the total area of (congruent) triangles UOV, VOW and UOW <em><strong>M1</strong></em></p>
<p>Area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\left( {\frac{1}{2}} \right)\left( {{{12}^{\frac{1}{6}}}} \right)\left( {{{12}^{\frac{1}{6}}}} \right){\text{sin}}\frac{{2\pi }}{3}"> <mo>=</mo> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{{12}^{\frac{1}{6}}}} \right)\left( {{{12}^{\frac{1}{6}}}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> and <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{2\pi }}{3}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\sqrt 3 }}{4}\left( {{{12}^{\frac{1}{3}}}} \right)"> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>UV<sup>2</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {{{12}^{\frac{1}{6}}}} \right)^2} + {\left( {{{12}^{\frac{1}{6}}}} \right)^2} - 2\left( {{{12}^{\frac{1}{6}}}} \right)\left( {{{12}^{\frac{1}{6}}}} \right){\text{cos}}\frac{{2\pi }}{3}"> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>UV <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt 3 \left( {{{12}^{\frac{1}{6}}}} \right)"> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>attempting to find the area of UVW using Area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{1}{2}} "> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></span> × UV × VW × sin <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span> for example <em><strong>M1</strong></em></p>
<p>Area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {\sqrt 3 \times {{12}^{\frac{1}{6}}}} \right)\left( {\sqrt 3 \times {{12}^{\frac{1}{6}}}} \right){\text{sin}}\frac{\pi }{3}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>×</mo> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>×</mo> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\sqrt 3 }}{4}\left( {{{12}^{\frac{1}{3}}}} \right)"> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> (or equivalent) <em><strong>A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> = 0 <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}\left( {{\text{cos}}\left( { - \frac{{7\pi }}{{18}}} \right) + {\text{i}}\,{\text{sin}}\left( { - \frac{{7\pi }}{{18}}} \right) + {\text{cos}}\frac{{5\pi }}{{18}} + {\text{i}}\,{\text{sin}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}} + {\text{i}}\,{\text{sin}}\frac{{17\pi }}{{18}}} \right) = 0"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>A1</strong></em></p>
<p>consideration of real parts <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}\left( {{\text{cos}}\left( { - \frac{{7\pi }}{{18}}} \right) + {\text{cos}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}}} \right) = 0"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </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="{\text{cos}}\left( { - \frac{{7\pi }}{{18}}} \right) = {\text{cos}}\frac{{17\pi }}{{18}}"> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </math></span> explicitly stated <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{7\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}} = 0"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi><mo> </mo><mo>\</mo><mo> </mo><mfenced open="{" close="}"><mi>k</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>≠</mo><mn>5</mn></math>. </p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the vertical asymptote on 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>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the horizontal asymptote on 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>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an algebraic method to determine whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is a self-inverse function.</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>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>, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.</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>The region bounded by the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>7</mn></math> is rotated through <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. Find the volume of the solid generated, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>)</mo><mo> </mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[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>y</mi><mo>=</mo><mi>k</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mfenced><mstyle displaystyle="true"><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></mstyle></mfenced><mo>-</mo><mn>5</mn></mrow><mrow><mfenced><mstyle displaystyle="true"><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></mstyle></mfenced><mo>-</mo><mi>k</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>k</mi><mfenced><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo>-</mo><mn>5</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced></mrow><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn><mo>-</mo><mi>k</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><mn>5</mn><mi>k</mi><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>k</mi></mrow><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn><mo>-</mo><mi>k</mi><mi>x</mi><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><mn>5</mn><mi>x</mi></mrow><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>x</mi><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfenced></mrow><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi></math> , (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is self-inverse) <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> The statement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>x</mi></math> could be seen anywhere in the candidate’s working to award <em><strong>R1</strong></em>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> 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 any stage.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfenced><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>-</mo><mi>x</mi><mi>k</mi><mo>=</mo><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>-</mo><mi>k</mi><mi>y</mi><mo>=</mo><mi>x</mi><mi>k</mi><mo>-</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><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><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math> (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is self-inverse) <em><strong>R1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img style="display: block;margin-left:auto;margin-right:auto;" 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"></p>
<p>attempt to draw both branches of a rectangular hyperbola <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>5</mn><mn>7</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong>EITHER</strong></p>
<p>attempt to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mi>q</mi><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mn>4</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup></math> and divide out <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo>+</mo><mfrac><mrow><mn>24</mn><mi>x</mi><mo>-</mo><mn>56</mn></mrow><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo>+</mo><mfrac><mn>24</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>16</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>5</mn><mn>7</mn></munderover><mfenced><mrow><mn>9</mn><mo>+</mo><mfrac><mn>24</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>16</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></mrow></mfenced><mo> </mo><mtext>d</mtext><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mn>9</mn><mi>x</mi><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mfrac><mn>16</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mrow></mfenced><mn>5</mn><mn>7</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced open="⌊" close="⌋"><mrow><mfenced><mrow><mn>63</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>4</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>45</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>-</mo><mn>8</mn></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>22</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>5</mn><mn>7</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>⇒</mo><mfrac><mrow><mtext>d</mtext><mi>u</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn><mo>=</mo><mn>3</mn><mfenced><mrow><mi>u</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mn>5</mn><mo>=</mo><mn>3</mn><mi>u</mi><mo>+</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>2</mn><mn>4</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>u</mi><mo>+</mo><mn>4</mn></mrow><mi>u</mi></mfrac></mfenced><mn>2</mn></msup><mtext>d</mtext><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>2</mn><mn>4</mn></msubsup><mn>9</mn><mo>+</mo><mfrac><mn>16</mn><msup><mi>u</mi><mn mathvariant="italic">2</mn></msup></mfrac><mo>+</mo><mfrac><mn>24</mn><mi>u</mi></mfrac><mo> </mo><mtext>d</mtext><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mn>9</mn><mi>u</mi><mo>-</mo><mfrac><mn>16</mn><mi>u</mi></mfrac><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>u</mi></mrow></mfenced><mn>2</mn><mn>4</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Ignore absence of or incorrect limits seen up to this point.</p>
<p><em><strong><br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>22</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math><em><strong> A1<br></strong></em></p>
<p><em><strong><br></strong></em><em><strong>[6 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>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</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>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. </p>
</div>
<div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>8</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> may be obtained by transforming the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> using a sequence of three transformations.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></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>Hence, given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> does not exist, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>a</mi><mi>c</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> exists.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>(</mo><mi>x</mi><mo>−</mo><mn>2</mn><msup><mo>)</mo><mn>3</mn></msup><mo>+</mo><mi>q</mi></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<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">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State each of the transformations in the order in which they are applied.</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 graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math> on the same set of axes, indicating the points where each graph crosses the coordinate axes.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em></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>since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> does not exist, there must be two turning points <em><strong>R1</strong></em></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> has more than one solution)</p>
<p>using the discriminant <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo>></mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>12</mn><mi>a</mi><mi>c</mi><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>a</mi><mi>c</mi><mo>></mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>a</mi><mi>c</mi><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>-</mo><mn>9</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> exists <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>6</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>×</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mo>=</mo><mn>36</mn><mo>-</mo><mn>36</mn><mo>=</mo><mn>0</mn><mo>⇒</mo></math> there is (only) one point with gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and this must be a point of inflexion (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> is a cubic.) <em><strong>R1</strong></em></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> exists <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>-</mo><mn>8</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>4</mn></math></p>
<p><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><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>4</mn><mo>⇒</mo><mi>q</mi><mo>=</mo><mo>-</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[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>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>4</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> 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 any stage.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>=</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mn>3</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mrow><mn>2</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></mrow><mn>3</mn></mroot><mo>=</mo><mi>y</mi><mo>-</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mroot><mrow><mn>2</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></mrow><mn>3</mn></mroot><mo>+</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mroot><mrow><mn>2</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></mrow><mn>3</mn></mroot><mo>+</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>…</mo></math> must be seen for the final <em><strong>A</strong></em> mark.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>translation through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math>, <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> This can be seen anywhere.</p>
<p><br><strong>EITHER<br></strong>a stretch scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis then a translation through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A2<br></strong></em><strong>OR<br></strong>a translation through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>8</mn></mtd></mtr></mtable></mfenced></math> then a stretch scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis <em><strong>A2</strong></em></p>
<p><strong><br>Note:</strong> Accept ‘shift’ for translation, but do not accept ‘move’. Accept ‘scaling’ for ‘stretch’.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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gOmQKBrR2BgUHgmKGwp9tG4TwKDM8Dw9OlTtS10r169GBfMjZ7RgA4ETK8TGBgUngmKaBRiW96DwOAMMFy9elUBwxCDI8Zt0Qrv0xmt2GRXUyDQtSMwEBgIDBZqgMDgTBE4fvw4T6q0MB5sKt7xuFcdCJheJzAwOAgMFmqAwOAMMGzdukUdPDV37lzGhYVxEY9ibMPfNAUCXTsCA4OCidFCDRAYnAGGRYsWSbp0aWXZsmWMCwvjwobiHY971IGA6XUCA4OCidFCDRAYnAGGGTNmqJMqN2zYwLiwMC7iUYxt+JumQKBrR2BgUDAxWqgBAoMzwDB69GjJkiWzHD58mHFhYVzYULzjcY86EDC9TmBgUDAxWqgBAoMzwPDdd99KjhzZ5dKlS4wLC+MiHsXYhr9pCgS6dgQGBgUTo4UaIDA4Awxdu34pb76ZSx4+fMi4sDAubCje8bhHHQiYXicwMCiYGC3UAIHBGWBo3bq1FCpUkDFhYUzEoxDb8jdNgUDXjsDAwGBytFADBAZngKFu3TpSteo7jAkLY8KW4h2P+9SBgOl1AgMDg8nRQg0QGJwBhnLlysqHHzZiTFgYE/EoxLb8TVMg0LUjMDAwmBwt1ACBIfrA8OjRIzUc0aZNG8aEhTFhS/GOx33qQMD0OoGBgcHkaKEGCAzRB4Zr165Jvnx55ZtvujEmLIyJeBRiW/6mKRDo2hEYGBhMjhZqgMAQfWD48+RJyZkzh/Dgqejb1pbC6tX71IGA6XUCg4XFwqui5nOZJ2oCg7mtTHW1a9dOyZgxg0yYMJ4QzbzoKQ2YAoGuHYGBgeGpwDAtDra3IzBEHxjWrFmjtoWeM2cOY4J50VMa0IGA6XUCAwPDU4FhOwiY3j+BIfrA8PPPi9XBUxs38hwJUx2yXfR16IRNTYFA147AQGAgMFioAQJD9BP1jBnTJU2aN+TYsWOMCQtjwolC65X31IGA6XUCAwODydFCDRAYog8MI0eOVMBw8+ZNxoSFMeGV4u7Ec5gCga4dgYGBweRooQYIDNEHhv79+6khiX/++YcxYWFMOFFovfKeOhAwvU5gYGAwOVqoAQJD9IGhc+cvpEiRwowHC+PBK4XdqecwBQJdOwIDg4MJ0kINEBiiDwzNmjWVd999l/FgYTw4VWi98r46EDC9TmBgcDBBWqgBAkN0geHZs2dSq1ZNadLkI8aDhfHglcLu1HOYAoGuHYGBwcEEaaEGCAzRBYZ7d+9KhQpvS4cOnzMeLIwHpwqtV95XBwKm1wkMDA4mSAs1QGCILjBcvXpVHTzVp09vxoOF8eCVwu7Uc5gCga4dgYHBwQRpoQYIDNEFhlOnTknmzJlk2LBhjAcL48GpQuuV99WBgOl1AgODgwnSQg0QGKILDL/9dlAyZEgvkyZNZDxYGA9eKexOPYcpEOjaERgYHEyQFmqAwBBdYNi2bZvag+GXX35hPFgYD04VWq+8rw4ETK8TGBgcTJAWaoDAEF1gWLFihdrlcefOnYwHC+PBK4XdqecwBQJdOwIDg4MJ0kINEBiiCwyzZ89WwHDixAnGg4Xx4FSh9cr76kDA9DqBgcHBBGmhBggM0QWGCRPGS5YsmeXSpUuMBwvjwSuF3annMAUCXTsCA4ODCdJCDRAYogsMgwYNlKJFi8jff//NeLAwHpwqtF55Xx0ImF4nMDA4mCAt1ACBIXrAgMOmunbtKpUrV5L79+8zHiyMB68UdqeewxQIdO0IDAwOJkgLNUBgiB4wvHz5Ulq2bCG1a9eWJ0+eMB4sjAenCq1X3lcHAqbXCQwMDiZICzVAYIgeMOAciWrVqkrLli3l+fPnjAcL48Erhd2p5zAFAl07AgODgwnSQg0QGKIHDOhVyJ8/n3Tq1FFevHjBeLAwHpwqtF55Xx0ImF4nMDA4mCAt1ACBIXrAcOfOHbVpU69ePQXzGbxSJPgc0dOI7bY0BQJdOwKDhcXCdvHy/iNPZASGyG0Y0OH58+ckbdo0MmLEcMIC86EnNaADAdPrBAYGiCcDJFAMvPqdwBA9YPjjj0Nq06ZZs2YyFpgPPakBUyDQtSMwMEA8GSBeBYXAcxEYogcMS5cuVUMS3BY6ejYN6JTf3WFTHQiYXicwEBgIDBZqgMAQvUQ8Y8YMyZUrp3Bb6OjZlKDgLluaAoGuHYHBwmLBYHRXMMbDHwSG6Gnghx9+kIIF35KrV68QnpkPPakBHQiYXicwMEA8GSDxKOKx/JsEhugBQ5cunaVUqZLy8OFDxgLzoSc1YAoEunYEBgaIJwMklsU7Hn+LwBA9YKhXt65Ur/6uYAOnePiSfzN6vqQtg9tSBwKm1wkMBAYmSQs1QGAInhjDLRjYdwHDEY0aNeSmTRbGQbj+9mt7UyDQtSMwMEgIDBZqgMAQHWB4/PixZMqUUVq3bs1NmyyMA78CQLjPrQMB0+sEBgYJgcFCDRAYogMM169fl4wZM8hXX3VlHFgYB+EWTr+2NwUCXTsCA4OEidJCDRAYogMMJ0+ekPTp08mgQYMYBxbGgV8BINzn1oGA6XUCA4OEidJCDRAYogMMu3btVJs2TZgwnnFgYRyEWzj92t4UCHTtCAwMEiZKCzVAYIgOMKxZs0YBw5IlSxgHFsaBXwEg3OfWgYDpdQIDg4SJ0kINEBiiAwzz58+TDBnSy8GDBxkHFsZBuIXTr+1NgUDXjsDAIGGitFADBIboAMPYsWPUpMdr164xDiyMA78CQLjPrQMB0+sEBgYJE6WFGiAwRAcYevfuJdmzZ+OmTRbGQLhF08/tTYFA147AwEAhMFioAQJDdIChZcsWUrx4MW7aZGEM+BkAwn12HQiYXicwMFAIDBZqgMAQOTC8fPlS3n77bXnnnSqCn8NNwmwfuQ9ow9jY0BQIdO0IDBYWCwZZbILMzXYmMESugQcPHkiJEiWkYcMGhAXmQU9rQAcCptcJDAwUTweKm4t+JPdGYIgcGDDRsVChgtKuXVvGAPOgpzVgCgS6dgQGBoqnAyWSouzm3yUwRA4Mf/75p7z5Zi7p0aMHY4B50NMa0IGA6XUCAwPF04Hi5qIfyb0RGCIHBuy9kC1bVhk2bBhjgHnQ0xowBQJdOwIDA8XTgRJJUXbz7xIYIgeGrVu3qj0YVq/+lTHAPOhpDehAwPQ6gYGB4ulAcXPRj+TeCAyRA8PKlSskbdo0cvz4ccYA86CnNWAKBLp2BAYGiqcDJZKi7ObfJTBEDgxTp06VNGnekPv37zMGmAc9rQEdCJheJzAwUDwdKG4u+pHcG4EhcmDo37+f5MuXV54+fcoYYB70tAZMgUDXjsDAQPF0oERSlN38uwSGyIDhn3/+kVatWkmZMmW4LTRzoBU5EJp9/vx5qjYZ04GA6XUCA4PFimBxc/GOx70RGCIDBvQq1KlTR6pXr66ScDx8yL8ZmQ9tsR8KPfSGjcLu3LkjN25cl4sXL8qZM2fk2LFjcuDAAdm9e5ds3LhRli9fLgsWzJdJkybJjz+OkG+/HSCdO3eW1q1bS716deX9998XTNYN99lNgUDXjsBAYAhbfOGKle2jnxgJDJHZFPMWKleuJE2bNOE5EsyBKebAFy9eyOPHj1Wxv379mly4cEGwhweW5W7etEkwefann36SH3/8UQYMGCDdun2tNgPDDqIA0tKlS6nzSooUKSwFC74lefPmkZw5c6hDz7JkySKZM2dSq3XSp0+nJuFiXg2+8H8s+82TJ7fkz59PChcupN5nxfLlKd5vsHyrAwHT6wQGBkvY4gsmSL4WWQEL134EhsjsfePGDbXLY4cOn6eqizdcf7F9ZP4Kx374RP/s2TN59OiR3Lt3T/7++2+5cuWKnD9/Xk6dOiVHjx6RAwf2q0/qq1atkvnz58n48ePku+++la+/RrFvJx991Fhq1qwhJUuWlAIF8quijSKfJUtmVcgDRT1DhvSq4OPE01y5cql2aF+sWFEpU6a0VKjwtlSpUkVq1KgujRo1lE8//US++uor6d+/vwwePFjGjx+v/j56FrZv3ya//XZQTpw4LpcvXxbsRHrr1i01Kffhw4eqlwLwEo4tAm1NgUDXjsBAYEiVAANC5PfYJcLEtiYwRGZ3FA/s8tivX19BgUlsW/4cmW2jaT8UfXTjX716Rf766y85fPiw7N+/X3bs2CHr16+XZcuWyaxZs2TUqJGq+/6LLzpJ06ZNpF69elK9+ruqYBctWkR9qoe/UdgzZcqY8Ekey2rxGia/ogegVKmS8nb58up369SprYp8s2ZN5ZNPPpHu3b+RYUOHyoTx42XevHmyatVKBR179uyRI0eOyOnTp9VQA2AU94whCPRMAF7irTEdCJheJzAQGJgsLdQAgSGyonbixAnJmiWLzJgxnfqPgf5RMPEpGZ+aUfgPHjggW7ZskdWrV8vixYtk5swZMnLkSOnXr5+g1wfd+fh0jgJerFgx1ZWfO/ebkiNHdsma9b9ufHTZBz7p45M/uu1LliwhlSpVlJo1a0qjRo2kVauP1ft9/fVXCg7xiX7RooWyYcN62bt3r0AHmEtw6dIldW/ojbh79648fPiq2Kf2U300wSnS9zIFAl07AkMMgiVSZ/P3IysOXrQfgSEyTeBTIYrNhg0bCAwGORAFH4UTs/QxgS9Q/NHFj7F82BGf9mfP/knGjh0jAwb0l44dO8iHHzaS8uXLqbF4fLLPmDGDoBsfxT5durTqC//HF64BBnAgWPly5dT4Pyb6tWjRQjp16ii9evWUgQO/V8MHS5cuEfgQ8IECj0/yT548UfeGT/S4T9wvvnB0ebw/4cc7B+lAwPQ6gcEgWOLtbP79yIqDF+1HYIhMEytWrFDF6uTJk74FBhR+FFuM76Pw//bbb7J9+3b1qR8z9adOnaJm6mO/CnT1N2nykbzzThX1SR5d+YFP94HveA2T9NC9X7Tof2P4775bTerXryfNmzeTNm0+UxMCv//+OzWEgI2zABm7du2SP/88qeYaBAp8oMj7vdBHK3eZAoGuHYGBwODbhBmtYIzH+xAYIgOGyZMnqzkMmMsQD/859Tcx5g8AOHr0qBpfX7lypZpUh2V6Q4cOlV69eqku+sCkPozXY+Y+ZuoHPvUnBgDM0Mcs/2rVqkqDBvWldetW6tN+jx7dZeDAgTJhwgSZO3euGs/ftm2bHD78h5w/d05N1sMn/sAn/QAAOPXcfN+U40EHAqbXCQwEBk8lTL8kDgJDyglSp4OePXsIJsNdv37d9fpH0b169ar88ccfsmbNGpkzZ7ZMmDBeBg8epGb1f/rpp1K3bh3V9Y/ufCzbw0Q+dPOj2z9xbwDG+gEA77//nnz8cUv58ssugh6EESNGyJQpU2Tp0qVqbsGhQ4fk3LlzalwfE/gwBJG4qx8AoLMxr0em0WjazxQIdO0IDAQGBr6FGiAwRJaM8Um5YsUKqks+mok52HuhWx0FFt3tiZf7YTY9Jt1t27ZVFi5c8B8EDBok3b7+Wj7++GOpVaumgprEn/j/G/dPp2b6o/s/V66cagigSpXKar5A+/bt1Fj/8OHDBV3+ixcvVpsCYWIfhh+C3R9fi0xLNthPBwKm1wkMFhYLGwTKeww/CaGgmI7ZEhjCt29iTWJdPD5lows/8evh/gx/BUAAn8LRY4GJeJiBjxUA2NAH6+2xvj+wWx9m8b/1VgH16R8wAAhAbwAmBaJ3oESJEmpTKYz9t2vXVgHADz/8INOmTZNFixap98X7AwJu375trJlwn43tI9OYm+xnCgS6dgQGAkNECdNNQWHzvWCDGRQVbBlr8hwEhtQnc3TxY9wek/jwcyh7J4YB7AwJEMD6/wULFqiZ+lgJgCWAmMWPNftYBogNe7DePzAMEIABbOZTteo7ao8AbPX73XffydixY9X4P8AC74thgLNnz6rJfxj/D3VffD31vver7XQgYHqdwEBgYGJygQamTJmsPmmii9okqREYUl80YDsU9S5dOicMFeDTOvbyx/a+mBjYvn17adCggZrsh70AsAcAhgCw9C8wRBD4jveqU3UtMosAACAASURBVLu2dO3aVYYP/0HtKfDLL8vUpEPMOwAEYh4AejPC6UUy0QHbpF4HfrKdKRDo2hEYXFAs/CRcPuvrCQ671/Xt21dNVCMwvG6fcDSDuQL4dI5d9rCtLjYKwpLBTZs2qTH9Hj16qJn++OSPQo9JghgOQPHHd0wKxBbAGBooWLCgVKxYUW3+A/9MnDhRfv75ZwUW6A24fPmSmgxoOowUznOwbWQ6oP2S2k8HAqbXCQwEBqNPtAzApAEYLXugqGFfeXRFYxc7AkPKdsYndMwVwI58586dlT/+OKQKOOYK9O7dSxV3bPaD/QKKFy+WMDyQ+CAfzB8AIKANehmwQgBzA/7b/W+PnDx5Qq1KwN8hDKTsj2jFAd/HWTubAoGuHYGBwEBgiJMGUPywPA6wgMNmCAz/qgKNHgKAFPbmx25+mDMwYsRw6dixo3zwwftq698SJYpL3rx51c6AmC8AIMCKAcAA9gxo3qyZGiKAfZcv/0VNQsTeBFgqiPdDbwLmDbBQOVuoaF932FcHAqbXCQxxKhYMJHcEUrz8gE+uu3fvlhUrlqtxdD8AQ2ASIXYYxCRPPDO69n/55Re1qRAmEGJnQBwChDkDibcRxs94DUf8YoUDlg9iHwKcC7Bv314FXTdv3kzYLyCljYJwngDg7Pjx4wQG5j9faMAUCHTtCAwMGF8ETLzAINTfxSS4Nm3aqLH1Xbt2KnDAPvr49Iseh1C/F3jdrZMeAQXY1x/FGxMJAQTr16+TyZMnyVdfdVXHBpctW0btHYDdBTE0gO/YURAbKWHJYe3ataVz5y/UBEScGYAlhBh+wBLCaKwewKZNmKMAYAnYk9/9DfBe978OBEyvExgIDEyacdAA1uvj2FycyocvdLVjIh426/npp1lan8QbGLABESYU4lhfbDyEyYBDhwyRli1bql4CnBqIHoEAFGDnQSw5xNLCRo0ayueffy6jR48WbF2Mw4sAF9jIKLCSwMkEjh0OMb8B0Obk3+F7E0LcogFTINC1IzDEoVi4RUS8j/glNHwSR9d84Atj6yiqWDGR0t4AAZ/FAhhwj5j4h7kEW7dulXnz5qnTAps2baKKPw4Zwj1jwyHMI8B3gMJ7770n2K64b98+snDhQjUxEZ/mMS8BqxcAG3jvwLPE8juGKnB6InZ5jEZvRSzvnX8rfvFqu+11IGB6ncBAYIhL4rY9AKN9//GawxBYdYADizB8sGrVKsGeEBg+qFSxYpI9BzCPIEeOHIIJhzVr1pRPPmktffr0kTlz5qjTBgECKc0diLbNUvN+6MHA0Ad6dOIFLam5b/4OYSESDZgCga4dgYHAQGBwgQbwyXv27NlqnN4kMYTbw4BCfuvvv9UphtigaNasmWq3wZYtW0i5cmVV7wCGRDCnAPsT4DVsTdy2bRt1KuHs2T/Jnt271dbHATAwuU+3tbl584aav9CpU0fq3gW6d5s+vHo/OhAwvU5gYNAwcVqogVDAgE/N+BSNJYQrVqyQiRMnyDfffCN169aVUiVLqmWIWIIIMMDEvxo1akibNp9Jv359ZdasWeqkQmyBjFUMNoNBqMR/+vRfasdGHPUcqg1f56d5r2nAFAh07QgMFhYLr4mZzxM8QeNIY+wjMGbMmNe+cBgRVhZgWADXR40aKd26dZPatT+Q/PnzqR0LAz0GmFuA5YqdOnWSH0eMkOXLl8vJkyfVpD+M4/tpu2KsuMDQyty5cwkMzH2+0YAOBEyvExgYNL4JGtvABLs+4kRFHF2M8wxwlgF6BpJ/YcIhiiB6DDCUADgIfGGfAk7uewVkWJUBe2FDKNv0wPt95UfaIjxbmAKBrh2BgcDAxOkSDWA4AZ/2sZpgz57dap4BtjvG8svSpUupZZcodjjvAHsZYMli8+bNZM6c2WppI3okmEhTTqSTJk1SwIVlnLRVyraifbxjHx0ImF4nMLikWDA4vROcpr4EIGCewKHffxfsPvjll19K3bp11JJFDCMEehKqV39XzUOYMmWKrF+/Xp03gYOPABHz589j4Qsjhrt3766WggLKTP3Edv6LTa/53BQIdO0IDGEkG6+JiM8Tu0SIVQrYXwGbBWEOQY8e3dV8A+xlgHMNMN8A30uVLCHdun2tVkxgvB2HLAEqki8BDDXpkT4N7VPYEJtGFSiQX+1GSVuFthVt4y3b6EDA9DqBgcDAT1oOaABFHsX+yOHDaltkLE+sXLmSWr4IMMDSRfQQ4HX0HGzZslmuXr2i9jEwSdYEhvATOrasxtyOSpUqvQZgJjZnm/BtTpu5w2amQKBrR2BwoFgwSNwRJLH0A3oPsFUytkkePHiwfPhhI7XrISYjYq5B4cKFpVGjRjJ8+PCEYQXsvZDaeyQwhK8xnEWBuR+tWrVKtd1T6y/+Xvj+os2iZzMdCJheJzAQGJg8U6EBdG9jr4KdO3eqDZBq1qwhBQu+lXDCIs5NwOmLCxbMlz/++EOdk2Cy5bNpkiQwhJ9ML126pHyEPSdM7cx24duZNnOfzUyBQNeOwJCKYsGAcF9AOO0TzEG4f/++YFOjadOmqd4CDCtg1QKWO2K74XZt28rixYsF50JEEw6CPRuBIXwN/vnnScmZM4dafRLMpnwtfJvSZnbYTAcCptcJDAQGftoKoYHnz5/J+fPnZd26ddKlS2e1xwGGGHAMdZkypaVFi+Yyb95ctWoBuyvGMnkSGMJP1JhEisml27Zti6mvYqkL/q3wdeEHm5kCga4dgSFEsfCDiPiMrycX9AwcP35MfQrFEkfMqEeRwfcGDerL6NGj1ImSWJaHXod42ZDA8LrvdL7AEdzw5cmTJ+LmN9098nr4fqXN9DbTgYDpdQIDgcH3yROF/6+/TsmECRPUSgb0IGCoAcMMHTp0kPXr1qnNlHAUdfLljfFKVgQGfZJM7puBA7+XXLlyCk4GTX6N/w/fnrSZPTYzBQJdOwIDgcGXyRM9CadPn5Y5s2fLB++/LziQCZsllS9fTr7+6ivZtWunABDcmhQJDOEla0AhDtkqVKig8YmgbvU97ys839Ne/+o4wPg6gYHA4Nqi6ESg37hxQ35ZtkxatfpYcubIobqoq1Z9R/r06S07d+xQG/q4pRchpecnMIRXNAB/GFKCr7EfQ0q25bXwbEt7ud9exkSgaUhgIDD4Inli8uKAAQOkePHiqjehSJHCakfFHTt2qNUPNkBC4sRMYAgvST98+FAqV64sjRt/6PgKlsR+4s/h+Yn2csZeGg4wvkxgIDB4EhgAAPi6efOmjB07VvLlzatAoU6d2mplAwouuqltA4VAQiUwhJdYsWcG5qR07fqltT4P+J7fw/M97cUhCU8WOQo7OokAEHD27FkZMmSw5M+fT4FC8+bNBcvq4rmyIZr+JTCEpxVs041TPidOnMjcwQ9JvtOAcReCpiF7GBg8ngkegAKOeAYoYBkkzmxo2bKFrFy5Qh3gFM2CHe/3IjCEBwzHjh1Tp3+uXbvGM3qPtwb598PTYDztpeEA48sEBgKDJxLoixcvZObMmVKyZElVGJo1aypbt26VJ0+eeOL5kicbAkN4yXrZsmXqTI8DBw54Ug/J9cH/h6cPr9vLmAg0DQkMBAarEyiGGE6ePCn169dTeydUqPC2rF692jNDD6ESGYEhvILw3XffqhNCoZVQNuXr4dmU9rLHXhoOML5MYCAwWJtAsVQOWzO/VSC/5MqZQ4YOHSK3bv1t7fOEk4AJDOEl644dOwhWxmDIKhw7s214dqa93GkvYyLQNCQwEBisTKCAhX79+qlu5urVq3tqQqNJ0iUwmCdmzG2pV6+uvP12eYnkSHETv7CNuV9oq9jZSsMBxpcJDAQG64ABsPDFF53UpkvYvQ9LJ21dHpnapElgME+2mN9SrlxZef/99zw/VJVaPfH3zPVko62MiUDTkMBAYLAKGAALffv0UUslsaber7v2ERjMEzw0gzMkunfvbpXWbSxMvGdzXcbSVhoOML5MYCAwWJNE0YuAEwdx5kP79u08uwLCJJEQGMwT85UrV9TKmSlTJlujdRMNsI25BvxuK2Mi0DQkMBAYrEmiGHooXbqU2uIXG/H4OQkQGMyLBZbX4ljrVatW+Vozfo4Xvz+7hgOMLxMYCAzWJNElS5aooYjly5f7bs5C8oRHYDAHhrFjx6heqT179lij9eT+5v/N/U1bvW4rYyLQNCQwEBisSaLNmzcT7LNw584da+7ZqeRFYHg9KYayda+ePSVbtqzy119/+V43oWzE1831ZKOtNBxgfJnAQGCwIok+f/5ccud+U1q0aC6Y9W5j0EbzngkMZgke815atmihNm0iaJrZLJo65Xu5w+bGRKBpSGAgMFhRfHE8cdasWQTLKP22hDJY0iUwmCVigGatWjWlRIni1A1znRW5Lli8R/qahgOMLxMYGERWBFEAGNq2bWPF/UYa4LrfJzCYAcOjR48ULLRu3Yq6Ya7zrQaMiUDTkMDAILIiiG7fvq12dfzyyy5W3K+u4Ed6ncBgBgxYUpknT24ZMmQIdcNc51sNaDjA+DKBgUFkRRARGJIWSAJDUnuEAjACg5mdQtmPr3vDfsZEoGlIYCAwWAEMFy9elAwZ0svAgQOtuF+nEy2BwSyRb92yRekGx1s77RO+v5lPaKfY20nDAcaXCQwEBisS6aFDv6vd+qZNm2bF/TqdFAkMZkl36tSpSje///47dcNc51sNGBOBpiGBgUFkRRCtX79e7db366+/WnG/BAazgu6knbCapm/f/84duXbtGnXDXOdbDWg4wPgygYFBZEUQ4RwALKs8evSoFffrZCHEe7OHQQ8k2K/j008/UXswcO8Ovb2c1izfP34+MCYCTUMCA4HBigLcsWMHKVAgP3d5/J9eCQz65PvkyROpXfsDqVunjhUaZ0HV+5Q2Sp2NNBxgfJnAQGBwfTJF4i9evJhUq1ZVnj175vr7jUVSIzDoEydW1pQtW0a++eYbaoZ5ztcaMCYCTUMCAwPJ9YGEMwBy5copnTt3dv29xgIW8DcIDHpgOHv2rOTMmUMmT+ax1rHSJf+OXpfxsJGGA4wvExgIDK4vwqtWreSEx2Q6JTDoE/POnTvVComVK1e4XuPxKCL8m3oNecVGxkSgaUhgSJaIvSIQrzzHy5cvBdv64uCphw8fMPH/T68EhpSTPVZIzJs3Vx2HfuDAfuqGec7XGtBwgPFlAgMDydWBhA2bChUqqE4c5KFTr4okgeGVLYLBMbTy/fffSZYsmeXcuXOu1niw++drKfuX9gnPPsZEoGlIYCAwuDaZIunPnv2T2qlv06aNrr3PeCQvAkPKCRM9Ux06fC6lSpWUO7dvUzvMc77WgIYDjC8TGBhIrg2kx48eSZUqlaVChbfl7t27rr1PAkPKxTse9sGx1nXr1pGGDRsIfo7HPfBvuk8XfvWJMRFoGhIYCAyuTKboXViy5Gc12XHmzJnC4YikyZc9DEntkbwQYClukSKFuaSS+c2V+S25Xp3+v4YDjC8TGBhQrgyoW7duSYkSxaV8+fLy999/u/IenQ7ylN6fwJAyMFy9elWtkBg/fjy1wxznew0YE4GmIYGBweS6YMI2vr1791Iz3JcsWcLehSAaJTCkDAxbt25VvVM8pTJlO6UEpbzmHdtpOMD4MoEhSDJmoMQvUDD0sHHjBsmePZs0bdpEnj596jqgcYM+CAwpa3Ty5EmSMWMGwV4MbvAX7yFlf9E+ztrHmAg0DQkMBAbXJFTAAnbnK126lBp/RrcyE0nwREJgCG6XgF569+4t+fLllfPnuaQyYBN+T1kzXraPhgOMLxMYCAyuKco3blyXDz54X23nu2nTJg5FpKBNAkPKyR+9U+XKlZXHjx+7Rt9eLkh8tpT1GG/7GBOBpiGBIYWkHG8n++nv37hxQ+rXry+ZMmUUrIrgccQpJyACQ8r2KVmyhDRp8hFhgfmNGvj3Xw0GmF8mMDCg4h5Q169fV2vm06ZNIz/++CNhwUCTBIbQwHDnzh01f6Fvnz5x17afoJ/PGlqT8baNORKk3JLAYJCc4+1sL//9EydOSI0a1SVz5kwyduwYJnhDPRIYQifnvXv3qhU206ZNo54M9eTlHMNnYw8DE4HliQBDDljyhs118ufPJwsXLmTPQhg+JTCEBoZZs2apJZWrV69mnghDUyysoTVlu21S7jcwv8oeBgZUzJPqzZs35euvv1a9Cm+/XV52794t2Pvf9qCM5f0TGEIn94EDB6ohiVOn/qSmmN+oAc5hCJ0sYpm0+bfM/YAlk9jTf926dWrZZPr06aRDhw6CHR257bO5HQOaIzCEtlm7dm0lV66c3MODsEBY+J8GzPsQUm7JHgYGleNBBVDYt2+ffPxxS8HExvLly8kvv/zCIYgItEdgCA4M6KmqVq2avPdeLcd1HYA3fg/uC9rFPXZJGQPMrxIYIkjaDIiUAwKgcPDgQXXMcI7s2SR37jflhx+GyZUrV5jMI9QdgSG49rA8t1ChgvL1119RYxFqjPktuMZstIs5EqTcksDAoIp6YsWERuzl37RJE8mWLauaq9CrVy85deoUhx+ipDcCQ/Bkjp4srLiZMIGHTtlY2HjPwXUdqV1SxgDzqwSGKCXwSB1q8+9jDgIg4fbt2zJz5gx5t1o1NUs9Z84c0r37N/LXX38RFKKsMwJD8MS6dOlSNey1fv26qIOwzTHKew+uF7/YxRwJUm5JYIhyIveLAPGcGC/GioctW7YIJpq99VYBNTu9bNmyMmrUKHUuBCc0OpOoCAzB7Tpq1Eh1rPXFCxcIDMxt1MD/NJAyBphfJTAwqMIKKgDA33//LTjroVOnTlKqVEnJkCG95M2TW9p89pl6HSsf/ARO8XhWAkNwYOjatauaw8BTToPbJx5a5d+Mvy/MkSDllgQGAoO2uAMScHLk4kWL1JHT2GgJkJAlS2apX7+e/PzzYrl+/RpXPcRQSwSG15Pws2fP1BbjH33UmFqMoRYJBK9r0W02SRkDzK8SGOIYWJjR/ccff7hq0yLAwZMnT+TSpUuyY8cOGTRokFSt+o6CA0ACZqA3atRQ5syeLVeuXHbVvbstSJ28HwJD0iR9+vRp2blzpxQokF8wwZZDYUnt46QW+d7ut7U5EqTcksAQR2BYvHiRVKxYQW1oFM+ge/TokeBMh7Vr18j333+njpjOly+v2o8fSyGrV39Xvvmmm2zcuEGuXbtGSIijZgI6ITAkTdIdO3aUMmVKq/kLEydOJDC4QKMBrfJ7Uq3Gwx4pY4D5VQJDHAMrHsCAiYo4HXL79u0ya9ZMad+undp5MVeuXAoQsmbNItWqVZUvvugkS5cukQsXLsiDBw+YgOOok2AJhsCQNAkDGLCcMk2aN9Q8mmA242tJbUZ7+Mce5kiQcksCQxwLgZPAgC5ZTPzCsAfWpi9YsEB69uwp5cqWVfMP0qVLq77nyZNbatWqKV991VWWL/9F7t27p8Z/2aXr7mRCYEjqHwADYAHzajA8wWKY1D60h532uH37lvz2228Rf2BLGQPMrxIYLAeGABhcvnxZDh44ICtWrFDHRH/xxRdSuXIlBQVIpBkzZpDChQvJ+++/J0iuY8aMFhwDjOEI9DoQEOxKKASGpP4KAEOFCm8r6GWBTGof2sNOe/z000+SPXs2wYdL5OnU+tEcCVJuSWCwCBiwORK2Vd6xY7s6DnrEiOHSqVNHqVmzphQs+JaCA5zVADjAKZDNmzeT3r17qaEHnAiJ+QeYSU44sDN5JE4WBIakPgwAQ4sWzblCIo45LbFG+XNSjabGHg8fPpT+/fv/b/fSCSHnuwEmzp07J2vXrhX8Dv7Wvn171eR1TGCP1j8CQxyDK9SQBJyPblX0FkycOEENJTRt2kRN6sIQQqZMGdVOiug5yJs3jzRs2EC6d+8u48ePky1bNivhYGgBZzkQDiIP2tQEutO/Q2BI6tcAMPTu3TuiT2JO+43vn9RvtIfeHviQhx5hzNEZNmyo+tCX3G7I9b///pvUq1dPTVw/fPiw1K79gYwbN059WQsM9+/flzt37vj+C2NTkydPlqJFi8jkyZNk8OBBardETDjEOCxgIPEXuqXefbeatG3bRgYNGigLFsyXo0eP+t6OftXSxYsXpWTJEjJt2jRq4M4dadOmjYqXsWPH0B7Mr57TADbDQ48ylrYPGDBALX1PDg34/6pVK9VQNPbGQY90oI21wNCiRQu1hTC2EfbiFzY1wpJE3ReWKwaAIH36dGqFQqjvEAnez4v24jOlLg6w3wAmruK8DtqwQAJkv/lmLtrDo7nV7zpHbUGNQN1Az3MABhJ/x/ADVrr9+eefSa5bCww47njz5s3WfmFL5I0bN8r69etl2bJl8uOPP0qHDh3UZkZYB549W7YEEAgAAb4juePkRvQgfPJJa/n222+lTZvPpFixorJhwwZr7WGzL22+919//VVtotW3b19qZ/NmteMohudgF5v9ynu3tzY46TvUneHDf1DDEvXq1ZXz588nAYIANOBE4JIliqsl8YHX8D1a/ziHIYU5DBj/x5bI6ObB2FHXrl9Ks6ZNpXr16lK8WDH16Q7zCRKDAcaZatasoeYdYAOZX375Re1AhxMb0a2EXRQDs11DzWFI7Gj+rB/j86ONOIchqS6qVKmsNhgLxJYfNcFnTqoJL9ljz57dahdTzGXDUvlgz4Zl9Avmz5cfR4yQbt26qdqFYQl8Reufr4ABAIDJIVhKiLFvrBo4e/aMOm1x0sSJ0qNHd2ndupVaeohDlQJzCTAkgG4eDCNgNUL58uXk008/Udsmz549W81M/f3339V5Cnj/YM4M9hqBwbsBHszf0XyNwPBKOzgMLUeO7Cp2Ocn3lV2iqTe+V3zsCgBevXq15MqVU+kbWk/uiwf370u3bl/L9OnT1aZ827Ztk3LlyqlecMQDvqL1z3PAAJrCshIcu3z27Fn57beDsnLlSjXm07nzF+rwpFq1aqkVB5gXABjAcAEmFaJLs2zZMmqZIoBgyJAhMm/ePDVkcODAfgUXd+/eTeghSO64cP9PYIhPEIbrp1i0R1CjO3HKlMkyc+YMtXw2pb9LYHilnf3796uVQ5gQnJLNeO2VzWgLO2yxZs1q9UEVtQsfcoP5DT0La9asUavjkEceP34s69atEywwCLT3NTDAQDAe1p3u3rVLbWqB8Z0OHT5XvQOVKlVUqw/y5MmT0EuAoQMsScTMcpyw2LFjBxk1aqQ6aRHdPcePH1fjQrdv31Y7JMLwAWM79Z3AYEfQOuX/xO+LszwwtwWQe/LkSWnWrGnIrkf8HoHhlXZ+/vlnBf0Ehlc2Sawt/myvXXA4IZZUolc8Ej96EhhQpNGlj3F+FG7M+ERXP7YsHjZsmHz+eXu1jXHRIkUUdaF3ALNG8R1zBwAERYoUkRo1qkvnzp1l+PDhsnz5cvnjj0PqvdA7gDWt6IWIBRDoHExgsDeQdb4N5zq02LZtW7VsCj+jGxKHfUG/oXRKYHilHRyYhnlEBIZXNglHf2zrXrsh/kPlgHD8Zi0wAAhQuLGOHJ+qcM7BL78sk5EjR0r79u3Up//SpUup4YHAYTKYS4DhguLFiwl6Dz744H35sksXGT16tCxdulS9Bz6ZxbJ3IBxnhWpLYHBvoIbymROvY/gMc2QwVon3R4LAhNn33ntPdS8G+5sEhv+0A/jHBjUEBsZSsDjha//pwlpgwCmIb5cvr2Z8Yg05eggwhwATCjHRECsQPvqoseohmDBhvBqLOXTod9VVi8SK+QlemQkN2MGKinAmSjIAvJcYDx48oIABS44D/l28eLFgTwFMzA28lvg7geE/HeDkVWx+hl5GnJ+S2Eb82XuxQp+mzqfWAgOWJmJpCL5jDwPsQYBtkBH4mJeAsRoU0Gh0w7hdXHhWQJAfntXtvojn/W3ZskUt0UWPW+A+AJP41IzjxQOvJf5OYPgvcWLIEvubDBz4fchJYYntxp9TV3BoN7vtZi0wuK044n5C9VjgWqw//eNe0M3KALU7QMPxHw4Gw7LAY8eOJfgdq3PwGvYBCfZeBIb/9IHzVnDg2p49e16zE2PJHTGky7HMd877yVpgCJb84vUaznOYN2+uYJZ18ntAj8eMGdMFx4suWrhQHv3vBLDk7aL1f4AJThrDkjrs7TB37lx+YkphU61o2d0N74NhB8xhwMFhuB8k2HFjx3IOg4H/ca4KhjaTLzkDLCxcuFDFkRt87Nd7wEq20aNHqRN2E9sAGgcg4ywU5NgNG9bH/MNZ4vvx+s8EBoNkkpIIMByANa6FCxeSgQMHJgEGzJNo1epj2bp1q1pVAcFjgiVEntJ7pvYa3hcT3rBxFNbQAh6mTp0i33//fcjej9T+Lf6e8zQfro3h/yZNPhIcnITfRbHDRixDhw4JqTn2MPyrYrNhgwZqu3XETWK7Hz9+TEqUKK5O7kv8On+Onf6xI+GsWTNVD9CSJUk/lF28eEEaNKivlsYj32KFC7bcdyrH+t3vBIYIgQHCxBdOuUsODL/+ukrKlyuXcK44xpZx5kOwXbaiIUTcx4AB/dU4bOD9tm/frgIKy0ADr/F77JJdrG39+2+/KWjAZivYJ77xh43k6tUrIRMogeFftdU6dl5t165tkmE8xAzO2MAZL1hyGWtf8u/9F6fIa4BfTGpPDAx47bvvvhNMgMfPaLdixXKpUYMTwJ3SDoEhQmAIOAZLORMDA8bTsKsWehggZrS7efOGFClSWJYtXepI8kHALFiwQJ2yh14NfFrq2bOHYKZ84D753buwEPAt5jJMmjRJJkyYEHKyY6AtgeFfdbx7xowZZNy4cQlxgliCDbHhDfbTJzDEP26SAwN6dwEHWDoc0DNWwqW0KijQjt9T508Cg0PAgF0kP/ywkeoSDogTn/rKlCkj/fv3SxB44Fq0vqNbrmvXrmp5KT4ZYW+KFy/Mz6WI1n3wfVIXkLG2G4HhX3WwG5ZTYi+XgP0BXQsXEJjkPQAAIABJREFULFA9DgQGd2g5OTBgv5xChQrKokWLEvyG45gBDMEmrwZ8y++p9yeBwUFgqFOnthoiCAg0AAz41B94zYnvO3bskJYtW6h98Xv16hnxdqBO3CPfM/VBG03bERj+le7du6uev8BKEpwGO2LECHnw4MH/5oGwhyGamkvteyUHBgztYt8d7MIbeM8AMAQm/gZe5/fo5BsCQwrA0K9fX2nT5rOgXyNH/pggUogx+ZAExj+bNGkiPXr0SBg/BhFjl0lsT50aAWPiT6j7wX4UmOR45MgRNaaHZIcjsXE6GSa9BYZFUvN3+TvRCTY32tHvwIC4KFOmtDo7JjDhEXvuD//hB1m2bJksW7ZUGjZsIC1aNFfj4270oV/uKTkwYEULNtv6+edXQ67IfzgA8PDhw6nKsX6xZWqfk8CQAjCEY9TkwIBE1KdPb2nevFnCRCqcafFWgQKye/cuR8SMv/n111/L5MmT1Pvj/1haWaxoUbXddTjPw7behYTEvvU7MKBXAVvHf/nllwlgjw8DWG2CL+wWi3lHWCkBaEhsO/4c2xhJDgw4KwgrJMaNG5vgFwwlFSiQXx0bQP9E3z8EBoeAAWLdu3ev+vQSWNu9b+9eeadKFceGCAAIX37ZRbAVdiBYsMPfu+9We219eeA6v0c/qGyyqd+BYdu2bWq5HvZaCOY3xBTnMLgjRpIDAyamYtl4q49byov/7eqLCZBYWgm/BfMnX4vMlwSGCIEBokVXJo66xmRDTHbEaxAmhggG9O8vM2bMUFs3Y9UEtu8NXI+2ePG++/fvV9SNJZw4nAvwgA1NGECRBUq0feWW9/M7MGB4EFtC//nnyaAFhsAQ/7iBD7ABHnbinDp1qsqrgRx67949tRJt165d6mgAzN3CAYJuiS+v3QeBIUJgwFyFo0ePqN0VsWEIxBoQM8QCaNi8aZPMnTtHzpw5neSaE2LC30avwvx589TukxjLIyzEP+k54etovKefgQFxgUPbSpQooSY4BrMn4uno0aMhgSLY7/C16MYb5mPhgxB2sMVycQwjJc6xuI6l6ji116k9bujT/3xKYIgQGCik6CYH2jO29vQzMGAHQYx3Y3Jy4gJEDcZWg7S3PfYmMBAY2H3nYw34GRhwDHiWLJmTTJpj8bKneNFXsfcVgcHHxYIBF/uAc5vN/QwMmDCHcfFQ8xfc5iveD+M13hogMBAY2MPgYw34FRiwdXvTJk3UToGB/RfinYz59wkEbtcAgcHHxcLt4uT9OZ9A/QoMmBxXunQpadasKecvMAfyQ5OhBggMhoZi8XK+eNHGsbexX4EBy46zZs2i1vFTd7HXHW1up80JDAQG0rWPNeBHYMCKiOnTpwtOqDxx4jj172P9E1zCAxcCA4OFCdPHGvAjMKj5C02bqO2euW4/vILBAutvexEYfFwsGPz+Dn7434/AgCPgcQhc8+bN1cZqjAPGATVgpgECA4GBPQw+1oAfgeHkyZOCcwnmzJnDCY8+1j4hwQwSEtuJwMCAITD4WAN+BIYhQwZLjhzZ1VHwiZMhfw6/gNBm/rIZgcHHxYLB7q9gD+ZvvwED5i/UqlVTSpUqyRNcmfv4YSlMDRAYwjRYsKTL11h4bdWA34Dh5s0bkj17NuncuTOHI5j7CAxhaoDAEKbBbC0MvG9CTTAN+A0YFi5cqOYvrFmzhsWCuY8aCFMDBIYwDRYs6fI1FmNbNeAnYMBx1u3atZVs2bLKpUuXWCyY+6iBMDVAYAjTYLYWBt43oSaYBvwEDPfu3ZWyZcvIBx+8L4CHYPbga4wTaiC0BggMBAYmTh9rwE/AsGvXLsmcOZPMmDGdmvex5gkEoYFAZxsCAwOHydPHGvALMGA76G+/HSAZMqSX06dPU/M+1ryuKPJ6aKAgMDBwmDx9rAG/AAN2d3zvvVpStWpV7u7oY70TBkLDgIltCAwMHgKDjzXgF2A4fvy4Op1y9OjRXE7pY72bFEW2CQ0VBAYGD4HBxxrwAzBgOGLEiOGSKVNGOXjwIPXuY70TBkLDgIltCAwMHiZQH2vAF8Dw8qUajqhWrao8ePCAevex3k2KItuEhgoCA4OHCdTHGvADMJw6dUrSp08nw4cPp9Z9rHWCQGgQMLUNgYEBxCTqYw34ARj69u0j2bJmlb1791LrPta6aVFku9BgQWBgADGJ+lgDXgeG58+fS5kypQXDEU+ePKHWfax1gkBoEDC1DYGBAcQk6mMNeB0Y1q5dq4YjJkyYQJ37WOemBZHtUoYKAgODiInUxxrwMjBgdcQXnTqpzZrOnTtHnftY5wSBlEHA1D4EBgYRE6mPNeBlYAAkFCz4lrRv344a97HGTYsh2+mhgsDAQGIy9bEGvAoM6F2YPHmSpEnzhmzZspka97HGCQJ6EDC1EYGBgcRk6mMNeBUYsN9CrZo1pWrVd+TRo0fUuI81bloM2U4PFgQGBhKTqY814FVg2LBhvaRNm0YmT57MraB9rG9CgB4CwrERgYHBRGDwsQa8CAwvX76Uxo0bS9GiReTixYvUt4/1HU4xZFs9XBAYGExMqD7WgBeBYdeuXWruQv/+/di74GNtEwD0ABCujQgMDCgCg4814DVgePbsmTRv1lTy5s0jp0+fprZ9rO1wiyHb6wGDwMCAYlL1sQa8BgybNm2SzJkzSffu3dm74GNds/jri39qbERgYFARGHysAS8Bw8OHD6Vu3TqSJ09uuXHjBnXtY12nphjyd/SQQWBgUDGx+lgDXgEG7LuwcOECtQ306NGjqGkfa5qFX1/4U2sjAgMDi8nVxxrwCjBcuXJFihUrKpUrV5K7d+9S0z7WdGqLIX9PDxoEBgYWk6uPNeAFYMCJlG3afCZZs2aRdevWcu6Cj/XMoq8v+pHYiMDA4CIw+FgDtgMDhiJmzJihNmnq3v0bATxEkhD5u84WHNrXbvsSGHxcLBi8dgdvNPxnMzBggybs6Jg9e3apXv1duXPnDmGB+YwacFADBAYHjRuNhM73YFF3UgO2AgN6FrZs3ixvvVVASpYsIX/99RcLBXMZNeCwBggMDhvYyWTP9yZMRKoBG4EBPQtz5syRN9/MJYULF5bff/+d8xaYxwgLMdAAgSEGRo40qfP3CQZOacCtwAAowHyE58+eydOnT9UXTqA8dOiQNGvWVNKlSyvly5WVEyeOs1Awh1EDMdIAgSFGhnYq4fN9CRORaCDewPDixXP5+++/1ZDCzp07ZPr06dKjRw/57LNPpWnTJtK48YdSu3Zt+eCD96VcuXKSMWMGte1zv3795ObNmywUzF/UQAw1QGCIobEjSez8XYKBExqIBzA8efJEjh07JpMmTZKGDRtI6dKl1PBCmjRvJABB4cKFpFy5svLOO1Xk/fffkzp16kjz5s3U75w69aegB8IJe/A9GWfUQGgNEBgIDEy8PtZArIABkxRv3bolkydPkkqVKqk9EwAIBQrkVyAwbtxY2blzp5w9e1auX7+mVjxgq+dHjx4JAAPDEi9evKBWfaxVFvLQhTxWtiEwMACZhH2sAaeBAaBw8eJFGTJ4sOTLl1cyZsggZcuWkd69e8lvvx1UQBCrZMe/E/+CQx/Y7QMCg4+LBYPX7uCNhv+cBIa7d+/IxIkTBcML2IXxo48ay9o1a+TevXtc1cC8ww8qFmqAwGCh06JRKPgehAVowClgOHLksNSu/YE6DAoTFrds2ayGFqg76o4asFcDBAYCA0nfxxqINjBgCGLFihWSM2cOyZUrp4wfP14tj8TrLBT2Fgr6jr6DBqL17/+i9Uam70MBU8DUQOQaiCYwYHLimDGjBZMZq1SpLAcOHODQg49hlPEZeXy6zYam9VnXjsDAxMBPkBZqIFrA8PjxYxk48Hs1BNGkSRM10ZG9Ct4rGG4rYLyf2GpMBwKm1wkMFhYLBltsg82N9o4GMGC547BhwyRDhvTStm0bwY6MhAVqy4165z1FpktTINC1IzAQGNjDYKEGIgUGgMGCBfMVLLRo0VzQ08CkHFlSpv1oP7dqQAcCptcJDBYWC7eKkvcVu4QZCTAAFnbs2CFZs2RWuzHeuHGDsMA8QA14WAOmQKBrR2DwsEhYwGNXwGNt60iA4dKlS1KmTGkpWPAt+fPPkywUzAHUgMc1oAMB0+sEBo8LJdaFjH8vNpCSWmDAVs3t2rVVZz9s3LiRcxYY/4QFH2jAFAh07QgMPhALi3hsings7ZwaYMBQxKpVqyRTpozSt29fwgJjn7DgEw3oQMD0OoHBJ4KJZTHj33IeUFIDDHfv3lVDEW+/XV44b8F5HzEOaGO3aMAUCHTtCAwEBn7KsFAD4QIDehf69+8nmTNnUr0M+L9bkhnvg4WVGnBWAzoQML1OYLCwWDC4nA0uG+wbLjCcOX1abfvcqFFDdeS0Dc/Ie6TOqYHoaMAUCHTtCAwEBn7StFAD4QADNmjq/EUntefC/v376W8L/c3CGZ3C6Vc76kDA9DqBgcmDBcRCDYQDDEePHpW8efNIhw6fy8uXL+lvC/3t10LH544OKJkCga4dgYHJgwXEQg2YAgN6F3r06K56F44dO0ZfW+hrFs3oFE0/21EHAqbXCQxMICwiFmrAFBguX76sNmhq0+YzATz4OWny2Vl4/aoBUyDQtSMwWFgs/Cp6PverhG8KDGPHjlG9C5s3byIsMNapAZ9qQAcCptcJDD4VEIvvq+Jroy1MgAE9CthzoWbNGnL//n0WC8Y6NeBTDZgCga4dgcGnArKxSPKeX0GOCTCsXr1a0qZNI1OnTuWujoxzwoKPNaADAdPrBAYfi4gF+FUBts0WOmDAxkyffvqp5M79ppw5c4bFgnFODfhYA6ZAoGtHYPCxiGwrkrzfV4CjAwacSAlY+OyzT1koGOPUgM81oAMB0+sEBp8LiUX4VRG2yRY6YJgxfboajlixYgWLBWOcGvC5BkyBQNeOwOBzIdlUJHmvr+AmJWDAcESrVh9Lnty55dGjRywWjHFqwOca0IGA6XUCg8+FxCL8qgjbZIuUgAFzFvLlyyvdu3dnoWB8UwPUgCkPaNsRGCgmJhQLNRAKGNC7MG/ePEmT5g3ZsGEDfWuhb20CV96rHR84tCRg2IDAwITComKhBkIBA86K6Nz5CylcuJDcunWLvrXQtyzCdhRhm/xkyAPaZgQGJhQWFQs1EAoYAAn58+eTjh07citoC/1qUxHivdoDNloSMGxAYGBSITBYqIFQwLB9+zY1HPHTrFncrMlCv7II21OEbfKVIQ9omxEYmFQIDBZqIBQwjBgxXHLmyCE8mZKFx6aCxnt1Vq9aEjBsQGCwsFgwuJwNLhvsGwoYKlWqKO+9V0uePn1KEGRsUwPUgNKAIQ9omxEYKCgmFQs1EAwYzp49q4YjevToweEIC31qA6jyHu38sKIlAcMGBAYmFgKDhRoIBgw///yz2t1x0cKF9KmFPmUxtrMY2+A3Qx7QNiMwMLGwuFiogWDA0KN7d8mZM4dcvXqVPrXQpzYUHt6jnVCjJQHDBgQGJhYWFws1kBwYnj17JmXLlpF3363G4QgL/clCbGchtsVvhjygbUZgYHIhMFiogeTAcPLkSdW70KVLF/rTQn/aUnh4n3aCjZYEDBsQGJhcWGAs1EByYFi1apWkT59OFi5cQH9a6E8WYjsLsS1+M+QBbTMCA5MLC4yFGkgODMOGDVPAcPv2bfrTQn/aUnh4n3aCjZYEDBsQGJhcWGAs1EBiYHj+/LnUq1dPKlaoQF9a6EsWYTuLsE1+M+QBbTMCAxMMi4yFGkgMDFgVUaRIYWnXri19aaEvbSo8vFc74UZLAoYNCAxMMCwyFmogMTD89ttvkjlzJhkzZgx9aaEvWYTtLMI2+c2QB7TNCAxMMCwyFmogMTAsXbpU7fAIcLApifFeWSipgdhoQEsChg0IDBYWCwZZbILMzXYOAMO8eXOlV69e8tZbBeTGjRsEBsYzNUANvKYBQx7QNiMwUFyvicvNhZL39h8sBYBh9uzZUrv2B1KrVi159OgRfcl4pgaogdc0oCUBwwYEBorrNXGxKLu/ByMADJMnT5Lcud+Utm3byosXL+hLxjM1QA28pgFDHtA2IzBQXK+Ji8BgDzB8++0Atf/CuHFjuSU0Y5mxTA0E1YCWBAwbEBgosKACIzS4GxoCPQzNmzeXdOnSytatW+lHxjI1QA0E1YAhD2ibERgosKACIzDYAQwFCuSXfPnyyvnz5+lHxjI1QA0E1YCWBAwbEBgosKACIzDYAQxp0rwhFSq8Lffv36cfGcvUADUQVAOGPKBtRmCgwIIKjMBgDzD07duHPmQcUwPUQEgNaEnAsAGBgSILKTJCg3uhITCHAT0Mc+fOpQ8Zx9QANRBSA4Y8oG1GYKDIQoqMwOB+YMCR1n/8cYg+ZBxTA9RASA1oScCwAYGBIgspMgKD+4EhZ84cwiOt3esnxhB94wYNGPKAthmBgcBAYLBQA4EhidKlS8mzZ8/oQwt96IZCwnvwB9BoScCwAYGBiYbFxkIN3Lp1S4oXLyYfftiI/rPQfyzU/ijUbvGzIQ9omxEYmGxYcCzUwOnTpyVPntzSpw9XSLglKfM+CAFu1YCWBAwbEBgsLBZuFSXvK3YJ88CBA5I5cyaZPfsnAh9jmBqgBlLUgCEPaJsRGCi0FIVGCIgdBIRj659//lmwpHL+/Hn0H2OYGqAGUtSAlgQMGxAYKLQUhRZOEWNbM7jA/IM5s2fLxIkTZN68uXLnzp2wfdCzZ08CA2M3bN0wRs1i1Gt2MuQBbTMCA5MOk04MNfDgwQNp2LCBnDhxQl6+fCl79uxR/w9npcPz58/V77CHwZ/J32vFjM/jvI61JGDYgMAQw2LBwHA+MNxu4/3796uzH1D0ca84AyJv3jxy5cplY3ADdFSuXJk9DIxdY824PS54f87mRkMe0DYjMDDpMOnEUANnzpxWp0v+9NMsefHihezevVvq168nT58+NfbD9evXpXDhQgSGGPqNBc3Zgkb7OmtfLQkYNiAwMOkYFyoGdeRBjWGIMWNGS5YsmaVbt27Sr29fAQAEs+3x48fl/fffk2rVqib5qlSpomBLaAxJFCtWNMm15G23b98e9L2D/T2+Frl/aUPa0I0aMOQBbTMCA4GBBSXGGkBvQqtWH0uOHNmlQ4cOISc9Xrp0SYYMGSzfffdtkq9PP/lEwQKA4aOPGie5lrwtoMONCYz3xMJKDcROA1oSMGxAYIhxsWCQxC5IYm3ry5cvy8iRP8qPP4547QuvY6IjYOHbb7+VtWvXyvr161QPQYsWzcMakpgyZUoCMHBZpXf1FGv98u95V0uGPKBtRmAgMPATaJQ0gJUON27cCPn1+PFj2bFju9SsWUPNX/jnn3/UHIZs2bLKxg0bjP3w5ZddBL+DYQ0Cg3eTPAs4fRstDWhJwLABgSFKxSJajuX7eDtJLF68SOrUqS2ABfga37HMcuHCBUbAgPYYhsDchQIF8hMYGL9GumFe8XZe0fnXkAe0zQgMTDhMODHUwLVr16Ra1aqyb98+efLkiZw5c0YdIGW6eRNWVpQpU1ow8bFEieIEhhj6TpeUed3fRdnN/teSgGEDAgMTDoEhxho4ffovmT59mmB75yVLfpYLF84b+wDDGnnz5JFPP/1EgQOHJFik3FyoeG/u0KchD2ibERhiXCwYQO4IIFv9cPXqVUmbNo2MGTOGwMDYNQZNW/XO+45OvtSSgGEDAgOTDpOORRrYunWr2oNhxYrlBAaL/MbCF53CRzumzo6GPKBtRmBg0iEwWKQBbPqEFRI7duwgMFjkNxa61BU62i06dtOSgGEDAgOTDoHBIg1gSWXu3G/KkSNHCAwW+Y2FLzqFj3ZMnR0NeUDbjMDApENgsEQDWFLZoEF9dY7EpUsXCQyW+I1FLnVFjnaLnt20JGDYgMDApENgsEQD2BiqQoUK8vbb5eXu3TsEBkv8xsIXvcJHW6bOloY8oG1GYGDSITBYogEchV2w4FvSvn07uX//HoHBEr+xyKWuyNFu0bOblgQMGxAYmHQIDJZo4Pz585I9ezYZPnw4gcESn7HoRa/o0Zapt6UhD2ibERiYeAgMlmhg7949kjFjBlm2bCmBwRKfscilvsjRdtGznZYEDBsQGJh4CAyWaGDOnDnqlMrdu3cRGCzxGYte9IoebZl6WxrygLYZgYGJh8BgiQaGDh0q6dKlVedPcA5D6pMnCw9t5zcNaEnAsAGBwZJi4TeB83mTJnUsqWzfvr1kzpxJcFAVgSGpfagX2oMaCK0BQx7QNiMwEBjYw2CBBrCksmHDhlKoUEF1JDaBIXRyZOGgbaiBpBrQkoBhAwKDBcWC4k8qfj/a4+HDh/LOO1Xko48aK8AjMFATfowDPnPqdG/IA9pmBAYCA3sYLNDA9evXpVixovLttwMIDBb4i4UtdYWNdnPGbloSMGxAYGDyITBYoIFTp/6UXLlyyuyffiIwWOAvFj5nCh/tmjq7GvKAthmBgcmHwGCBBnbs2K6WVK5fv47AYIG/WNhSV9hoN2fspiUBwwYEBiYfAoMFGliwYIFaUvnHH38QGCzwFwufM4WPdk2dXQ15QNuMwMDkQ2BwuQawpPL777+XLFkyC7aHRtLkpMfUJU4WHNrNjxrQkoBhAwKDy4uFH8XNZ06a1F+8eCGdOnWSvHnzyJ07twkMjFlCPjUQlgYMeUDbjMBA4YUlPBbzpMU8FvbAHgxNmzaRd9+tJo8fPyYwMGYZs9RAWBrQkoBhAwIDhReW8GJRIPk3kkLJo0ePpGrVd6RlyxaC3gbYh0MSSW1EzdAe1EBoDRjygLYZgYHAQGBwuQbu3bsnRQoXln79+iX4isAQOjmycNA21EBSDWhJwLABgcHlxYLCTyp8P9rj6tWraoXElCmTCQyM1wQN+DEW+Mypy4eGPKBtRmBgAmICcrkGDhzYL2nTppGVK1cm+Io9DKlLnCw4tJsfNaAlAcMGBAaXFws/ipvPnDSpL1gwXzJlyii7du0iMDBeEzTAOEkaJ7RHaHsY8oC2GYGBCYgJyGENPH/+XObOnSOrV69+zdY4qnrkyJEyY8Z0ARhgRUTyxDd06FDJnj2bHDt2LOEaexhCJ8fk9uP/aSu/a0BLAoYNCAwOFwu/C9Xvz48VDsuWLZUCBfLLuHHjEgo+7IITKD/95BPZvn2bOrJ62LChMmXKFPVzYrt17dpVihUtKtevXUv4fQIDi2BijfBn6iElDRjygLYZgYHAkFCEUhIcr0WWkBo2bPAaMOBcCBxZ/fDhA+WDffv2SfHixQW9Dont3bhxY6lYsYIAPgKvExgi80fAjvxOO/pBA1oSMGxAYCAwJBQhPwROvJ6xceMPkwDDy5cv5fPPPxe8jiEL3BdWQxQs+JYsXrw4iU8qVaooH37YKMlrBAYWunhpmX/XPu0Z8oC2GYGBwJCkEDEZOJMMkgPDkydPpFatmvLpp58I4AF2f/DggZQoUVwGDRqY4BOcI5EtW1bp1atXwmtoS2Bwxk/UP+3qRQ1oScCwAYGBwJCkEHkxWJx4JhTyx48fqWECDBUk/3r69GkSuyYHBrTH7o3t27dLaIc5DSVLlpABA/onvHbt2jV1rPXYsWMSXsPzEBhY2JzQNd/Tm7oy5AFtMwIDgSFJIWLCMEsYGEYYPHiQ9OjRPejX/Pnzk9g1OTCgh6F27Q+kTZvPEiY53rp1S4oUKSzDhw9P+N1t27aqTZsWL16U8BqBwcxH1DLtRA38pwEtCRg2IDAQGJIUIgaYWZJFDwN6CdArEOwLQJDYlsmB4Z9/Xso333RTcxMCcxjOnDkjefLklm1btyb87vTp0yRDhvSyceOGhNfwvuxhMPNTYh/wZ9rMrxow5AFtMwIDgSFJIfJrQDn93MmBAX8PqyIwofH+/fvKB1u2bFarJhIPZ6AXI0uWzHL06JEkfiIwsPg5rVm+v3c0piUBwwYEBgJDkkLEJBHdJIGeCCyTrFmzhvTt21dNbMRrsDPAoGfPnrJw4QK5ePGi9OzZQ9atW5cwRIE2X3zRSXLlyik3b95M4icCQ3T9RN3Tnl7WgCEPaJsRGAgMSQqRl4MmHs+G46hPnTola9askU2bNsmFCxeSAAGGNTZs2KCg4eTJkwkrJgL32rRpEylUqOBrrxMYWOACGuF3akGnAS0JGDYgMBAYCAwu1kCVKlWkXr16r/mIwMAioSsSvE6NBDRgyAPaZgQGFxeLgLP53Z+B//jxY9W70KVLFwID4/Q1DTAv+DMvpMbvWhIwbEBgYCJiInKpBs6cOa3mL4xItMwykCzYw8BiEdACv1MLOg0Y8oC2GYHBpcVCJwBe936SwJyH9OnTyZw5s1+DOgKD9/3PGKePo6UBLQkYNiAwEBheK0bREinfJ7KEN2/ePLXL46ZNG1/zEYEhMttSm7SfnzRgyAPaZgQGAsNrxchPgeTWZ8XSyzFjxihgwCqL5PdJYGDBS64J/p+aCKUBLQkYNiAwEBheK0ahRMfXY5eQAAw4cCpNmjcEkx+T257AEDtfJLc9/0/b26YBQx7QNiMwEBheK0a2BYMX7xfA0K5dWylevFhQ/xAYWLS8qHs+kzO61pKAYQMCA4EhaEFi4DoTuKZ2xZHX9evXk48+ahzUPwSG+PrH1I9sRz+5QQOGPKBtRmAgMAQtSG4QuZ/v4fbt21KuXFnp2aNHUP8QGFiI/BwffPbw9K8lAcMGBAYCQ9CCxIAMLyCjba+zZ89Kzpw5ZNSokUH9Q2CIr3+i7W++H/3ppAYMeUDbjMBAYAhakJwUL99bnxxxkiUmPAbbgwH2IzDobUid0UbUwH8a0JKAYQMCA4GBwOBCDaxatep/ezBsCuofAgOLIYshNWCqAUMe0DYjMLiwWJiKgO28mTCwQmLq1KkKGI4ePUpgYIwG1QDj35vx74RftSRg2IDAwGTEZOQyDQAYBg8eJNmyZZXLly8H9Q97GFgsnCgsfE92rIpfAAAIYElEQVRv6sqQB7TNCAwuKxYMWG8GbDh+xZLKrl2/lIoVK8j9+/cJDIzRoBoIR1Ns6++8oiUBwwYEBiYjJiOXaeDFixfy4YeNpHHjxoKfgyV79jD4uwAE0wRfoyZCacCQB7TNCAwuKxahHM7X/ZMMnjx5IiVLlpAuXToLhieC+Z7A4B89BPM/X6P/w9GAlgQMGxAYCAxBC1I4YmTb6CavO3fuSLp0aWXgwIEEBsYn45MaiFgDhjygbUZgoBgjFiOBIbrAcOLECbVCYtKkSQQGxifjkxqIWANaEjBsQGCgGCMWI4EhusCwbt06BQzLl/8S0jcckoiuzalh2tPLGjDkAW0zAgOBIWRR8nIAufnZZs2aKRkypJddu3aF9A2BgQXOzRrmvblLn1oSMGxAYCAwhCxKDPr4BP2QIUMkX768cu7c2ZC+ITDExzeMCdrdRg0Y8oC2GYGBwBCyKNkYGF64ZwIDi5IXdMxncI+OtSRg2IDAQGAgMLhMAx06fC5ly5YRLK8MlXTZw+CeZBzKR3ydPnKLBgx5QNuMwOCyYuEWgfE+4pfsatSoLrVrfxASFuAbAkP8/MPYoO1t04CWBAwbEBgIDCkWJtsCw/b7ff78ueTIkV3atGmTol8IDCxatmud9x87DRvygLYZgYHAkGJhYlDHLqhh66tXr6gVEn379knRLwSG2PqFcUB726wBLQkYNiAwEBhSLEw2B4mN975v3z61y+OYMWNS9AuBgQXMRn3znuOjW0Me0DYjMBAYUixMDPDYBjg2a8K20IsWLUrRLwSG2PqFcUB726wBLQkYNiAwEBhSLEw2B4mN9z5t2jQ1JLF///4U/UJgYAGzUd+85/jo1pAHtM0IDASGFAsTAzy2Af7dd99KpkwZ5fbt2yn6hcAQW78wDmhvmzWgJQHDBgQGAkOKhcnmILHx3tu0+UwKFnxL6xMCAwuYjfrmPcdHt4Y8oG1GYCAwaIsTgzw2Qf7PP/9ItWpVpWbNGlqfEBhi4xNqn3b2gga0JGDYgMBAYNAWJy8EjA3P8PDhQylZsoS0aNFc6xMCAwuZDZrmPbpDp4Y8oG1GYCAwaIsTgz42QX/jxg0pXLiQdOzYUesTAkNsfELt085e0ICWBAwbEBgIDNri5IWAseEZLlw4r06p7NWrp9YnBAYWMhs0zXt0h04NeUDbjMBAYNAWJwZ9bIL+yJEjkj17Nhk8eLDWJwSG2PiE2qedvaABLQkYNiAwEBi0xckLAWPDM+zZs0cyZ84k48aN1fqEwMBCZoOmeY/u0KkhD2ibERgIDNrixKCPTdBv3rxZMmbMIAsWzNf6hMAQG59Q+7SzFzSgJQHDBgQGAoO2OHkhYGx4hl9/XSXp06eTbdu2an1CYGAhs0HTvEd36NSQB7TNCAwEBm1xYtDHJugXLlwgadOmkWPHjml9QmCIjU+ofdrZCxrQkoBhAwIDgUFbnLwQMDY8w7RpUyVNmjfk6tWrWp8QGFjIbNA079EdOjXkAW0zAgOBQVucGPSxCfpRo0aqIYknT55ofUJgiI1PqH3a2Qsa0JKAYQMCA4FBW5y8EDA2PEO/fn0lf/588vTpU61PCAwsZDZomvfoDp0a8oC2GYGBwKAtTgz62AT9F190krJly8izZ8+0PiEwxMYn1D7t7AUNaEnAsAGBgcCgLU5eCBi3PwMOnmrS5COpUaO6PH/+XOsTAgMLmds1zftzj0YNeUDbjMBAYNAWJwa+84GPYYjatT+Qhg0byIsXL7Q+ITA47xPqnjb2iga0JGDYgMBAYNAWJ68EjZuf48GDB1KlShVp3bq1vHz5UusTAgOLmZv1zHtzlz4NeUDbjMBAYNAWJwa/88F/69YtKV68mHTp0pnAwJhkTFIDUdWAlgQMGxAYKMyoCpNwkTq4uHLliuTKmVN69uwpmM+gsyN7GFJnZ51deZ129aIGDHlA24zAQGDQFicvBpDbnunMmTOSLVtWGTjwewIDY5IxSQ1EVQNaEjBsQGCgMKMqTLcVYlvu59ixo+qkygkTJhj5gz0M/CRsi7Z5n/HXqiEPaJsRGAgMRgWKQe9s0B84sF8yZEgvixcvMvIHgcFZf1DvtK+XNKAlAcMGBAYCg1GB8lLwuPFZtm7dqg6e2rRpo5E/CAwsaG7UMe/Jnbo05AFtMwIDgcGoQDEROJsIVq1aqQ6eOnjwoJE/CAzO+oN6p329pAEtCRg2IDAQGIwKlJeCx43PMm/ePMmYMYOcOnXKyB8EBhY0N+qY9+ROXRrygLYZgYHAYFSgmAicTQSTJ0+W3LnflAsXLhj5g8DgrD+od9rXSxrQkoBhAwIDgcGoQHkpeNz4LEOHDpWiRYvI9evXjfxBYGBBc6OOeU/u1KUhD2ibERgIDEYFionA2UTQrVs3KVu2rNy+fdvIHwQGZ/1BvdO+XtKAlgQMGxAYCAxGBcpLwePGZ/nss0+latV35OHDh0b+IDCwoLlRx7wnd+rSkAe0zQgMBAajAsVE4GwiqFu3juDryZMnRv4gMDjrD+qd9vWSBrQkYNiAwEBgMCpQXgoeNz5LuXJlpWnTpvLs2TMjfxAYWNDcqGPekzt1acgD2mYEBgKDUYFiInAuEeCwqZw5cwiGJV68eGHkDwKDc/6g1mlbr2lASwKGDQgMBAajAuW1AHLT8zx48ECyZMksX3zRyejgKdw7gYFFzU0a5r24W4+GPKBtRmAgMBAY4qyBmzdvSqZMGaV//37GviAwuDtBs4DSP27SgJYEDBsQGOJcLNwkKt5LfJLcuXPnJH36dDJq1EgCA+PRWAOM1/jEq412N+QBbbOYA4P2jtiAFqAFaAFagBagBVxnAQKD61zCG6IFaAFagBagBdxngf8HT/K36avIjpAAAAAASUVORK5CYII="> <em><strong>A1A1A1M1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for correct ‘shape’ of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> (allow non-stationary point of inflexion)<br>Award <em><strong>A1</strong></em> for each correct intercept of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math><br>Award <em><strong>M1</strong></em> for attempt to reflect their graph in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math>, <em><strong>A1</strong></em> for completely correct <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> including intercepts</p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the 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>Hence or otherwise, show that the series is convergent.</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 <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.iii.</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"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[8]</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><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;"><strong>EITHER</strong></p>
<p style="text-align:left;">since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>p</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo><</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;">since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>p</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo></math> the geometric series converges. <em><strong>AG</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.<br>Award <em><strong>R0</strong> </em>if both values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> not considered.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><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="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.iii.</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"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced></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><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced><mo>=</mo><mo>-</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>3</mn></msup><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><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;">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;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced><mo>=</mo><mo>-</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>3</mn></msup><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><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;">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>[8 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>Part (a)(i) was well done with few candidates incorrectly using the value of <em>p</em> to verify rather than to 'show' the given result. In part (a)(ii) most did not consider both values of <em>r</em> and some did know the condition for convergence of a geometric series. Part (a)(iii) was generally well done but some had difficulty in simplifying the surd. Part (b) (i) and (ii) was generally well done. Although many completely correct answers to part b (iii) were noted, weaker candidates often made errors in properties of logarithms or algebraic manipulation leading to an incorrect quadratic equation.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[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>The cubic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>k</mi><mo>=</mo><mn>0</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>0</mn></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>β</mi></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mi>β</mi><mo>=</mo><mo>-</mo><mfrac><msup><mi>k</mi><mn>2</mn></msup><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>=</mo><mi>k</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>β</mi><mo>=</mo><mfrac><mi>k</mi><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mi>β</mi><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn><mi>k</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><msup><mi>k</mi><mn>2</mn></msup><mn>4</mn></mfrac></mrow></mfenced><mfenced><mfrac><mi>k</mi><mn>2</mn></mfrac></mfenced><mo>=</mo><mo>-</mo><mn>3</mn><mi>k</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mfrac><msup><mi>k</mi><mn>3</mn></msup><mn>8</mn></mfrac><mo>=</mo><mo>-</mo><mn>3</mn><mi>k</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p>attempting to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><msup><mi>k</mi><mn>3</mn></msup><mn>8</mn></mfrac><mo>+</mo><mn>3</mn><mi>k</mi><mo>=</mo><mn>0</mn></math> (or equivalent) for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>2</mn><msqrt><mn>6</mn></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>24</mn></msqrt></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow></mfenced></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>k</mi><mo>=</mo><mo>±</mo><mn>2</mn><msqrt><mn>6</mn></msqrt><mo> </mo><mfenced><mrow><mo>±</mo><msqrt><mn>24</mn></msqrt></mrow></mfenced></math>.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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> has a local maximum at A. Find the coordinates of A.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there is exactly one point of inflexion, B, on 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>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The coordinates of B can be expressed in the form B<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{2^a},\,b \times {2^{ - 3a}}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>a</mi> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mn>3</mn> <mi>a</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> where <em>a</em>, <em>b</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \in \mathbb{Q}"> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> </math></span>. Find the value of <em>a</em> and the value of <em>b</em>.</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> showing clearly the position of the points A and B.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to differentiate <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = - 3{x^{ - 4}} - 3x"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for using quotient or product rule award <em><strong>A1</strong> </em>if correct derivative seen even in unsimplified form, for example <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{ - 15{x^4} \times 2{x^3} - 6{x^2}\left( {2 - 3{x^5}} \right)}}{{{{\left( {2{x^3}} \right)}^2}}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>15</mn> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>×</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>5</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{3}{{{x^4}}} - 3x = 0"> <mo>−</mo> <mfrac> <mn>3</mn> <mrow> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </mrow> </mfrac> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {x^5} = - 1 \Rightarrow x = - 1"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>x</mi> <mn>5</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( { - 1,\, - \frac{5}{2}} \right)"> <mrow> <mtext>A</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 12{x^{ - 5}} - 3\left( { = 0} \right)"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>12</mn> <mrow> <msup> <mi>x</mi> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct derivative seen even if not simplified.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = \sqrt[5]{4}\left( { = {2^{\frac{2}{5}}}} \right)"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mroot> <mn>4</mn> <mn>5</mn> </mroot> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>hence (at most) one point of inflexion <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> This mark is independent of the two <em><strong>A1</strong> </em>marks above. If they have shown or stated their equation has only one solution this mark can be awarded.</p>
<p><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> changes sign at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt[5]{4}\left( { = {2^{\frac{2}{5}}}} \right)"> <mi>x</mi> <mo>=</mo> <mroot> <mn>4</mn> <mn>5</mn> </mroot> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>R1</strong></em></p>
<p>so exactly one point of inflexion</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt[5]{4} = {2^{\frac{2}{5}}}\left( { \Rightarrow a = \frac{2}{5}} \right)"> <mi>x</mi> <mo>=</mo> <mroot> <mn>4</mn> <mn>5</mn> </mroot> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {{2^{\frac{2}{5}}}} \right) = \frac{{2 - 3 \times {2^2}}}{{2 \times {2^{\frac{6}{5}}}}} = - 5 \times {2^{ - \frac{6}{5}}}\left( { \Rightarrow b = - 5} \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mn>3</mn> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>5</mn> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">⇒</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for the substitution of their value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<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><img src="data:image/png;base64,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"><em><strong>A1A1A1A1</strong></em></p>
<p><em><strong>A1</strong></em> for shape for <em>x</em> < 0<br><em><strong>A1 </strong></em>for shape for <em>x</em> > 0<br><em><strong>A1 </strong></em>for maximum at A<br><em><strong>A1 </strong></em>for POI at B.</p>
<p><strong>Note:</strong> Only award last two <em><strong>A1</strong></em>s if A and B are placed in the correct quadrants, allowing for follow through.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<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> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {x^2} - {a^2},{\text{ }}x \in \mathbb{R}">
<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>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<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="a">
<mi>a</mi>
</math></span> is a positive constant.</p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = x\sqrt {f(x)} ">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<msqrt>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| x \right| > a">
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>></mo>
<mi>a</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{f(x)}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span>;</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left| {\frac{1}{{f(x)}}} \right|"> <mi>y</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f(x)\cos x{\text{d}}x} "> <mo>∫</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </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>By finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> is an increasing function.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_08.15.01.png" alt="M17/5/MATHL/HP1/ENG/TZ2/09.a.i/M"></p>
<p><strong><em>A1 </em></strong>for correct shape</p>
<p><strong><em>A1 </em></strong>for correct <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts and minimum point</p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_08.17.28.png" alt="M17/5/MATHL/HP1/ENG/TZ2/09.a.ii/M"></p>
<p><strong><em>A1 </em></strong>for correct shape</p>
<p><strong><em>A1 </em></strong>for correct vertical asymptotes</p>
<p><strong><em>A1 </em></strong>for correct implied horizontal asymptote</p>
<p><strong><em>A1 </em></strong>for correct maximum point</p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_08.20.22.png" alt="M17/5/MATHL/HP1/ENG/TZ2/09.a.iii/M"></p>
<p><strong><em>A1 </em></strong>for reflecting negative branch from (ii) in the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis</p>
<p><strong><em>A1 </em></strong>for correctly labelled minimum point</p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt at integration by parts <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {({x^2} - {a^2})\cos x{\text{d}}x = ({x^2} - {a^2})\sin x - \int {2x\sin x{\text{d}}x} } "> <mo>∫</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mn>2</mn> <mi>x</mi> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span> <strong><em>A1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ({x^2} - {a^2})\sin x - 2\left[ { - x\cos x + \int {\cos x{\text{d}}x} } \right]"> <mo>=</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mo>∫</mo> <mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>]</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ({x^2} - {a^2})\sin x + 2x\cos - 2\sin x + c"> <mo>=</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo>−</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> </math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {({x^2} - {a^2})\cos x{\text{d}}x = \int {{x^2}\cos x{\text{d}}x - \int {{a^2}\cos x{\text{d}}x} } } "> <mo>∫</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </mrow> </math></span></p>
<p>attempt at integration by parts <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{x^2}\cos x{\text{d}}x = {x^2}\sin x - \int {2x\sin x{\text{d}}x} } "> <mo>∫</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mn>2</mn> <mi>x</mi> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span> <strong><em>A1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {x^2}\sin x - 2\left[ { - x\cos x + \int {\cos x{\text{d}}x} } \right]"> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mo>∫</mo> <mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>]</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {x^2}\sin x + 2x\cos x - 2\sin x"> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \int {{a^2}\cos x{\text{d}}x = - {a^2}\sin x} "> <mo>−</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {({x^2} - {a^2})\cos x{\text{d}}x = ({x^2} - {a^2})\sin x + 2x\cos x - 2\sin x + c} "> <mo>∫</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = x{({x^2} - {a^2})^{\frac{1}{2}}}"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = {({x^2} - {a^2})^{\frac{1}{2}}} + \frac{1}{2}x{({x^2} - {a^2})^{ - \frac{1}{2}}}(2x)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Method mark is for differentiating the product. Award <strong><em>A1 </em></strong>for each correct term.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = {({x^2} - {a^2})^{\frac{1}{2}}} + {x^2}{({x^2} - {a^2})^{ - \frac{1}{2}}}"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></span></p>
<p>both parts of the expression are positive hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is positive <strong><em>R1</em></strong></p>
<p>and therefore <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> is an increasing function (for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| x \right| > a"> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>></mo> <mi>a</mi> </math></span>) <strong><em>AG</em></strong></p>
<p><strong><em>[4 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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{1 - 3x}}{{x - 2}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>x</mi> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span>, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.</p>
<p><img src="images/Schermafbeelding_2018-02-07_om_17.42.06.png" alt="N17/5/MATHL/HP1/ENG/TZ0/06.a"></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><img src="images/Schermafbeelding_2018-02-07_om_17.44.18.png" alt="N17/5/MATHL/HP1/ENG/TZ0/06.a/M"></p>
<p>correct vertical asymptote <strong><em>A1</em></strong></p>
<p>shape including correct horizontal asymptote <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{3},{\text{ }}0} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0,{\text{ }} - \frac{1}{2}} \right)"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{3}"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - \frac{1}{2}"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> marked on the axes.</p>
<p> </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>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>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>a</mi><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> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced></math>, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">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>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>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><mi>y</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfrac></math></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong> exchange <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 attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>y</mi><mo>-</mo><mi>x</mi><mi>y</mi><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>x</mi><mo>+</mo><mi>x</mi><mi>y</mi><mo>=</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>y</mi><mo>-</mo><mn>4</mn></mrow><mrow><mi>y</mi><mo>+</mo><mi>a</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>+</mo><mi>a</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Condone use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo></math></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>≡</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfrac><mo>≡</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>+</mo><mi>a</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mo>-</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfrac></math></p>
<p>attempt to find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mi>g</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced></math> and equate to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>a</mi><mfenced><mstyle displaystyle="true"><mfrac><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfrac></mstyle></mfenced><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mfenced><mstyle displaystyle="true"><mfrac><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfrac></mstyle></mfenced></mrow></mfrac><mo>=</mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mfenced><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mn>4</mn><mfenced><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mfenced><mrow><mn>9</mn><mo>-</mo><mn>3</mn><mi>x</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mfenced><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mn>4</mn><mfenced><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mn>5</mn><mo>-</mo><mfenced><mrow><mn>3</mn><mo>+</mo><mi>a</mi></mrow></mfenced><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mfenced><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mn>4</mn><mfenced><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mi>x</mi><mfenced><mrow><mn>5</mn><mo>-</mo><mfenced><mrow><mn>3</mn><mo>+</mo><mi>a</mi></mrow></mfenced><mi>x</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>equating coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> (or similar)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>3</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">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{{x^2} - 10x + 5}}{{x + 1}}{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \ne - 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>10</mn>
<mi>x</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the co-ordinates of all stationary points.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the vertical asymptote.</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>With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{\left( {2x - 10} \right)\left( {x + 1} \right) - \left( {{x^2} - 10x + 5} \right)1}}{{{{\left( {x + 1} \right)}^2}}}">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>10</mn>
<mi>x</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0 \Rightarrow {x^2} + 2x - 15 = 0 \Rightarrow \left( {x + 5} \right)\left( {x - 3} \right) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>15</mn>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong> M1</strong></em></p>
<p>Stationary points are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 5{\text{,}}\,\, - 20} \right)\,\,{\text{and}}\,\,\left( {3{\text{,}}\,\, - 4} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>20</mn>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>and</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Looking at the nature table</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXEAAAA5CAYAAADA+xsRAAANbElEQVR4Ae1dS2gVSReuDFkqLiLim5Bo3IkiuhAfYGLiaxlFXcQsRI0rUVEEwY0zQyRRYVBBXJhNoiar0fgAFSQuREgEEcRHFESjMHGhcQYH/Lk/X82cO3XrVpLurqp7q+89By7Vj6pTX33V9/Tp09VVFZlMJiNYmAFmgBlgBlLJQKWKuqKiQt3lbWaAGWAGmIHAGND97hwjDqx6hsDwe4GDm1fI7WZ88bo9dL7itaa4uUPk0jcm3/ptehTYdPlJP8D7zAAzwAwwA+lhgI14evqKkTIDzAAzkMcAG/E8SsrzwK5duwQe1ejX2dlZnkRwq5mBlDHARjxlHeYD7pMnT0R/f39WdVVVlVi3bl12nzeYAWYgXAYq1CGGIQf0fVIYert944MXvnXrVrFly5ZENPvGFxdUaHji4g8pf4hc+sbkW79N/5qwsSduw2gJlP348aN4+PChaG1tFWfOnBHfvn0rgVaF24QNGzbIkFW4CEsH2Zs3b7LhQRi/BQsWlE7jlJawEVfIKMfN7u5ugYv98+fP4uDBg6K6ulogvMLingEYkjt37rhXzBqNDDQ2Norh4WE5fJiGEO/fv9+YN80H2YinufccYD906JAYGxsTQ0NDAl4ijHlzc7OAh87ijgFw29TUJHp6etwpZU0TMvD69WtRU1OTzQOjDoel1MTaiN+4cUM+skyfPj3rwV25ckVgv6GhgR/PU3DFTJkyRSxdulTcunVLdHR0yAt9cHAwBcjTA/H27dsCP5biMADjfeHCBRk2LA4Cf7VaG3G8DMOjyrFjx8TZs2fFpUuXxKJFi8To6Ki4e/eugIFgKSwD+nBBPMbrv/GGEO7du1fU19eLFy9eFBY018YMOGZAjYnX1tbKp6Dt27c7rqX46qyNODUBQ9IePXokli1bJr06Os5p4Rno6urKxgFxgzX9EEYxCW66c+bMkTdi03k+xgykhQGEUtRr//jx4zJkmBb8UXE6M+ILFy4U8+bNEx8+fIhaN+cLkAGMTpkxY4a8GQcIL3hIp06dynnqQWiRJQwGLl++LF8sl1pc3JkRpzh4b29vGD3GKCZlAAYbwwpfvXol82L/xIkTYu3atWLWrFmTlucM+QwcOXIkx/srxcf3/FbzkWIy4MSIw4DPnDlTHD16VHz//l2+zLx27RqPcChmz0as+/nz56Kurk56j/jgZ9++fYk/+olYJWdjBrwzAG9bHxd+8uRJOUJIHbHiHUgBKrAy4ng5hlEo8ODwghMhFQxRQ3wc2+zNFaAHLapA/PvixYtZzxGjU9BvLO4ZoI98duzYIZXTi+ZSe7R3z1wyjTDUGFJIPCOFlOIIIf7sXgjZ0XgBEqrgAmR80XsndL6it6T4OUPk0jcm3/ptetWEzcoTtwHDZZkBZoAZYAbsGWAjbs8ha2AGmAFmoGgMsBEvGvVcMTPADDAD9gzkxcTtVbIGZoAZYAaYAV8M6O/HeKFkfrFpfa2ZXrZYK7VQEBoei6YUvWiIXPrG5Fu/TacCmy4cTtEZ4X1mgBlgBlLEABvxFHUWQ2UGmAFmQGeAjbjOCO8zA8wAM5AiBtiIp6izGCozwAwwAzoDwRnxJHOuYBUalGNhBkJgAC+fsG5pHEEZ/PB5fiEE+EwvyQpRN9fhloGgjDgM8erVq2PPuYI5WjDnBxakKHehOTrIKPDcHOFfEVj3EUu3YegY5vag6WzDRx4+wnL4P3g14jCqmCBLXboNy7lh5R99MV7sf/36NbYBp8sIy4tBdL10vhxS8uJgDPBra2sTWNGEJWwGcKPFpHEsbhkol/+DNyOOJcIGBgbEs2fPxJIlS8T9+/dlD2G+8R8/fsipa6nLMAsi1r/bvHkzHUqUojz0QJ9PgecU2qrZMARYSR2rl5AcPnxYbvLCBMRImCkW9FWF5iRXj/F2PAYK+X8otj3wYsThbWOFn3PnzkljjbU2sRwYjCuOo9HqNLVYlGDq1Kk5x+J12T+5oePLly/ZRQ6S6EhaBo/A+vzFetwRxpTCHEhdGtfHjx9L6KtWrco2AdNxwhN/9+5d9lhIG7gOyFsiXOBE5ZFCC8Rb3Fgz6TWlqAf6VUE9VAdhUftNxUblCNt4fUrXAeWjOmFocGx4eFjOxY9t5KX6SD8ZCaSkA9u60DmkOq86hvfv3+vFne4Xu2+L/X8opD1wbsRhqLFg8saNG/MWSR4bG5MX4c6dO3MuGHjps2fPzjmWZAfzY69YsSLr9SfRkbRMc3Oz/DOSAYCe7u5uGdLANv6YmEsaf1iEOpBiX82ftO60lsN1gKcHGDMSLKG1Z88euYs/AhYaofAQnuzwzkTNT+V8pegnYCIMqEc1oDDqFM9GHuRVBf0LzMBOOtAmXA+0BiRutO3t7fK8ehNW9eAJs7q6WuYBJuxDBwkMN+lAPRDCCb6AQT1P85pTeddpGvrWdZtVfQW1BxlFhMA1YicdHR2ZqqqqzNDQUJ6ikZGRzNWrV3OOj42NZerr6zPXr1/POZ50B3qgD3qjSpR2t7e3459h/PX09MiqmpqaMm1tbdlqkX9gYCB7DjpU0fOr59TtKPiAwZSvtrY2o9er6naxbao3ql4V3/DwsGwDUgjOEbekT81Px/Q0Kh6TLpSlPjNxiv5Fv0EIL+UnHKoO5FevCeRRdWBfx6HXq+fXyyA/dKii6lC3KQ8wR+EpSh7Sqadqu4gr275FHVEwmdqMsiomHS/tR9FfLHtgwubME4cH3tDQIBCHxeo+69evz3vJiBDKtm3b1BuWgHeOF5pYYd0kCM3QYyK2IXh5CY8b9Zni32/fvpV6TfqSHqM4Ja4hvDDEjzwrWkextbVVekioA14SPCzyrBD3hAdGbUEKL7RcBF612nbyIuF1Y3UhSF9fX87yWfS0opbDsWIKvGGSkZERuTnRUyS8YHjNahuwbytqWAfhMvCi1qF62jjv8wV3qfRtnD4JyR44M+IIZSD23dLSIn+jo6OCRoxMRM6nT59kHHu8PFj2DYYeehGmwWMpxoUj5oX6UK8quBlUVlYK6C20kDHHI7QaFgAO/OnUx1m6AZw/f94JzLlz50o9angGBgR/7pUrVzqpw0aJetGj7cQVPXYCK4w5boQkMDw9PT3ZmyVxBl0uRDWESfRNZLxJH0Im6g2f2uBymbD58+dLI0261RQ4cN6nhNi3Ifwf6Br3bQ+cGXFcJPTicvHixU6vGRhqLOJ77949Ge/etGmTU/0uleEPi1g4vGwYKBIMIYMn7kvg8cPoYTFYko6OjpynAToeUgojh5gysOKGQxc+MGKNRHW0jWvcqJueAqBbfxk4WX0oD86BnUS/MaxZs0Z64rhJ+RI8lYI7errR69HPAwti5L6lmH0byv+hIPaA4kBITfEW9fxk24iD19XVGePh45VFDHu8GLpaBvH0mpqaSWPnwAB9cWLstu1WcVKskeKm6jlTHE2Pp6r5aTsOPuSlnx4nJX2u0zj4THVT/FKPHSMvjlF7KKW4qkkXjsXBQzqRAgc4oz4hXGo96EO1bynWS3pQVtWBsqSH8lBdpFeP01J+Om+KiQOD+q5Dx4E6VD5JJ2Gg/FTHeCny2wjVq2IhfUn6FmXjYKL2Io36f4ijn9oyXuraHpiw5fSQKcN44EzH8VIThhYGN6pENfxkxFtaWiZUHVWfqsS23aouH9uMLx6rofMVrzXFzR0il74x+dZv06MmbE7DKU+fPpUv8tQx4K4e2W7evCnj4ogvISYemoxWVIj//fvS7a9ffxXYRwqJsv93b29oTfKOhznzTnHRKij3vrVuf19f9L5T7womK6+en2ibPGV443GEyk0U/sAj2YMHD2SYhsI1p0+fNnr88MSXL19uPDceLpt2k84/f/4586WxMfP3779n/hAi89dvv8VKf7x6RaryUhf48pQ6PJAUny/OkuJxSEnJqErKpa++BbFJMUXtFBf6fbXfhM1ZOCVJGAOkTjROnMacw4ireWGkX758aewTX+PEjZVpB8f275eG+89ffpFnkMKgR93X1GV3TR2XPRnAhg0+H5zZ4AmAzqAg2HDpo29Bjg2mKOS60u+j/SZseQsl/8NRdE+ecnZ2dgqEU7q6uuhQ5BRlIfg031aS6ML42qTttsUbpTzji8LSf3lC5+s/pOFvhcilb0y+9dv0ugmbk5g4hhZi3PaBAwcS4cPwO5Q3fbgTVyE+wOAZ4eKyxvmZAWYgrQwkNuJ4uUhfZcL7xVjUKB/3mIjCXODTpk2z/soSmPBhEPSxMAPMADNQDgwkNuIgBzMGwpDjK0mbUAg+5sGg+P7+fivOUR569K84rZRyYWaAGWAGAmagMik2DCOk6R6T6lDLwYsfHByU86Ik8ehpMYgkZVUcvM0MMAPMQJoYsPLEXTd09+7dci5wzC8eR5AfP5RnYQaYAWagnBhwNjolzaSZ3viG1B7GF683QucrXmuKmztELn1j8q3fpkdN2ILyxG0ax2WZAWaAGShHBvI88XIkgdvMDDADzEBaGNC/acl5samfTEujGCczwAwwA+XKAIdTyrXnud3MADNQEgywES+JbuRGMAPMQLkywEa8XHue280MMAMlwQAb8ZLoRm4EM8AMlCsD/wfs0rHOcwJjDwAAAABJRU5ErkJggg=="> <em><strong>M1</strong><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 5{\text{,}}\, - 20} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>20</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is a max and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {3{\text{,}}\, - 4} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is a min <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <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> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{ln}}\left| x \right|">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> \ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ 0 \right\}">
<mrow>
<mo>{</mo>
<mn>0</mn>
<mo>}</mo>
</mrow>
</math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {\text{ln}}\left| {x + k} \right|">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>k</mi>
</mrow>
<mo>|</mo>
</mrow>
</math></span>, <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> \ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ { - k} \right\}">
<mrow>
<mo>{</mo>
<mrow>
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
<mo>}</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k > 2">
<mi>k</mi>
<mo>></mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> intersect at the point P .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe the transformation by which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> is transformed to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right)"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graphs 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 <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> on the same axes, clearly stating the points of intersection with any axes.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>translation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> units to the left (or equivalent) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>range is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {g\left( x \right) \in } \right)\mathbb{R}"> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>∈</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p>correct shape 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> <em><strong>A1</strong></em></p>
<p>their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> translated <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> units to left (possibly shown by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - k"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> </math></span> marked on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis) <em><strong>A1</strong></em></p>
<p>asymptote included and marked as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - k"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> intersects <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right)"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> intersects <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - k - 1"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - k + 1"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right)"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> intersects <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{ln}}\,k"> <mi>y</mi> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>k</mi> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not penalise candidates if their graphs “cross” as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \pm \infty "> <mi>x</mi> <mo stretchy="false">→</mo> <mo>±</mo> <mi mathvariant="normal">∞</mi> </math></span>.</p>
<p><strong>Note:</strong> Do not award <em><strong>FT</strong> </em>marks from the candidate’s part (a) to part (c).</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at P <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {x + k} \right) = {\text{ln}}\left( { - x} \right)"> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + k = - x"> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </math></span> (or equivalent) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \frac{k}{2} \Rightarrow y = {\text{ln}}\left( {\frac{k}{2}} \right)\,\,"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> <mo stretchy="false">⇒</mo> <mi>y</mi> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math></span> (or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{ln}}\left| {\frac{k}{2}} \right|"> <mi>y</mi> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>|</mo> <mrow> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>|</mo> </mrow> </math></span>) <em><strong>A1</strong></em></p>
<p>P<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \frac{k}{2},\,\,{\text{ln}}\frac{k}{2}} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> (or P<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \frac{k}{2},\,\,{\text{ln}}\left| {\frac{k}{2}} \right|} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>|</mo> <mrow> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>)</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>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><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The region <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is bounded by the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>6</mn></msqrt></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> be the area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> divides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> into two regions of equal area.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> be the gradient of a tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>6</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>32</mn></mfrac><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p style="text-align:center;"><img src="data:image/png;base64,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"></p>
<p>a curve symmetrical about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis with correct concavity that has a local maximum point on the positive <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis <strong>A1</strong></p>
<p>a curve clearly showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>→</mo><mn>0</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>±</mo><mo>∞</mo></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p>horizontal asymptote <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis) <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="[" close="]"><mrow><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mi>arctan</mi><mfrac><mi>x</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>M1A0</strong> for obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mi>k</mi><mo> </mo><mi>arctan</mi><mfrac><mi>x</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>≠</mo><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac></math>.</p>
<p style="text-align:left;"><strong>Note:</strong> Condone the absence of or use of incorrect limits to this stage.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mfenced><mrow><mi>arctan</mi><mo> </mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi>arctan</mi><mo> </mo><mn>0</mn></mrow></mfenced></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>×</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mi>k</mi></munderover><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle></mrow><mn>4</mn></mfrac></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math> <strong>(M1)</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>k</mi><msqrt><mn>6</mn></msqrt></munderover><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle></mrow><mn>4</mn></mfrac></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mfenced><mrow><mi>arctan</mi><mo> </mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arctan</mi><mo> </mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mi>k</mi></munderover><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><munderover><mo>∫</mo><mi>k</mi><msqrt><mn>6</mn></msqrt></munderover><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mfenced><mrow><mi>arctan</mi><mo> </mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>d</mo><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mn>3</mn></mfenced><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> <strong>A1</strong></p>
<p style="text-align:left;">so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>6</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts product rule or quotient rule differentiation <strong>M1</strong></p>
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfenced><mrow><mo>-</mo><mn>6</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>+</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>6</mn><mi>x</mi></mrow></mfenced><mfenced><mn>2</mn></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>4</mn></msup></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>A0</strong> if the denominator is incorrect. Subsequent marks can be awarded.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>THEN</strong></p>
<p style="text-align:left;">attempts to express their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> as a rational fraction with a factorized numerator <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>4</mn></msup></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>6</mn><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>3</mn></msup></mfrac></mrow></mfenced></math></p>
<p style="text-align:left;">attempts to solve their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>±</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math> <strong>A1</strong></p>
<p style="text-align:left;">from the curve, the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math> <strong>R1</strong></p>
<p style="text-align:left;">(the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math>)</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>R1</strong> for any equivalent valid reasoning.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mn>6</mn><mfenced><mrow><mo>-</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></mrow></mfenced></mrow><mstyle displaystyle="true"><msup><mfenced><mrow><mstyle displaystyle="true"><msup><mfenced><mrow><mo>-</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></mrow></mfenced><mn>2</mn></msup></mstyle><mstyle displaystyle="true"><mo>+</mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mrow></mfenced><mn>2</mn></msup></mstyle></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;">leading to a maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>32</mn></mfrac><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[7 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Sketch the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{x}{2} + 1"> <mi>y</mi> <mo>=</mo> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mn>1</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left| {x - 2} \right|"> <mi>y</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>|</mo> </mrow> </math></span> on the following axes.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><img 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"></p>
<p>straight line graph with correct axis intercepts <em><strong>A1</strong></em></p>
<p>modulus graph: V shape in upper half plane <em><strong>A1</strong></em></p>
<p>modulus graph having correct vertex and <em>y</em>-intercept <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A 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><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</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><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</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><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>The inverse of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfrac><mi>x</mi><mn>2</mn></mfrac></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>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p>Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mi>q</mi><mn>2</mn></mfrac><msqrt><mi>r</mi></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</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><strong><img src="data:image/png;base64,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"></strong></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept an indication of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<p><br>vertical asymptotes <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>x</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p>horizontal asymptote <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>uses a valid method to find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the local maximum point <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> For example, uses the axis of symmetry or attempts to solve <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>.</p>
<p><br>local maximum point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)A0</strong></em> for a local maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> and coordinates not given.</p>
<p><br>three correct branches with correct asymptotic behaviour and the key features in approximately correct relative positions to each other <em><strong>A1</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"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>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> (this can be done at a later stage).</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempts to complete the square <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>3</mn><mo>=</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mfenced><mrow><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>4</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>1</mn><mo>=</mo><mo>±</mo><msqrt><mn>4</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><mo>±</mo><msqrt><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mi>x</mi></mfrac></msqrt></mrow></mfenced></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>-</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> for <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><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msqrt></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>even if <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo></math> (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo></math>) is missing</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>±</mo><msqrt><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi></msqrt></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>±</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>3</mn></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math> is rejected <em><strong>R1</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for concluding that the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> must have the ‘<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo></math>’ sign.<br>The <em><strong>R1</strong> </em>may be awarded earlier for using the condition <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>3</mn></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow><mn>2</mn></mfrac></mfenced><mi mathvariant="normal">=</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></mfenced><mi mathvariant="normal">=</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p>attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><mn>2</mn><mo> </mo><mo> </mo><mfenced><mrow><mfrac><mn>1</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mfrac><mo>=</mo><mn>2</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mn>2</mn></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mn>2</mn></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mfenced><mn>2</mn></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award all available marks to this stage if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is used instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>a</mi><mo>-</mo><mn>7</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempts to solve their quadratic equation <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><mfenced><mn>2</mn></mfenced><mfenced><mn>7</mn></mfenced></msqrt></mrow><mn>4</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>4</mn><mo>±</mo><msqrt><mn>72</mn></msqrt></mrow><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award all available marks to this stage if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is used instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msqrt><mn>2</mn></msqrt></math> (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>3</mn></math>) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>2</mn></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>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mn>18</mn></msqrt></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>18</mn></mrow></mfenced></math></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;">
<p>Part (a) was generally well done. It was pleasing to see how often candidates presented complete sketches here. Several decided to sketch using the reciprocal function. Occasionally, candidates omitted the upper branches or forgot to calculate the <em>y</em>-coordinate of the maximum.</p>
<p>Part (b): The majority of candidates knew how to start finding the inverse, and those who attempted completing the square or using the quadratic formula to solve for y made good progress (both methods equally seen). Otherwise, they got lost in the algebra. Very few explicitly justified the rejection of the negative root.</p>
<p>Part (c) was well done in general, with some algebraic errors seen in occasions.</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>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^{2x}} - 6{{\text{e}}^x} + 5{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \leqslant a">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>a</mi>
</math></span>. 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> is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAW8AAAJXCAYAAABc/dVJAAAgAElEQVR4Ae2dC5BV1Znvvz5aJhBMhMtDEBXFy8OLCmlUEkSIIoZJWoFcYmJGNC2TsbFLLhJEHTERNDEoWmh72hnHNpLRyaBAOySFGiQgcItQtCjhCt1RRJ5RqJaQTpNo7H1rbdxNv/uc/Vxr7d+uOsXpfdb6Hr9v8+/d66y9VoHjOI5wQAACEICAUQQyRkVLsBCAAAQg4BJAvLkQIAABCBhIAPE2sGiEDAEIQADx5hqAAAQgYCABxNvAohEyBCAAAcSba8AiAn+Xg8tvkYKCAiko6Cdfe3iz1LnZNcjRNXdLv36lsnz/xxblSyppJoB4p7n61uV+svSd8qQ4n+6QpyeIrP3NdjnQoJLMSLehX5ObBn8kR/7inrAucxJKHwHEO301tz/jTH+56KrBIu/Wyp8/0+pMnzPlvHMvlYv6nWJ//mSYCgKIdyrKnLYkT5ZTe/QSefcdef/Q30XkY9lf+Yxs/sa1MqIbl3zargZb8+VKtrWyqc7rZDn1tB4nCNS9KUu3fFXmTTpbuOBPYOGd2QS4ls2uH9G3SSAjXzith/SVatm19x1Zs/A3cvatE+UMrvY2aXHSTAJczmbWjag7JJCRrl/qIV3lsGx94klZO+4mmXQGY90dIuND4wgg3saVjIBzIZA5tYcMFJGTx06XO644g+GSXKDRxigCiLdR5SLY3AmcJP3vekQevGmYdMu9Ey0hYAwBxNuYUhFozgTqNsvipzMy5+4rpS9XeM7YaGgWgZPNCpdoIdAOgbrN8nDRLDl4wyzpX31YLp/3AxnCtMB2YHHaBgLcl9hQRXIQaaiTQ9XvyRvVx+Ty/1MshQg3V4XlBArYScfyCpMeBCBgJQHuvK0sK0lBAAK2E0C8ba9wivPbt2+fHDt2LMUESN1mAoi3zdVNcW5KuKdMmSK33XYbAp7i68Dm1BFvm6ub0tw84a6qqpKKigp5+umnU0qCtG0mgHjbXN0U5tZUuL30Z86cKWVlZd6P/AsBKwgg3laUkSQUATW+fd9994m64168eHEjlMLCQkHAG3HwxhICiLclhUx7Gkq41fi2GiZRwl1aWtqIZPny5eIJuHrPAQEbCDDP24YqpjyHpsI9e/ZsWbhwoUvkpJNOcv/99NNPpelwyuuvvy6jR49OOTXSN50A4m16BYlfXn31VZk4caIUFxfLY489Jl26dHGpNBVvdcITcPV+3bp1je1ACAETCSDeJlaNmFsRUAI+ZsyYZoLcUrxVp5qaGunatav079+/lQ1OQMAkAoi3SdUi1rwItCXeeRmgMQQ0JsAXlhoXh9AgAAEItEcA8W6PDOchAAEIaEwA8da4OIQGAQhAoD0CiHd7ZDgPAQhAQGMCiLfGxSE0CEAAAu0RQLzbI8N5CEAAAhoTCLyHZXl5ucbpERoERLhGk70KPvzwQzlw4IAMHz482UA0815SUhIsIrUNWpAjm80G6e6rb5w+4/SlYODP1yXRZqdMJuOoV5yHzfXzm9uyZct81cGvP7/1jtNfGL4YNgn2u4/eEIAABBIhgHgngh2nEIAABIIRQLyD8aM3BCAAgUQIIN6JYMcpBCAAgWAEEO9g/OgNAQhAIBECiHci2HEKAQhAIBgBxDsYP3pDAAIQSIQA4p0IdpxCID0E+vTp4yardjLiCI8A4h0eSyxBAAJtEOjVq5d7tr6+vo1POeWXAOLtlxz9IAABCCRIAPFOED6uIQABCPglgHj7JUc/CEAAAgkSQLwThI9rCEAAAn4JIN5+ydEPAhCAQIIEEO8E4eMaAhCAgF8CiLdfcvSDAAQgkCABxDtB+LiGAAQg4JcA4u2XHP0gAAEIJEgA8U4QPq4hAAEI+CWAePslRz8IQAACCRJAvBOEj+t8CRyWNXNHSkFBwWevfnJ1xU5pyNcM7SFgAQHE24IipiWFhv2vy3O/qGqS7mVy3WUDhIu4CRLepobAyanJlEQNJ3BEtv7nS9LzPw6Jc0VPw3MhfAgEJ4B4B2eIhTgIHH1Dlj6yRBYOPl0GH5kol024XAZ14547DvT40JMAV7+edSGqZgT+KjUvPikLD4rI2oVy87e+JoOL7pHlNUebteIHCKSJAOKdpmobm+vnZVDxUnGcv8mBLSvl6TsmiKz9qXzrn5+WqroTX1ee+CLz+BeaDQ0Nol4tzxuLgcAh0IRAgeM4TpOf835bXl6edx86QCAYgb/Ln6pfkYrFv5dzZpbKpMHdXHMzZszIyWw2m82pHY3CIfDhhx/K/Pnz5d5775XevXuHY9QCKyUlJcGyUOId5Mhms0G6++obp884fSkY+Mv1kjjkvHZHodP3jtecP7XTJZPJOOoV52Fz/fzmVl1d7dZB/ZvP4ddfPj6ato3TXxi++MIy2O8+eidGoJv0HzxYLpB+cvy+O7FAcAyBRAgEFu+//OUvsnz58sbghw0b1vh+0KBBje95A4FwCdTJvncukLl3DmKed7hgsWYIgVDEe+rUqR2mW1xcLN27d5dRo0ZJnz595Oyzz5b+/ft32IcPIdBIoOGgVK3cKnLJeCnse4pIw0HZvPhxWTv+VvnxF/nOvZETb1JFILB4qy8gPv3000ZoNTU1je+3b9/uvt+0aZN89NFH0lLkZ8+e7Qq6ulvnLr0RG29aEWiQP/1usVw56RsiMkymPXSP3HjNnTJ/0BdbteQEBNJCILB4twTVVIS991OmTHGbPfXUU7Jv3z55//335d133xUl7k0FXYn5+PHjZeTIkdKjR4+Wpvk5rQQyZ8gVP3lFnJ+kFQB5Q6A1gdDFu7WL5mfUcIl6jR492v1g4cKFou7W1d35+vXrZeLEie55NdSihB0hb86PnyAAAQgoAloMGKo79GnTpom6Mz906JCsWrVKzjnnHFfIe/XqJf/0T/8kGzdulGPHjlE1CEDAMAI9ex5fi2b37t2GRa53uFqId1NEarhkwoQJcvfddzcKufr88ssvl7Fjx0pZWZmoGS4cEICAGQS8IdC6ujozAjYkytiHTfLh4gm5EvMf/ehHUllZKUuWLJGqqipRT21Nnz6dLzrzAUpbCEDAGgLa3Xm3R1aNk5eWlsq6detc0Vbj5EOHDm0cUmmvH+chAAEI2EjAGPH24Hfp0kWGDx/u3oW//vrr7tCKGlKZNGmS+8Wn145/IQABCNhMwDjxbloMNWNFDaUoEVeHuhO/4447EPGmkHgPAQhYScBo8fYq0lTE165d64q4+mKztrbWa8K/EIAABKwiYIV4exVRIq7GxF944QWZOXOmfP3rX2+27orXjn8hAAEImE7AKvFWxVBj4uqJTjVffNy4ce6DPmqeeNPH9k0vGvFDAAIQsE68vZKqaYbq6U01rfCtt95yh1LUNEMe9PEI8S8EIGAyAWvF2yuKmpmihlKeeeYZ+f73vy/f/e53uQv34PAvBCBgLAHrxVtVRg2lqMfvd+zY4RZKzUppuga5sdUjcAhAILUEUiHeXnXVGir/+Z//KQsWLGgcC2dGikeHfyEAAZMIpEq8VWHUXbhaN0XNDVdj4WpGyptvvmlSzYgVAhCAgB6rCiZRBzWtUA2dXHTRRVJYWOiumZJEHPiEAAQg4IeA1gtT+Ukonz5qvZTHHnvMFXD1ZabaHOK+++5z787zsUNbCEAAAnETSLV4K9hqGEUteDVixAh32Vk1H1w9namEnQMCEICArgRSN+bdXiHUMIranu3AgQPuQz6Mg7dHivMQyI+A2vpQHeeee25+HWndIQHEuwkedbf98ssvN46Dq917OCAAgWAE6uvrXQNdu3YNZojezQgg3s1wiLvxsRoHV5shq6Vm1QM+HBCAAAR0I1DgOI4TJKjy8vIg3bXu6y1ypTZCVluwcZhFQH2XoQ71HQZHcgTUrlfz58+Xe++9V3r37p1cIJp5LikpCRaREu8gRzabDdLdV984fd54441OJpNxpk+f7tTX1/uKN59Oceam4rLZn6qbesV52MzTb27V1dVuHdS/+Rx+/eXjo2nbOP2F4Ythk05+91166aXuAz0VFRVy2223sbBVJ7z4GAIQiIdA6qcK5oJZzURRT2SqMXB1qDFxNcWQAwIQgEBSBBDvHMkj4DmCohkEIBALAcQ7D8wIeB6waAoBCERKAPHOEy8CnicwmkMAApEQQLx9YEXAfUCjCwQgECoBxNsnTgTcJzi6QQACoRBAvANgRMADwKMrBCAQiADzvAPhE/EEXM0Df/TRRwNaozsEIACB3Ahw550bpw5bKQF/4YUX3K3VvvjFL7pLzHbYgQ8hAAEIBCTAnXdAgF73KVOmyOLFi2XmzJnsyuNB4V8IQCAyAtx5h4jWWwhJ7cozcOBAd0glRPOYggAEINBIgDvvRhThvLn55puluLjYfZSeDR3CYYoVCECgNQHEuzWTQGfUmidq7ZOioiL5wQ9+IN4uIoGM0hkCEIBACwKIdwsgYfyoBNxbQ1qNhR87diwMs9iAAAQg0EgA8W5EEe4btaXaf/zHf0hVVRVLyYaLFmsQgICIIN4RXgaDBg2SVatWiZoD/vTTT0foCdMQgEDaCCDeEVd8woQJjVMIX3311Yi9YR4CEEgLAaYKxlBpNYXwrbfekokTJ8qOHTtE3ZFzQAACEAhCgDvvIPTy6KtmoBQWFso//uM/Sm1tbR49aQoBCECgNQHEuzWTSM6oGSjeF5hz586NxEd6jH4s+5eXSr+rK6SmIT1ZkykEmhJAvJvSiPh90y8wlyxZErE3e8037P+V3Fv6hBy0N0Uyg0CnBBDvThGF20B9gblgwQJRj9DzBKYPtnWb5ZG7X5eeVw3z0ZkuELCHAOKdQC1nzZrV+AQm49/5FOCIVD35vMhtM+Tq0z+XT0faQsA6Aoh3AiX1nsBUD/Aw/p1rARqkrmqJPCbXyy2FPXLtRDsIWEugwHEcJ0h25eXlQbqnuq+aNvjEE0/IDTfcIJdeemmqWXSa/N92y29++Z4M/s44Oetzf5HqyjJZvPdy+XHpV6V3wfHeM2bM6NSMapDNZnNqR6NwCHz44Ycyf/58uffee6V3797hGLXASklJSbAslHgHObLZbJDuvvrG6TNqXw888ICTyWScrVu3uiyi9tcSuBn+PnK2PPJTZ9m+v30W/iHntTsKHZnwtFP96YmMRETdiHT6OtEj/Hdm8PSXt9/cqqur3Wtc/ZvP4ddfPj6ato3TXxi+GDYJ9rsvcG81/q3mf6sVCFnAqi2carjkl7Ls7O/KpDNOaatB4zn1R2TTVyaTEfVqei7gH5qNvngDgaQJIN4JV6Dp/O8f/ehHCUejo/ta2bz03+Wn3xogJxUUSIH76iVXLqwSefVmGXxSP7m6Yqcw3VvH2h2Pqb6+Xt/gDI4M8dageGr+9zPPPCOLFi1yH5/XICSNQugpV/xsS4u750Py2h2FIhOelupPD8grxUNYYU2jirUMZdeuXe4ploVoSSbYz4h3MH6h9Z42bZq7A89///d/8/h8aFQxBAF7CSDeGtVWDZvs3buX6YMa1YRQIKArAcRbo8qoDRymT5/urv+9fPlyjSLTLZTPhlJeKZZBXMG6FYd4YiLApR8T6FzdDB8+3B0+mTp1Kvtf5gqNdhBIIQHEW8Oiq+ETNX3wvvvu0zA6QoIABHQggHjrUIUWMajhkzvvvJPhkxZc+BECEDhBAPE+wUKrd2rX+eLiYmH4RKuyEAwEtCGAeGtTitaBeA/tqF14OCAAAQg0JYB4N6Wh2Xs1fPLCCy+4D++webFmxSEcCCRMAPFOuACdufeGT+655x4e3ukMFp9DIEUEEG8Dij1nzhxRa38/+eSTBkRLiBCAQBwEEO84KAf04a19Mm/ePLZOC8iS7hCwhQDibUgl1awTNff7xz/+MUvHGlIzwoRAlAQQ7yjphmhbLR376KOPysqVK90vMUM0jSkIQMBAAoi3QUUbPXq0zJ492915ft++fQZFTqgQgEDYBBDvsIlGbE89eakO5n5HDBrzENCcAOKteYFahtejR4/Gud8bN25s+TE/QwACKSGAeBtY6IkTJ0pRUZGo/S/Z99LAAqYs5J07d7pLPaQs7cjTRbwjRxy+A/XlpZp1ouZ+r1q1KnwHWIRAiASOHDki3bt3D9EiphQBxNvQ60Ct+62+vFRTCGtraw3NgrAhAAG/BBBvv+Q06Hfbbbe5UTz44IMaREMIEIBAnAQQ7zhph+xLLVy1ePFid+GqmpqakK1jDgIQ0JkA4q1zdXKI7eabb3afvLzjjjtyaE0TCEDAFgIFjuM4QZIpLy8P0p2+IRB488035d///d/l1ltvlaFDh4Zg0Q4TpaWlbiJlZWV2JGRoFitWrHAjnzx5sqEZRBN2SUlJMMNKvIMc2Ww2SHdffeP0GacvBcOvv2uvvda5+OKLnfr6+ryY+vWXl5MmjeP0l8lkHPWK84gzP5VXnP78+pozZ46jXvkefv3l68drH6e/MHwxbBLsd582vZk6qE0pCAQCsRBAvGPBHL0Tpg5GzxgPENCJAOKtUzUCxjJ9+nTXwvPPPx/QEt0hAAHdCSDeulcoj/jUpg0LFiyQmTNnCqsO5gGOphAwkADibWDROgr5lltucT9m1cGOKPEZBMwngHibX8NmGahVB3lwpxkSfoCAlQQQbwvLev3117sP7qi53xwQSJrARx99lHQIVvpHvC0sq7r7Vps2LFq0iA2LLayvaSlVVFTIqFGjTAtb+3gRb+1L5C9Atea3t2GxPwv0ggAEdCaAeOtcnQCxqTW/1d232rCYHXcCgKQrBDQlgHhrWpgwwvLuvh966KEwzGEDAhDQiADirVExwg5F3X3ff//93H2HDRZ7ENCAAOKtQRGiDGHChAnufpfcfUdJGdsQiJ8A4h0/89g9zpkzh7vv2KnjEALREkC8o+WrhfXRo0dz961FJQgCAuERQLzDY6m1Je6+tS4PwUEgbwKId97IzOzA3beZdSNqCLRHAPFuj4yF57n7trCopJRaAoh3ikrP3XeKik2q1hNAvK0vcfMEuftuzoOfoiVQW1vrOujTp0+0jlJoHfFOWdG5+05ZwRNO9/Dhw24EvXr1SjgS+9yfbF9KZNQZgRkzZoh6dJ41TzojxecQ0JcAd9761iayyMaMGeOuOMhTl5EhxjAEIieAeEeOWD8HTVccfPfdd/ULkIggAIFOCSDenSKys4G34uC2bdvsTJCsIGA5AcTb8gK3l5539/3aa69JTU1Ne804DwEIaEqgwHEcJ0hs5eXlQbrTN0ECn3zyicyaNUuuvPJKmTx5coKRROO6tLTUNVxWVhaNA6x2SuDDDz+U+fPny7333iu9e/futH2aGpSUlARLV4l3kCObzQbp7qtvnD7j9KVgxO3vuuuuczKZjLN3715ftci3U5z5qbzUK84jzvxUXnH68+OrurrarYH6N9/Dj798fTRtH6e/MHwxbBLsd5/xvUeOHOnmUFlZaXwuJACBNBFAvNNU7TZy/cIXviCLFy+WmTNnivc0XBvNOAUBCGhGAPHWrCBJhKN221HH888/n4R7fFpMoL6+3uLskk0N8U6WvxbeBw0aJLNnz5YlS5bIsWPHtIiJIOwgsGvXLjcRdY1xhEsA8Q6Xp7HWrr/+eqmqqpJVq1YZmwOBQyBNBBDvNFW7g1yHDx/ubpWm7r45IAAB/Qkg3vrXKLYIWS42NtQ4gkBgAoh3YIT2GFDLxRYWFsrPf/5ze5IiEwhYSgDxtrSwftO68847paKigkfm/QKkHwRiIoB4xwTaFDfjxo1zQ33xxRdNCZk4IZBKAoh3KsveftI9evRwH9qZN28eD+20j4lPIJA4AcQ78RLoF8CkSZPcoH71q1/pFxwRQQACLgHEmwuhFYH+/fu7D+2o1fh4aKcVHk5AQAsCiLcWZdAviGuvvdZ9aOeNN97QLzgiMobAzp073ecHjAnYoEARb4OKFWeo7DIfJ217fR05ckR4ND6a+iLe0XC1wuq0adNk5cqVTBu0opokYRsBxNu2ioaYj9rnUh1MGwwRKqYgEBIBxDskkDaaUftcqrW+mTZoY3XJyXQCiLfpFYw4fqYNRgwY8xDwSQDx9gkuLd3UtMHi4mJhE9+0VJw8TSGAeJtSqQTjvOmmm9xpgxs3bkwwClxDAAJNCSDeTWnwvk0CrDbYJhZOQiBRAoh3ovjNcV5aWuquNrhv3z5zgiZSCFhMAPG2uLhhpvbNb37TNVdZWRmmWWxZTmDRokUybNgwy7NMJj3EOxnuxnlVqw2ySbFxZdMi4G7dumkRh21BIN62VTTCfFjvJEK4mIZAngQKHMdx8uzTrHl5eXmzn/nBbgL/+q//KupO6nvf+572iapxenUwzTG5UqkaTJ8+XdQG1xzNCZSUlDQ/ke9PSryDHNlsNkh3X33j9BmnLwVDd3/Lli1zMpmMs3fvXu1rp+JUrzgP3esXhIWf3BR/dc34Ofz48+PH6xOnvzB8MWyS72+7lLf3tknji8uUXwiknzgBxDvxEpgVAF9cmlUvorWXAOJtb20jy4wvLiNDi2EI5EwA8c4ZFQ09Ajxx6ZHgXwgkRwDxTo690Z554tLo8sUSvPc0bp8+fWLxlzYniHfaKh5Svt4Tl2vWrAnJImZsI1BfX++m1KtXL9tS0yIfxFuLMpgXhPrikqVizasbEdtDAPG2p5axZ8JSsbEjxyEEGgkg3o0oeJMvgS9/+ctSWFgo69aty7cr7SEAgYAEEO+AANPcXe1xqXaYZ4/LNF8F5J4UAcQ7KfKW+J0wYYKbyZYtWyzJiDQgYAYBxNuMOmkb5aBBg6SoqEiy2ay2MRIYBGwkgHjbWNWYc5oxY4asXLlSvHm9MbvHHQRSSQDxTmXZw0165MiRrkEWqwqXq+nWtm/f7qZw5plnmp6KlvEj3lqWxayg1JzvBQsWyJIlS8wKnGhjIaC+2OYInwDiHT7TVFocO3asVFVVycaNG1OZP0lDIG4CiHfcxC31x5xvSwtLWtoSQLy1LY1ZgTWd833s2DGzgidaCBhIAPE2sGi6huzN+V6/fr2uIRIXBKwhgHhbU8rkE/HmfL/wwgvJB0MEELCcAOJteYHjTm/KlClSUVEhtbW14bquq5HfPHyj9CsokIKCfvK1ub+QqoMfh+sDaxAwiADibVCxTAjVW+f7V7/6VXjhNrwvLz21RuSax+WA48inB16Qa/64UEZe/4RU1TWE5wdLoRLYuXOn+/RtqEYx1kgA8W5EwZswCHjrfC9fvjwMc66Nhvf2yZe+XSxXDfqi+3Om72iZ+S+zZMLa52Tp5pDv8EOLGkNHjhwRNZTGEQ0BxDsarqm2OnXq1FAfl88MHC3jzjilGdPM6QNkeN9mp/gBAqkigHinqtzxJBvb4/JdR8mlg4/fjceTGV4goA+BAsdxnCDhlJeXB+lOX0sJvPzyy/LWW2/J3LlzI8jQkb9W/1qe2jNMfnDVAPncZx7UAlm5HKyAmAul4G1WrFjhGpk8eXJwYxZaKCkpCZaVEu8gRzabDdLdV984fcbpS8Gwxd+GDRucTCbjbN26tVmNQ8nvz79zHpq22Nny50+b2RYRdSPS6atZp5B/CCW/PGKK01++vubMmeOol98jX39+/Xj94vQXhi+GTYL97qN3OwTU4/Lq2LBhQzst/J4+IlVPrZIed90khd2aX77qj8imr0wmI+rV9FzAPzT9Bk0/CIROoPnVH7p5DKaVgHpc3ltpMLzH5T+W/S/9UrZPnCXFQxjrTuu1Rd7HCSDeXAmREfiHf/gHd6XBN954IwQfH8vBNRWy9NQiuaFRuI/Kzmd/IWuPMtc7BMChm1i7dq2cddZZodvF4HECiDdXQmQEhg8fHtLu8kelZvmP5forS+T2K/vLSe5TlupJyy/J0Oc/kX4thk8iSwjDeRFQSwT369cvrz40zp0A4p07K1r6IODtLu9/6ORj2b/8bhn3rZ/K2lb++8qE674q53EVtyLDCfsJcNnbX+NEM7zssstc//6HTk6RM6aUuY/Ft/zi0XEOyCvFQ4SLONES4zwhAlz3CYFPi9vwhk7SQow8IZAbAcQ7N060CkAg+NBJAOd0hYClBBBvSwurU1rBh050yoZYIKAHAcRbjzpYHYUaOikqKpJ169ZZnSfJnSDg/wvqEzZ41zEBxLtjPnwaEgG1ScO8efPkk08+CckiZnQmsHfvXje8YcOG6Rym0bEh3kaXz5zgr7jiCjfYPXv2mBM0kUJAYwKIt8bFsSm0/v37u0Mnf/jDH2xKi1wgkBgBxDsx9OlzrIZO1PZojIemr/ZkHD4BxDt8plhsh4A3dOL/gZ12DHMaAikkgHinsOhJpayGTi644AJmnSRVAPxaRQDxtqqc+iejpg2qWScMnehfKyLUmwDirXd9rIvunHPOcXNi6MS60jZLaPv27e7PPXv2bHaeH8IjgHiHxxJLORDo3bt3SMvE5uCMJokT6NGjR+Ix2BoA4m1rZTXOi7VONC4OoRlDAPE2plT2BMpaJ/bUkkySI4B4J8c+tZ5ZJja1pSfxEAkg3iHCxFTuBBg6yZ0VLSHQFgHEuy0qnIucgDd0Ul1dHbkvHEDARgKIt41VNSAnb+hkw4YNBkRLiPkS2LRpk7uWTb79aJ87AcQ7d1a0DJnApEmTZMmSJSFbxZwuBAYNGqRLKFbGUeCoXV0DHOXl5QF60zXNBN5991159NFH5c477xT16HzYR2lpqWuyrKwsbNPY64TAihUr3BaTJ0/upGV6Py4pKQmWvBLvIEc2mw3S3VffOH3G6UvBSJO/+vp6J5PJOI8//riv66CzTsq2esV52Fy/fHKbM2eOo15Bjnz8BfHj9Y3TXxi+GDYJ9ruP3gEIdOnSRRYsWCCrV68OYIWuEEgnAcQ7nXXXJuuxY8fKypUrZd++fdrERCAQMIEA4m1ClSyOcejQoW52a9assThLUoNA+AQQ7/CZYjEPAmrhouLiYlm/fn0evWiqO4FFixYJmw9HWyXEO1q+WM+BwNSpU6Wioi57n5QAACAASURBVEJqa2tzaE0TUwh069bNlFCNjBPxNrJsdgV9/vnnuwlt2bLFrsTIBgIREkC8I4SL6dwIeDvLM+skN160goAigHhzHWhBQO0sr8ZJ2R5Ni3IQhAEEEG8DipSGEEeNGuWmyfZoaag2OYZBAPEOgyI2AhNQ62AUFhays3xgkskb4IvneGqAeMfDGS85EFBrfFdWVubQkiY6Ezh8+LAbHlMFo60S4h0tX6znQUCt8V1VVSVvvvlmHr1oCoF0EkC801l3LbMePHiwG9e2bdu0jI+gIKATAcRbp2qkPBa1UNXs2bNl+fLlKSdB+hDonADi3TkjWsRIYPz48SxUFSNvXJlLAPE2t3ZWRj5y5Eg3r7ffftvK/EgKAmERQLzDIomdUAiohaqKiopY4zsUmskY2b59u+v4zDPPTCaAlHhFvFNSaJPS5GlLk6rVfqzqOwyO6Agg3tGxxbJPAjxt6RMc3VJFAPFOVbnNSNZ72nLr1q1mBEyUEEiAAOKdAHRcdk5g0qRJsmTJks4b0gICKSWAeKe08Lqnrfa2VE9bsrel7pUivqQIIN5JkcdvhwS8vS03b97cYTs+1I/Azp073RlD+kVmV0SIt131tCYbb2/LVatWWZNTWhI5cuSIqO8tOKIlgHhHyxfrAQhMnDjR3duSDRoCQKSrtQQQb2tLa35il1xyiZsEGzSYX0syCJ9AgeM4ThCz5eXlQbrTFwIdEvjZz34mat63+gIz36O0tNTtUlZWlm9X2gcgsGLFCrf35MmTA1ixv2tJSUmwJJV4Bzmy2WyQ7r76xukzTl8KBv6aXxKPP/64c/HFFzc/meNPmUzGUa84D5vrl2tuc+bMcdQr6JGrv6B+vP5x+gvDF8MmwX730TtiAiNGjGDKYMSMwzZfU1Mjp512WthmsdeCAOLdAgg/6kWAKYN61SOXaFauXClDhgzJpSltAhBAvAPAo2v0BJgyGD1jPJhJAPE2s26pinrMmDFMGUxVxUk2FwKIdy6UaJMoAW+Vwerq6kTjwDkEdCKAeOtUDWJpk4C3yuCGDRva/JyTEEgjAcQ7jVU3MGe1yuDq1asNjDxdIXsLifXp0yddiSeQLeKdAHRc5k9A7W2pZjF44pC/BXrEQaC+vt5106tXrzjcpdoH4p3q8puTPBsTm1MrIo2HAOIdD2e8BCTgbUy8ZcuWgJboDgE7CCDedtQxFVmojYnnzZuXilxJEgKdEUC8OyPE59oQuPDCC91Y1OPXHBBIOwHEO+1XgEH5Dx482I1206ZNBkWdrlC3b9/uJnzmmWemK/EEskW8E4COS38EunTpIrNnz5b169f7M0Cv2AioWnFESwDxjpYv1kMmoJ62rKioEHbXCRks5owjgHgbV7J0Bzxs2DAXALvrpPs6IHsRxJurwCgC3qPyW7duNSpugoVA2AQQ77CJYi9yAjwqHzli3w7q6up896VjfgQQ7/x40VoDAt6j8rW1tRpEQwhNCajZJupLZY7oCSDe0TPGQ8gEvEfledoyZLCYM4oA4m1UuQhWEeBRea4DCPCFJdeAoQTGjx8vlZWVhkZP2BAIToA77+AMsZAAAXaVTwA6LrUigHhrVQ6CyZWAt6v822+/nWsX2sVAYO3atXLWWWfF4AkXiDfXgJEEvF3l2V1Hr/JVVVVJv3799ArK0mgQb0sLm4a01K7yixYtSkOq5AiBVgQKHMdxWp3N40R5eXkerWkKgfAIqC3RHnzwQbn33nuld+/erQyXlpa658rKylp9xoloCCjm06dPl+HDh0fjwCKrJSUlwbJR4h3kyGazQbr76hunzzh9KRj4y/2SqK+vdzKZjLNs2bI2O6nP1CvOw+b65ZJbR/XItw65+MvXZkft4/QXhi+GTYL97qN3ggTUsqPFxcXC+t4JFqGJa++J127dujU5y9uoCCDeUZHFbiwEvHFvloiNBXeHTg4fPux+PmDAgA7b8WE4BBDvcDhiJSECan1vdVRXVycUAW4hkAwBxDsZ7ngNiYBaIlYd27ZtC8kiZiBgBgHE24w6EWUHBNgarQM4fGQtAcTb2tKmJzG2RtOj1vX19XoEkpIoEO+UFNrmNL2t0Rj3TrbKu3btcgPwhrKSjcZ+74i3/TW2PkNPLBj3tr7UJNiEAOLdBAZvzSXAuLe5tSNyfwQQb3/c6KUZAca9NSsI4UROAPGOHDEO4iDAuHcclPGhEwHEW6dqEItvAox7+0YXWke1TIFaroAjHgKIdzyc8RIDAca9Y4DciYvu3bt30oKPwyKAeIdFEjuJE2DcO/ESEECMBBDvGGHjKloCjHtHyxfrehFAvPWqB9EEIMC4dwB4dDWOAOJtXMkIuCMCjHt3RCfaz9SWdN5fP9F6wroigHhzHVhFgHHvZMvJRgzx8Ue842ONpxgIeHd+rHMSA2xcJEoA8U4UP87DJsC4d9hEsacrAcRb18oQl28CjHv7RkdHgwicbFCshAqBnAioce+pU6fm1JZG4RCoqalxDXnDVuFYNdvKxo0b3QRGjx4dSSLceUeCFaNJEkBAkqSPb0WgtrZWLr/8cvfliXjYZBDvTojOmDFDCgoKOmnFxzoR8Ma9dYqJWPIjoP7Pqf97ph49evSQHTt2SGFhoSvgS5YsCT0VxDt0pBjUgYAa9+aAQJIE1E3E8uXLXQH//ve/L2VlZR2Es1eW33iee6NYUPBNebjqiEjDQdn86I3Sr6BAznu4qlVfxLsVEk7YQECNe3PER4D9K9tm3b9//0YBnzlzZgcCfqZMefYd+XTfMinu+2t5ZOkr8pvF/ybbJz4uBxxH3vlhYSsHfGHZCgknbCDAuHe8VWT/yvZ5ewJ+3333iRLwo0ePSnsPM2XOuFy+d0OhVCy8X3657Ffy1JAvtmu4wHEcp91Pc/jgpJNOyqGVuU0aGhrc4DMZ/kgxt4pEbhoB2//fffOb35SXXnqpjbI0yNE198iQK9+R+6uXSPGgz7fR5vgpFKldNHwAAQhAIBoC7Q/r/V3+fOSoiGyQ/9qwW47fOrYTg7rzDnJks9kg3X31jdOniKi/THzF6adTnLmp+Gz2l8lkHPWqr6/3UwpffWzm2VFuy5Ytc1n7gtZGp7j/36kQOsqvjRBzPqWuv+nTp7t81L/q5/Z8fbpvmfPPxbc7t4/r6/S94zXnTx144c67nV9qnLaHAOucRF/LAwcOuLMqovdklodjx47JbbfdJhUVFe4WcY899ph06dKl7SQa3pfKBb+TCfN/JP98w2Vy8BerZcv+TfLo3S/J/jZuwRHvtjFy1iIC27ZtsygbPVPZs2ePjBs3Ts/gEooqZ+H+e5U8fF6BFFz5ryK3/4tMOaOb9LvoKzLu4M9lwWMH5Bt3F8kZbSh1G6cSylRTt9lsVo2ZaBpdusJqOLhZnp17tTsXtt+NT8jmgx/nBGD9+vU5taORPgTU/zn1f8/UQz1h+d3vfte9416wYIF0eMd9cqH88B1HnN/+RKYMUrNLMtKtcJb81jkgv/3ZFBnUrW2ZbvusqcSI214CdZvlkevnS/XVFfKp8zepuvGwzL3+Camqa+PvyRYU1J+s6i6IAwJxEVBPWI4fP14WL14sd999d/tDJQECQrxzhndUan7zqNzYr+D4U1BfmyvPVh3s+NvgnG3TsGMCf5WapQ/LI5fcLndecYZk5BTpe8WtMu+SF+XupTU51YBx744Ja/NpXY385uHjTxXOmHGXfG3uL6Qqx7+wtMnhs0BKS0tFvaI6EO+cyH4s+1/6hfxaviFPHHDE+fSA/O6aP8pdI/9JHlGPsXJES6Bht2z4rzfkgsH9pFujpx4y8uqx8vv/+r/yTuc33+I9RNLYnTehEghlC7SG9+Wlp9aIXHP8qcInfjpdrvnjQhmZ419YoSZkgDHEO5ciNeyXP3zpGpl51aDj4pHpK5fMvEvun/CGPLL0DVGzMjmiI9Dwzv+V/3r1NBk+oGfrfftefVk2vPPXDp0XFxfLpk2bOmzDh8EJtPfUYK6WG97bJ1/6drFc5Y77ihR8aaDM/JdZMmHtc7J0c22uZlLTDvHOpdSZc2TcuDObC0empwwY3i+X3rQJRKBB6va9I7+Xc2Vw/xP33fmYHDNmjKg7Qw69CWQGjpZxZ5zSLMjM6QNkeN9mp/jhMwKId6BL4X/IxEsHNvlTPpAxOkdE4MILL3QtexsGROQGs1ER6DpKLh3c/hofUbnV3S7i7bdCR7fJK29+XWZMaHFH7tce/dohkJFu/c+TC2SXVO+ra6dNx6cHDx7sNti+fXvHDfnUFwE1LS6aw5GjW9bKm7dMkwkt7sij8WeWVcTbV72OSNW//bf0+sn3pbCdOZi+zNKpTQKZ874q1034XIvPPpY/7n5HDk74ulx2XvuL96hO6om2oqIixr1bEAzrx8OHD7umQl/J8W/vy78921N+cstI/rpto1gpFu+/y8Hlt3y2+Pln0/8KWv87497fSNXfm5JrkLqqX8rSHtPllsLTmn7A+6gIZAbIZdedIb94ZVuTL4frZF/1fplw3VflvByuYjXnlnHvqAoUhd0jsmfj/5Med93EDVI7eHO47Nvpafzpk6XvlCfdpyfV01ztvbLzr5LCJqueN+x/WZ7afpnMKx7G3UBs18DnZdC3fyi3b35EHlyzXxrkYzm45glZsPl/y0++Paj5F8ntxDRixAj3E8a92wGk1Wk1NfeXcuB/XSnFHaxnrVXICQSTYvHOn3bDwTWyeOnn5Ts3eMLdIHU7X5SKtcf/bMzfIj1yJtDtErn9+buk17Nfl5MKBsj1r5wrDz9/a853ZUOHDnVdMe6dM/GEGqpfzBWy9NQiubSPNxx2VHY++wtZezSHCf0JRZ2EW8Q7J+oNUlezXO66/h/l9tuvlH4necMrJ8mpQ5eK9PM3hS0n1zRqJJDpO1pmPft7cdw1H26Qwr7Np5U1NmzjjXpcWY1779y5s41PORWEgPcL8cwzzwxiRkSOSs3yH8v1V5bI7Vf2l1s/2/y7oOBLMvT5T6Qf3y8144t4N8PR9g8N+ytl5rhvycK1B1s3yOELs9adOJMEATXuXVlZmYTrVPhsd6nTnLL/WPYvv1vGfeunsrZV+745f7fRqqvFJxDvHIqbOWOKPK0ei29rbPyVYhkExRwoJt9E7eZdVVUl+/btSz4YImhB4BQ5Y0qZu9mu9//MW9FT/aX1SvGQnL7baGHU6h+RHavLS3JNCZx//vnuj2+//XbT07yHgJEEEG8jy0bQfgioXbwLCwtly5YtfrrTpx0C7KLTDpiITyPeEQPGvF4EJk2aJJs3b9YrKMOjYRedZAqIeCfDHa8JERg5cqSsXLlSonukO6HEcJs6Aoh36kqe7oS9ce8dO3akGwTZG0+gQO0sHySL8vLyIN3pC4HICHi7mJSVlTXzoc5PnTpVxo4d2+w8P/gj8Nxzz0nXrl1l8uTJ/gyktFdJSUmwzJV4Bzmy2WyQ7r76xukzTl8KBv58XRJtdspkMo56tTzmzJnjXHvttS1Ph/KzzfVrLzfFeNmyZaHwa2qkPX9N24T5Pk5/Yfhi2CTY7z56G0hg1KhR7rg3mxIbWDxCbiSAeDei4E1aCHhLl7IpcVoqbmeeiLeddSWrDgioJy3VsW3btg5a8REE9CaAeOtdH6KLiMDs2bNl/fr1EVlPj1lvid1zzz03PUlrkinirUkhCCNeAmropKKiQhj3Doe7mm3CES8BxDte3njThIC3KfHevXs1iYgwIJAfAcQ7P160toQAmxJbUsgUp4F4p7j4aU5drT1dXFzMpsQBL4LwNmIIGEgKuyPeKSw6KR8ncNFFF7EpcUgXQ7CNGEIKImVmEO+UFZx0TxBgU+ITLHhnHgHE27yaEXFIBLxNiXfv3h2SRcxAID4CiHd8rPGkGQFvU2I2Z/BfmE2bNrnfHfi3QE+/BBBvv+ToZwWBSy65hE2JA1aye/fuAS3Q3Q8BxNsPNfpYQ0BtzsCmxNaUM1WJIN6pKjfJtiTgbc7w/vvvt/yInyGgNQHEW+vyEFzUBLxNibdu3Rq1KyvtL1q0SLxVGq1MUOOkEG+Ni0No8RAYN26crF69Oh5nFnrp1q2bhVnpnxLirX+NiDBiAt7mDGxKHDFozIdKAPEOFSfGTCTg/dm/Z88eE8Mn5pQSQLxTWnjSPkGAzRlOsMjnHWt550Mr/LaId/hMsWggATZn8F801vL2zy5IT8Q7CD36WkOAzRmsKWVqEkG8U1NqEu2IAJszdESn7c/q6+vb/oCzsRBAvGPBjBPdCbA5Q/4V2rVrl9vJ+84gfwv0CEIA8Q5Cj77WEGBzBmtKmZpEEO/UlJpEOyPA5gydEeJznQgg3jpVg1gSJcDmDPnhP3DgQH4daB0qgQLHcZwgFsvLy4N0py8EIiNQWlrq2i4rK8vJx1/+8heZO3eu3HrrreJt1JBTx5Q2WrFihZv55MmTU0ogWNolJSXBDCjxDnJks9kg3X31jdNnnL4UDPz5uiTa7JTJZBz1yue49tprnQceeCCfLs3a2ly/lrnNmTPHUa+ojpb+ovLj2Y3TXxi+GDYJ9ruP3pYRYHMGywpqcTqIt8XFJbX8CbA5Q+7M1q5dK2eddVbuHWgZKgHEO1ScGDOdAJsz5F5BtQNRv379cu9Ay1AJIN6h4sSY6QTU5gzqYHMG0ytpf/yIt/01JsM8CahFqticIU9oNI+dAOIdO3Ic6k7A25zh2LFjuoeaWHwsB5sY+kbHiHcjCt5A4DgBb3OG6upqkHRCgOVgOwEU4ceId4RwMW0mAW+hpW3btpmZAFGnggDinYoyk2S+BIqLi2X9+vX5dktN+0OHDrm5cuedXMkR7+TY41ljAmPGjJGKigqNI0w2tA8++MANwJudk2w06fSOeKez7mTdCQFvcwbvi7lOmvMxBGIngHjHjhyHJhDwnhzcvn27CeESYwoJIN4pLDopd06gR48eUlRUJJs2beq8cQpbKC7qewGO5Agg3smxx7PmBMaPHy9q/Q6Otgl079697Q84GwsBxDsWzDgxkYDanEGt37Fv3z4Twydmywkg3pYXmPT8Ezj77LPdzm+//bZ/I5b2VF/knnbaaZZmZ0ZaiLcZdSLKBAioaXCFhYWyZcuWBLzr7XLlypUyZMgQvYO0PDrE2/ICk14wApMmTZLNmzcHM0JvCERAAPGOACom7SGgNmdQd5m1tbX2JEUmVhBAvK0oI0lERWDAgAGu6R07dkTlwji73oNL5557rnGx2xQw4m1TNckldALeIlVsztAaLeuatGYS5xnEO07a+DKSgNqc4a233jIydoK2lwDibW9tySwkAmpzBrVIFZszHAe6e/du903Pnj1DIowZPwQQbz/U6JMqAt7YLpszHC97XV2d+0YtIcCRHAHEOzn2eDaEwODBg91I2ZzBkIKlJMwCx3GcILmWl5cH6U5fCERGoLS01LVdVlYW2Mdzzz0n6gu6yZMnB7ZluoF169a5C3bNnTvX9FQSjb+kpCSYfyXeQY5sNhuku6++cfqM05eCgT9fl0SbnTKZjKNeYRzPPvtsTrZsrp+X25w5cxz1ivrw/EXtx7Mfp78wfDFsEux3H71TQmDgwIFupt4c55SkTZoaE0C8NS4OoelDYOjQoW4wbM4g8tFHH+lTmBRHgninuPiknjsBb3OGnTt35t7J0pZq2qSaPsmRLAHEO1n+eDeIgNqcobKy0qCICdVmAoi3zdUlt1AJqEfl2ZwhVKQYC0AA8Q4Aj67pInD++ee7Cad5cwbvC1vvwaV0XQF6ZYt461UPotGYgLc5gydgGocaeWgsShU54k4dIN6dIqIBBE4QUJszrF69+sQJ3kEgIQKId0LgcWsmAbX1V5o3Z/CmSrIoVfLXL+KdfA2IwCACw4YNc6NN++YMLEqV/EWLeCdfAyIwiIC3OcO7775rUNSEaiMBxNvGqpJTpATU5gzr16+P1IeuxtVDSkVFRbqGl6q4EO9UlZtkwyCQ5s0Zjhw5It5fH2GwxIZ/Aoi3f3b0TCkBb44zmzOk9ALQJG3EW5NCEIY5BLzNGXbt2mVO0CFFunbtWjnrrLNCsoaZIAQQ7yD06JtKAl26dJHi4mJ3Q4K0AVDLA/Tr1y9taWuZL+KtZVkISncCY8aMkUWLFukeJvFZTADxtri4pBYdgTRuzqC+rFRHnz59ogOL5ZwJIN45o6IhBE4QSOPmDB9//LELoFevXidA8C4xAoh3YuhxbDIBb3OGTZs2mZwGsRtMAPE2uHjpCr1B6mpekYdvvEAKCgqkoOBqmfvsZjnYkBwFtTmDmn2RluPAgQNuqqxrokfFEW896kAUnRBo2P+yPPVrkWueeEsc529y4HfXyB/vmiTXP7JZ6jrpG9XHI0aMSOXmDKxrEtUVlZ9dxDs/XrROhMBf5b0/9JBvz7xaBnVTl+wp0veSm+Vf7r9M1j5SKZuPJnP7ffbZZ7s00rw5QyKXA05dAog3F4IBBD4vA8eNkjOaXa2nyOkDzpO+CUbvbc6wZcuWBKOIz/V7773nzm+PzyOeOiLQ7L9DRw35DAI6Eug6caQMdu/Gk4lObc6Qpk2Ju3fvngxovLYigHi3QsIJMwjUypZX9sstM65ocUceb/QjR450x71ra2vjdZyAt/r6+gS84rI9AgWO4zjtfZjL+fLy8lya0QYCIRJw5G971sovq8+R71w1QD73meUZM2bk5CObzebULpdG6sGVe+65R2bNmiXegzu59DOxTWlpqUyfPl2GDx9uYvjaxVxSUhIsJiXeQY5sNhuku6++cfqM05eCkSp/B5Y500TUzUMHr3HOQ1v+3Pw6+fPvnEfuWOLs+POnzc53bOeEj2adQvghk8k4jz/+uGvJ5vqpPJctWxYCsdxM2MwyjNwYNgn2u4/eQQj0nSLPOo6oP/7af/1WfljY7YSXhvflpafek4nzvidDWox1t7SRyWREvVqeP2EsnHdqcwbbNyVOw7BQOFdDfFYQ7/hY4ykogYb9smbxKjn1O5NPCHfddnm2YqMcDWo7QH+1OYPtmxIfPnzYJeTt4RkAF11DIoB4hwQSMxETqNspy+8qlitvL5Er+33us6csC6Tg1AnyvPwPaXJvHnEgrc17grZnz57WH3IGAhERODkiu5iFQHgEGt6X5TOnyrcqtrdh8zK57rIBkuRdiLct2LZt29qIz45Tu3fvdhPp2rWrHQlZkEWS17wF+EghFgKZs2XK079vNXZ9fCx7qRQP+nwsYXTkxPZNievqji9CoB5M4tCDAOKtRx2IwnACauikoqJCPvnkE8MzIXxTCCDeplSKOLUmcOGFF7rxffDBB1rH6Tc4tfTtV77yFb/d6RcBAcQ7AqiYTB8Bb1Pi/fv3W5s84916lRbx1qseRGMoAW9TYm/Na0PTaDfsjz76qN3P+CAZAoh3MtzxaiEBtSnxa6+9ZmFm4o7nn3POOVbmZmpSiLeplSNu7Qh44941NTXaxUZA9hFAvO2rKRklRMAb996+va356AkFFYLbffv2uVZOPfXUEKxhIiwCiHdYJLGTegJq3PuCCy4Q2zYl9paCRbz1usQRb73qQTSGExgyZIgsWrTI8Cyah++Jd/Oz/JQ0AcQ76Qrg3yoC3hOINo1779q1y61R7969raqV6ckg3qZXkPi1InD66ae78dg27q0VZIJxCSDeXAgQCJHAF77wBSkqKrJq3Hvnzp1SWFgYIiVMhUEA8Q6DIjYg0ITA+PHjZe3atU3OmP1WbfU2btw4s5OwMHrE28KiklKyBEaMGOFuSuxNsUs2muDeeboyOMMoLCDeUVDFZqoJDB061M3/7bfftoKDWi1R7RbEoRcBxFuvehCNBQR69Ojhjntv2bLFgmxIQVcCiLeulSEuowmoce/Kykqjc1DBe0M/ffr0MT4X2xJAvG2rKPloQUBtjVZVVdUofloE5SMI7wGdXr16+ehNlygJIN5R0sV2agmcf/75bu6mj3sfOnQotTXUPXHEW/cKEZ+RBNSTlmputOnj3t7OQN4my0YWw9KgEW9LC0tayROYNGmSFePeyZMkgrYIFDhqC+4AR3l5eYDedIVAdARKS0td42VlZdE56cDyjh075IknnpD7779fTjvttA5a6vvRihUrRI17f+9739M3SEMjKykpCRa5Eu8gRzabDdLdV984fcbpS8HAn69Los1OmUzGUa84j6b127t3r+v/lVdeiSyEpv6icDJnzhxHvdQRta+W8dvsL4zcGDYJ9ruP3hBol4AN497qMf+zzjqr3Rz5IDkCiHdy7PGcAgKmj3ur6Y79+vVLQaXMSxHxNq9mRGwQgZEjRxo737u2ttYl3a1bN4OIpydUxDs9tSbTBAiYPN/78OHDLrEBAwYkQA6XnRFAvDsjxOcQCEDA5HFv7+nKAOnTNUICiHeEcDENAUXA1HFvb/szHtDR8zpGvPWsC1FZRMDkcW+LymBdKoi3dSUlId0ImDruvWnTJikuLtYNJ/F8RgDx5lKAQMQETB737t69e8R0MO+XAOLtlxz9IJAHARPHvRctWsQDOnnUOO6miHfcxPGXSgKmjnvzgI6+lyvirW9tiMwiAqaNe7ODjv4XH+Ktf42I0AICaty7qKjImPW9vTne7KCj78WHeOtbGyKzjIBJ+1ru3r3bpd+zZ0/LqmBPOoi3PbUkE80JjBgxwph1Turq6lyaPXr00JxqesNDvNNbezKPmcDQoUNdjybsa7lz5053mCdmRLjLgwDinQcsmkIgCAF1F6vGvVevXh3ETCx9jxw5IjwWHwtq304Qb9/o6AiB/AmocW81f1r3g00YdK+QCOKtf42I0CICatxbHTU1NVpnxSYMWpfHDQ7x1r9GRGgRAW/ce/v27dpmxRxvbUvTLDDEuxkOfoBAtATUuLda7Ekt+qTrwRxvXSvTPC7EuzkPfoJA5ATGjBnjjnsfO3Yscl9+HDDH2w+1+Psg3vEzx2PKCVx44YUugerqai1JMMdbFHnZgQAACsFJREFUy7K0CgrxboWEExCIlsDgwYNdB9u2bYvWkU/rrOPtE1zM3RDvmIHjDgJdunRxx73Xr1+vLQzW8da2NI2BId6NKHgDgfgIqHHviooK0XHcW81DHzZsWHww8OSLQIHjOI6vnp91Ki8vD9KdvhCIjEBpaalru6ysLDIffg1/+OGHMn/+fJk1a5YMHDjQr5nQ+33yySduTNOnT5fhw4eHbh+DJwiUlJSc+MHPOyXeQY5sNhuku6++cfqM05eCgT9fl0SbnTKZjKNecR751E/F9vjjjwcKLx9/uTiqrq52mal/Wx5h+2ppv+XPNvsLIzeGTfz8xqMPBEIgMHv2bO3WOTl06JCbWdeuXUPIEBNREkC8o6SLbQh0QECtc7Jy5Uqpra3toFW8H33wwQeuQ7V5BIfeBBBvvetDdBYT8LZG27JlizZZshSsNqXoNBDEu1NENIBANATU3W1hYaFWW6OxFGw0tY7CKuIdBVVsQiBHAtOmTZPKysocW0ffjGmC0TMOywPiHRZJ7EDABwFvazQdloj15px369bNRyZ0iZsA4h03cfxBoAkBnZaI3bt3rxsZD+g0KZDGbxFvjYtDaPYT8JaIXbVqVeLJsppg4iXIKwDEOy9cNIZA+AR0eVSe1QTDr22UFhHvKOliGwI5EBg1apTb6o033sihdXRNWE0wOrZRWEa8o6CKTQjkQcDbpX3r1q159Aq/qfrS9JxzzgnfMBYjIYB4R4IVoxDIj8CCBQtkyZIl+XUKubV62nPIkCEhW8VcVAQQ76jIYhcCeRAYOXKkqB3bvc1/8+gaSlNvquK5554bij2MRE8A8Y6eMR4g0CkBJd7q2Lx5c6dto2jAglRRUI3WJuIdLV+sQyAnAmrKYFFRkSQ1ZdBbkMobf88paBolSgDxThQ/ziFwgsCUKVMS212HmSYn6mDKO8TblEoRp/UEvF3lk5gyyEwT8y4vxNu8mhGxpQTUtmNqlcEkpgwy08S8iwrxNq9mRGwxgUmTJsU+ZZCZJmZeUIi3mXUjaksJjB07NvYpg6xpYubFhHibWTeitpSAt8rgmjVrYsvQu/Nm67PYkIfiCPEOBSNGIBAOAW+VwfXr14djMAcre/bsEbUZModZBBBvs+pFtCkgMHHiRHfKYFwbE7N7jpkXFeJtZt2I2mICl1xyiZtdHBsTe4/jDxw40GKidqaGeNtZV7IymIAae1ZPW65evTryLN5//33XR69evSL3hYNwCSDe4fLEGgRCIaCetlTDGd6+kqEYbcPIu+++657lsfg24Gh+qsBxHCdIjOXl5UG60xcCkREoLS11bZeVlUXmIyrDajjjwQcflFmzZkmUQxorVqxwU5g8eXJUqWC3HQIlJSXtfJLjaSXeQY5sNhuku6++cfqM05eCgT9fl0SbnTKZjKNecR5h1u/iiy92HnjggQ7DD+pP8Xn22Wc79OF9GNSXZyfXf232F0ZuDJvk+EuOZhCIm8C0adNk3rx5kQ2d8GVl3BUN1x/iHS5PrEEgNAKXXXaZa6u6ujo0m00Nvf322+6PZ599dtPTvDeEAOJtSKEIM30EvIWqNmzYEEny6slKtRAWT1ZGgjdyo4h35IhxAAH/BNTQSVR7W6qpiGohLA4zCSDeZtaNqFNCQA2dqL0t33zzzVAzVk9vsgxsqEhjN4Z4x44chxDInUBUQyc7duxwgxg2bFjuwdBSKwKIt1blIBgItCaghk5mzpwZ6qwTb8MHHs5pzduUM4i3KZUiztQS8GadhLk9mhrvZiVBsy8pxNvs+hF9Cgh4Qyfr1q0LJVtvvHvUqFGh2MNIMgQQ72S44xUCeRHwHtgJY5lYb7zbW70wr0BorA0BxFubUhAIBNon4E3pC2OZWHUHz/zu9lmb8gnibUqliDPVBLxlYrPZbGAOlZWVou7kOcwmgHibXT+iTxGBGTNmuHOzvTVJ/KSu5oureeMjRozw050+GhFAvDUqBqFAoCMCY8aMcT9Wd85+D+9R+y9/+ct+TdBPEwKItyaFIAwIdEagS5cusnjxYvdxeb+bNKhH7RcsWCDKFofZBBBvs+tH9CkjMGHCBHfYw8/u8t6QydixY1NGzc50EW8760pWlhJQT0Sq/S39fHGphkzULBOGTOy4OBBvO+pIFikiMGfOHPeLy40bN+actRpmUY/Yq1kmDJnkjE3rhoi31uUhOAi0JjB69Gj3Dvqhhx5q/WE7Z1atWuV+ooZdOOwggHjbUUeySBmBO++8M69pg2oz4+LiYmEhKnsuFMTbnlqmK5OG92X5zSPl6oqd0pCuzN1sJ06c6N59//rXv+40++XLl7tfct50002dtqWBOQQQb3NqRaSNBD6W/ZUPSWnFgcYzaXujxq0fffRR+f3vfy+vvvpqu+mrsW51162+5FTDLRz2EEC87allSjJpkLqqJ+XujZ+Tq/qmJOV20lRi/JWvfEXuueceae+pSyXw6onKhQsXtmOF06YSQLxNrVxa467bIk8+JnLb7V+X09PKoEne3/jGN1xxvu+++1pt1qCGS+bNmyfPPPMMY91NmNny9mRbEiGPNBA4IlVPPi9y24+l8NQ3ZGkaUu4kx9NOO01ef/11ufzyy+XQoUOiphH26tVLXnzxRVe41ZeULELVCURDPy5wHMcJEnt5eXmQ7vSFQI4EHPnbnrXyy+pz5DtXDZDP/bVaKu/7uewt+j9S+tU+UiAiauGmXA4/D7jkYjfJNmrY5LnnnpO9e/c2hnHDDTfIpZde2vgzb/QiUFJSEiwgJd5Bjmw2G6S7r75x+ozTl4KBv3YuiT//znnkrkpn36efff6n15w7+vZ1Jjy9w/FOiYi6Een01Y6HUE4nXb+9e/c61dXVTn19fSj5NDWSdG5NY4nifZz5heGLMe9gv/voHYTAweVyY0GBFHT4+po8XLVfqp5aI2ffOlHO6OCKVX9ENn1lMhlRr6bnAv6hGSTbWPqqdb/VXG6eoowFd6JOOvivkGhcOE8Dgb5T5NkWgttSaB3nt/LD/1ktSx+6S77V/3MnhP5LV8rCgwfl1ZuHykkF35aKmr+mgRg5QqCRAOLdiII32hL44hXyswPN76qdP70md/TtKxOe3iGfOkuleNDntQ2fwCAQBQHEOwqq2IQABCAQMQHEO2LAmIcABCAQBQHmeUdBFZvRE3CHUtL7eHz0gPGgOwHuvHWvEPFBAAIQaIMA4t0GFE5BAAIQ0J0A4q17hYgPAhCAQBsEEO82oHAKAhCAgO4EEG/dK0R8EIAABNoggHi3AYVTEIAABHQngHjrXiHigwAEINAGAcS7DSicggAEIKA7AcRb9woRHwQgAIE2CCDebUDhFAQgAAHdCSDeuleI+CAAAQi0QQDxbgMKpyAAAQjoTgDx1r1CxAcBCECgDQKIdxtQOAUBCEBAdwKIt+4VIj4IQAACbRBAvNuAwikIQAACuhNAvHWvEPFBAAIQaIMA4t0GFE5BAAIQ0J0A4q17hYgPAhCAQBsEEO82oHAKAhCAgO4EChzHcYIEWV5eHqQ7fSEQGYHS0lLXdllZWWQ+MAwBvwRKSkr8dnX7BRbvQN7pDAEIQAACvggwbOILG50gAAEIJEsA8U6WP94hAAEI+CKAePvCRicIQAACyRJAvJPlj3cIQAACvggg3r6w0QkCEIBAsgQQ72T54x0CEICALwL/HyRaPKYWpts+AAAAAElFTkSuQmCC"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> has an inverse function.</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>For this value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}\left( x \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, stating its domain.</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>attempt to differentiate and set equal to zero<em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 2{{\text{e}}^{2x}} - 6{{\text{e}}^x} = 2{{\text{e}}^x}\left( {{{\text{e}}^x} - 3} \right) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span><em><strong> A1</strong></em></p>
<p>minimum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{ln}}\,3">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = {\text{ln}}\,3">
<mi>a</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</math></span><em><strong> A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> Interchanging <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> can be done at any stage.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\left( {{{\text{e}}^x} - 3} \right)^2} - 4">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x} - 3 = \pm \sqrt {y + 4} ">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mo>=</mo>
<mo>±</mo>
<msqrt>
<mi>y</mi>
<mo>+</mo>
<mn>4</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p>as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \leqslant {\text{ln}}\,3">
<mi>x</mi>
<mo>⩽</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{ln}}\left( {3 - \sqrt {y + 4} } \right)">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>−</mo>
<msqrt>
<mi>y</mi>
<mo>+</mo>
<mn>4</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>R1</em></strong></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}\left( x \right) = {\text{ln}}\left( {3 - \sqrt {x + 4} } \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>−</mo>
<msqrt>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>domain 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> is <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4 \leqslant x < 5">
<mo>−</mo>
<mn>4</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo><</mo>
<mn>5</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let <em>f</em>(<em>x</em>) = <em>x</em><sup>4</sup> + <em>px</em><sup>3</sup> + <em>qx</em> + 5 where <em>p</em>, <em>q</em> are constants.</p>
<p>The remainder when <em>f</em>(<em>x</em>) is divided by (<em>x</em> + 1) is 7, and the remainder when <em>f</em>(<em>x</em>) is divided by (<em>x</em> − 2) is 1. Find the value of <em>p</em> and the value of <em>q</em>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to substitute <em>x</em> = −1 or <em>x</em> = 2 or to divide polynomials <em><strong>(M1)</strong></em></p>
<p>1 − <em>p</em> − <em>q</em> + 5 = 7, 16 + 8<em>p</em> + 2<em>q</em> + 5 = 1 or equivalent <em><strong>A1A1</strong></em></p>
<p>attempt to solve their two equations <em><strong>M1</strong></em></p>
<p><em>p</em> = −3, <em>q</em> = 2 <em><strong> A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the polynomial <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q(x) = 3{x^3} - 11{x^2} + kx + 8">
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>11</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>k</mi>
<mi>x</mi>
<mo>+</mo>
<mn>8</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q(x)"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> has a factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x - 4)"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, factorize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q(x)">
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> as a product of linear factors.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q(4) = 0"> <mi>q</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="192 - 176 + 4k + 8 = 0{\text{ }}(24 + 4k = 0)"> <mn>192</mn> <mo>−</mo> <mn>176</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>8</mn> <mo>=</mo> <mn>0</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mn>24</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = - 6"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> </math></span> <strong><em>A1</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{x^3} - 11{x^2} - 6x + 8 = (x - 4)(3{x^2} + px - 2)">
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>11</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mn>8</mn>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p>equate coefficients of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>: <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 12 + p = - 11">
<mo>−</mo>
<mn>12</mn>
<mo>+</mo>
<mi>p</mi>
<mo>=</mo>
<mo>−</mo>
<mn>11</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 1">
<mi>p</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x - 4)(3{x^2} + x - 2)">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x - 4)(3x - 2)(x + 1)">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow part (b) marks if any of this work is seen in part (a).</p>
<p> </p>
<p><strong>Note:</strong> Allow equivalent methods (<em>eg</em>, synthetic division) for the <strong><em>M </em></strong>marks in each part.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></math>.</p>
<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> is shown below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWUAAAD5CAYAAAD/ViQ3AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAA9KSURBVHhe7d1/bJXVGcDxB2PMcKhbA4hQbixZ+dVsVdwQWpbZFYeBQQrdxg87+TFIZIWONkugxWZ/QAskDEiRuKgDZI2iG3IDQmiow6gtIBOBWCarWimFQmYaYSiGGFif0/dC1QKltPc+73u/n3/ee99LiEu6bw/nnnPebpebCQDAhNu8KwDAAKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogyDTkh49o8kFEqU0JA/SPjUxZbb596QxUMmyuqDn7W8BwKIKMOg/pL1/CF5f9Ms+e7n78qB/5xrud1joDz66zukpv5/Le+BACLKMOo2uXvwMEmXc3LmswverQTpN2CojErp1fIeCCCiDLt6hiQloUn+Vfdf+ar57aVTu6Ts45/JpOTvtHwOBBBRhk98Jof+USuZT6bJ3d4dIIiIMuy6vZck/ThBmmo+kbqDr0g4NFkm9L3D+xAIJqIM+xq2yZ/D98mTE0L8wCLw+BmHYd+XUMp9zdcfyJTfj5G+/LQiDvBjDsO+lHOfJkp+6Tx5pA/TFogPPA4KRl2U029slFfklzLvkb6MHhA3iDIMuSTnDz4tv3q8XqaU3Cu158ZK4YwU6eF9CsQDBiAw5dK5T+X458ek9rYJBBlxiZEyABjCSBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZZgWDodl7ty5cuGCd9A9EHBEGWY1NTXJ6tWrZMeO7VJRUeHdBYKNKMOslStXytix49zrvLx5cvLkSfcaCDKiDJOqq6ulquptmTNnjnufmztfli5d6l4DQUaUYY7OHxcWLpLi4mJJSEhw9/Ly8uTo0RqprNzt3gNBRZRhTllZmaSnj5LRox/17oh07969+f5aWbJkiZtrBoKKKMMUnTdet26t5ObmeneuSk1NdbF+8803vTtA8HBKHMzRkXBk2kKFQolSX9/gvfv250CQMFKGOTcKLkFGkBFlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMIQoA4AhRBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMIQoA4AhRBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGdLvczHsNmBQKJUp9fYP3zt9qa2ulpqZG3nnnHfnwww9l375q75OrkpIGSHr6KBk6dKhkZGRIv379vE8QD4gyzPNzlC9cuCDvvfeevPXWW7Ju3Vrvrsi4cePl/vvvl0GDBnl3rmpsbJQjR47Ijh3b3fsRI9JkwYIFkpaW5t4j2IgyzPNjlHVEXFlZKcuWlbj3GlYd9WpYBw4cKN27d3f3rycS9DVr1rgRtf4dJSUlkpyc7P0JBBFRhnl+inJ1dbW89tprUl6+yb0vLFzsQpyamured5T+vZE4l5U9LVlZWd4nCBqiDPP8EOXDhw+7UaxGU+eE8/MLZMyYMe0aEbeXjpzLysrcNIjGfu7cud4nCBKiDPMsR1mnKTZs2OBGxhrjZcuWd/nc7zPPPOOmRQhzMLEkDuiApqYmF8fMzAypqnrbTSns2lURlS/jNMQaZA1zeXm5dxdBwUgZ5lkbKVdW7pYlS5ZIXd3HLo4zZszo1GmK9oqMmLdv33HLc9awgyjDPCtRPnnypCxdutQtVdMlbQUFBTFdCaFzzNOnT5czZ067UXosfjGg8zF9AbRDOByWkSMflqNHa9xUhY5SY700TSOsXy7qiF2/AEQwEGXgOnTuuKioSPLy5rnR8datYVPL0fQXQ2npcrciQ790hP8RZeAadJnbxIlZbmVFZHSckJDgfWpHdna2W/mhq0Dgf0QZaIOuahg/fpzce28f2bt3v+nNGjqNoeui9ZcHo2X/I8pAKzpdoUvOiooWSW7ufHnhhRd8cSCQblRhtBwMRBnw6ChTpyt0dcX69Rtk4cKFvlnRoP+dU6ZMdaNl/cUC/yLKQDNde6wbQZROV4we/ah77SeTJ0921507d7or/IkoI+7pF3izZs10qyt0va9fzy/WLyH1f8O2bdu8O/Ajooy4pZsvdP44co6ExtnvGzCysye5Q5F0owv8iSgjLmm0dEdeZP44KAf7DBv2kLvu2bPHXeE/RBlxR4M8bdpUtztv8+ZXfDl/fC1MYfgfUUZc0cPiNcjqxRdfCuQjlnR5nE5hsArDn4gy4oYGecqU37gNIRrkoD6QNCUlxV0/+OADd4W/EGXEBV3ypkHWf9r7ZUNIR0UOStJt4vAfoozAa73kbdWqVb5fYdEeOTlP8GWfTxFlBJoGWZe86ZbpeAmyGj58uJtX1mV/8BeijMCKBFnXIPtpy3RnSEpKcteGBn88BRxXEWUEUusgx+PDRfv37++uNTU17gr/IMoInHgPsoqc+9zY2Oiu8A+ijEAhyFfpl30nTpzw3sEviDICgyB/3T333CNVVW977+AXRBmBQJC/bdCgQe6hqvAXogzfI8ht6927t7tyYpy/EGX4GkG+tl69ernrF1984a7wB6IM3wqHwwT5Ou688053Jcr+QpThS3q4UF7ePLd1miC3LXK+R11dnbuii5wOy+xQooRCw+XJcL1ckoty+sBfZPaQ5nsPrJKDX3l/rp2IMnwnctpb5CwLIKb6ZMnz9e9LOL+f7Fy/Ww4d2Chra34qa/7dIPWHCmTY7d6fayeiDF/RJ063DnI8bZ2GZd+TYVOnyy8O/Uke/2tvyX0iRXp4n9wsogzf0FUEs2f/TpKSBhDkdtINJMeOHfPeoS36ZbH+sr9lvYfIqAf6SvpjD0nfWygrUYYv6FM0Wj8xhCC339mzZ71XaMtdd90lmZkZUl5e3gmn6p2VqgMfyTnvXUcQZfjC4sWL3UaIID8xBLGRk5Mje/ful6qqKnnssTEdfDjARTm1bavU/nCkyKv/lIPnLnn3bx5Rhi/oU6f1IacEGV1Bf650GiM/v0Dy8ubLihUrbuIZh5fk/LG/y7r6TClcOEMmyUfyyZnzcipcJn+r/dL7M+3X7XIz73WbQqFE7xUAxJdDh45cOXGvLZdqN0pW5jK5+NsVsrZwgiT3aJI3FmfJE68Okj9uKpF5P+lz0yPfG0YZt05/sZWWLnf/TEL7RXbrqfp6DmtH19M55S1btkhR0aKY/X+W6Yso6dGjowtk4lPr3XpANOhc8vTp093c8uuv74nZIIoow5zIbj1dzsVuPXQ1HR3rHPL48eNk2rRp7l9okSeCxwJRhimtd+sVFxd7d4GuoysudNmgzh9nZWV5d2OHKMOMb26fZi0yokGXWZaWll73C71oIspRwK6qGyPIiBVryyyJcpSwq+raCDJwFVFGTBFk4OuIMmKGIAPfRpSjYPjw4VJevsl7B0WQgbYRZURdZeVuggxcA1GOAp4qfJUuzJ81ayZBBq6BKEcBTxVuETnLQrdOE2SgbUQ5ChITW07aq6mpcdd41DrIunWaIANtI8pRoAHSRxg1NjZ6d+KHniugEW4dZADXRpSjJD19lBw5csR7Fx80yAUFBe6A+rKypwky0A5EOUp0WZzG6dafAeYP+qWmHoMYeWKIhYNeAD8gylGSlJTkrg0NwT+sXYOsDzk9c+a0bN++Q9LS0rxPANwIUY6SgQMHuuv+/fvdNah0U8jIkQ+713r6VmpqqnsNoH2IcpTol326NlefahBU+oj2yKaQrVvDPOQU6ACiHEXp6elujrX9T8n1h8iTG/S5ZnpMqa5BtnI2LeA3RDmKMjIy3PXgwXfdNQj0F4yusFi3bq1b8qaHhbMGGeg4ohxF+s/5ESPSZMuWV707/lZbWysTJ2ZdWWHBkjfg1hHlKJswYUIgpjD0adOZmS0j/71797PCAugkRDnKxo4d6647d+50V7+JzB9Hnja9a1cFX+gBnYgoR5l+AaYxe+65Z323kUSnK3RDiM4fl5YuZ/4Y6AJEOQZmzpwpdXUfS0VFhXfHvsh0RWRDSE5OjvcJgM5ElGMgOTnZjZZXr15lfrSsc99FRUVXpit0/TEbQoCuQ5RjJDJaLisr8+7Yo08I0dUV+igrPVBIpytYfwx0LaIcIzpa1nlZnZ89fPiwd9eGyOhYnxAydGiKW13BgUJAdHS73Mx7jSjTqQv94kznaXUVg4UvzXTuWKdVWkbxT5uIcSiUKPX1wT/ICVCMlGNII1xSUuICqLviYjm/rCsrdPOHzh0zOgZihyjHmE5j6G443VCyceNG72706FSFrjvWlRVHj9bI+vUb3KObWHsMxAbTF0boCWuRA32Ki4u7fCpDY/zyyy+7xzQpnd/Ozs42ue6Y6QvEE6JsiK52iDx+/6mnnuqS0eo3Y5ybO1/mzJljelUFUUY8IcrG6CHxhYWL3Ov8/IJOm9fVOePKysqvxXjSpElu+sQ6oox4QpQN0tHsypUr3fpgPVVuwYIF8uCDD9701IL+PXpM6LPPPif79lW7e3q85uTJk3213pgoI54QZcN01LxmzRoX1KSkATJlylS3m27w4MFtRlWfjXf8+HG37nnPnj1XQqzTIdnZk9wTtS3OGd8IUUY8Ico+oJHVQG/e/JJbPtce+oXhz3+eIcOGPeT7XXhEGfGEKPuMzg0fP/6JnD//uXfnKn1ids+ePQO3nI0oI54QZZhHlBFP2DwCAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMIQoA4AhRBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMIQoA4AhRBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMKTb5WbeawBAjDFSBgBDiDIAGEKUAcAQogwAhhBlADBD5P9XEnnjzfwFdgAAAABJRU5ErkJggg=="></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>f</mi></math> is an odd function.</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 range of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>y</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[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>attempts to replace <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mi>x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></msqrt><mfenced><mrow><mo>=</mo><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for an attempt to calculate both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced></math> independently, showing that they are equal.<br><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for a graphical approach including evidence that <strong>either</strong> the graph is invariant after rotation by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>°</mo></math> about the origin <strong>or</strong> the graph is invariant after a reflection in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis and then in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis (or vice versa).</p>
<p> </p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an odd function <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>attempts both product rule and chain rule differentiation to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mo>×</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>×</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac></math></p>
<p>sets their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p>attempts to find at least one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for an attempt to evaluate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> at least at one of their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> roots.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>≤</mo><mi>y</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></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><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2x + 6}}{{{x^2} + 6x + 10}}{\text{,}}\,\,x \in \mathbb{R}.">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>.</mo>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> has no vertical asymptotes.</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 equation of the horizontal asymptote. </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 exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {f\left( x \right)} \,dx">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</math></span>, giving the answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,q{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 6x + 10 = {x^2} + 6x + 9 + 1 = {\left( {x + 3} \right)^2} + 1">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mn>9</mn>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p>So the denominator is never zero and thus there are no vertical asymptotes. (or use of discriminant is negative) <em><strong>R1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \pm \infty {\text{,}}\,\,f\left( x \right) \to 0">
<mi>x</mi>
<mo stretchy="false">→</mo>
<mo>±</mo>
<mi mathvariant="normal">∞</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo stretchy="false">→</mo>
<mn>0</mn>
</math></span> so the equation of the horizontal asymptote is <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> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\frac{{2x + 6}}{{{x^2} + 6x + 10}}} \,dx = \left[ {{\text{ln}}\left( {{x^2} + 6x + 10} \right)} \right]_0^1 = {\text{ln}}\,17 - {\text{ln}}\,10 = {\text{ln}}\,\frac{{17}}{{10}}">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>17</mn>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>10</mn>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mn>17</mn>
</mrow>
<mrow>
<mn>10</mn>
</mrow>
</mfrac>
</math></span> <em><strong>M1A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>p</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mi>p</mi></math> has two real, distinct roots.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the possible values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></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>Consider the case when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>4</mn></math>. The roots of the equation can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>±</mo><msqrt><mn>13</mn></msqrt></mrow><mn>6</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></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.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use discriminant <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><mfenced><mrow><mo>></mo><mn>0</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>2</mn><mi>p</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mrow><mn>3</mn><mi>p</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow></mfenced><mfenced><mrow><mo>></mo><mn>0</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>12</mn><mi>p</mi><mfenced><mrow><mo>></mo><mn>0</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mfenced><mrow><mn>4</mn><mi>p</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mo>></mo><mn>0</mn></mrow></mfenced></math></p>
<p>attempt to find critical values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p>recognition that discriminant <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo><</mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Condone ‘or’ replaced with ‘and’, a comma, or no separator</p>
<p> </p>
<p><em><strong>[5</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"><mi>p</mi><mo>=</mo><mn>4</mn><mo>⇒</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></p>
<p>valid attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>b</mi><mo>±</mo><msqrt><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>a</mi><mi>c</mi></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math> (or equivalent) <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>8</mn><mo>±</mo><msqrt><mn>208</mn></msqrt></mrow><mn>24</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><mo>±</mo><msqrt><mn>13</mn></msqrt></mrow><mn>6</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</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">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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2x - 4}}{{{x^2} - 1}}{\text{, }} - 1 < x < 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>, </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>For 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>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, if <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>, then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept.</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>show that there are no <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercepts.</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>sketch the graph, showing clearly any asymptotic behaviour.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}"> <mfrac> <mn>3</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The area enclosed by 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 the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 4"> <mi>y</mi> <mo>=</mo> <mn>4</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,v"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>v</mi> </math></span>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule (or equivalent) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{\left( {{x^2} - 1} \right)\left( 2 \right) - \left( {2x - 4} \right)\left( {2x} \right)}}{{{{\left( {{x^2} - 1} \right)}^2}}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 2{x^2} + 8x - 2}}{{{{\left( {{x^2} - 1} \right)}^2}}}"> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>simplifying numerator (may be seen in part (i)) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {x^2} - 4x + 1 = 0"> <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>1</mn> <mo>=</mo> <mn>0</mn> </math></span> or equivalent quadratic equation <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>use of quadratic formula</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = \frac{{4 \pm \sqrt {12} }}{2}"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mo>±</mo> <msqrt> <mn>12</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>use of completing the square</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - 2} \right)^2} = 3"> <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>3</mn> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> (since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + \sqrt 3 "> <mn>2</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> is outside the domain) <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not condone verification that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 \Rightarrow f'\left( x \right) = 0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> <mo stretchy="false">⇒</mo> <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>Do not award the final <em><strong>A1</strong></em> as follow through from part (i).</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>(0, 4) <em><strong>A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - 4 = 0 \Rightarrow x = 2"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A1</strong></em></p>
<p>outside the domain <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>award <em><strong>A1</strong></em> for concave up curve over correct domain with one minimum point in the first quadrant<br>award <em><strong>A1</strong></em> for approaching <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pm 1"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </math></span> asymptotically</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to combine fractions (using common denominator) <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\left( {x - 1} \right) - \left( {x + 1} \right)}}{{\left( {x + 1} \right)\left( {x - 1} \right)}}"> <mfrac> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{3x - 3 - x - 1}}{{{x^2} - 1}}"> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{2x - 4}}{{{x^2} - 1}}"> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 4 \Rightarrow 2x - 4 = 4{x^2} - 4"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo stretchy="false">⇒</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> or) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p>area under the curve is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int_0^{\frac{1}{2}} {\frac{3}{{x + 1}} - \frac{1}{{x - 1}}{\text{d}}x} "> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mn>3</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><strong>Note:</strong> Ignore absence of, or incorrect limits up to this point.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {3\,{\text{ln}}\,\left| {x + 1} \right| - {\text{ln}}\,\left| {x - 1} \right|} \right]_0^{\frac{1}{2}}"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\,{\text{ln}}\frac{3}{2} - {\text{ln}}\frac{1}{2}\left( { - 0} \right)"> <mo>=</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\frac{{27}}{4}"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>27</mn> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>area is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 - \int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <mn>2</mn> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{1}{2}} {4\,{\text{d}}x} - \int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2 - {\text{ln}}\frac{{27}}{4}"> <mo>=</mo> <mn>2</mn> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>27</mn> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\frac{{4\,{{\text{e}}^2}}}{{27}}"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>27</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { \Rightarrow v = \frac{{4\,{{\text{e}}^2}}}{{27}}} \right)"> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>27</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows 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>. The graph has a horizontal asymptote at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span>. The graph crosses the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>On the following set of axes, sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\left[ {f\left( x \right)} \right]^2} + 1">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>[</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span>, clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima.</p>
<p><img 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"></p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><img 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"></p>
<p>no <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> values below 1 <em><strong>A1</strong></em></p>
<p>horizontal asymptote at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> with curve approaching from below as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \pm \infty ">
<mi>x</mi>
<mo stretchy="false">→</mo>
<mo>±</mo>
<mi mathvariant="normal">∞</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>(±1,1) local minima <em><strong>A1</strong></em></p>
<p>(0,5) local maximum <em><strong>A1</strong></em></p>
<p>smooth curve and smooth stationary points <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 4\,{\text{cos}}\,x + 1">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \leqslant x \leqslant \frac{\pi }{2}">
<mi>a</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a < \frac{\pi }{2}">
<mi>a</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - \frac{\pi }{2}"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, sketch 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>. Indicate clearly the maximum and minimum values of the function.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the least value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> has an inverse.</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>For the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> found in part (b), write down the domain of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}"> <mrow> <msup> <mi>g</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> found in part (b), find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}\left( x \right)"> <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>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQwAAAEaCAYAAADkA2kVAAAgAElEQVR4Ae2dCbiOZRrH72M9dmEwHYeDBsl6ZRvZKltElo4s2UWUiEgowkj2IVMTdZSlSNYhU1Mhiiyh0VjSZedYGo2jY//m+j/1Hmf3re/3Ps/7f65L512f5Xe//b/3fZb7jvB4PB5hIgESIAEvCGTx4hpeQgIkQAKKAAWDDwIJkIDXBCgYXqPihSRAAtmIgAS8JXD27FmZO3euHDp0SAoXLixdu3aVatWqeXs7rzOAAN8wDDCiHU2YP3++NGzYUBo3bizYhlBUr15dihQpInXq1JGbN2/aUQ2WEWYCFIwwG0CH4jds2CA9e/aUXr16Se3atVWV8XaB1KJFC9m6datkzZpVh6awjgESoGAECNANt69du1Y1s3Xr1knNPXjwoNqOjIxMOsYN8wlQMMy3ccAtzJEjh8ojb968SXmtWrVKbcfGxiYd44b5BCI4cct8Iwfawp9++klatWolDRo0kH79+smWLVvkvffek0GDBkmXLl0CzZ73a0SAgqGRscJZ1Z07d8qcOXOkbdu2UrFiRSldurRkycIX1HDaJBxl0+LhoK5ZmUeOHFEjIbVq1VJvGmXLlqVYaGbDYFWXghEskhrmg3kVERERapg0s+pfvXpVcuXKJf3795eYmBhp06aNTJ06Vc6cOZPZbTxnIAEKhoFGDWaTduzYIWPGjJFjx47JgQMHpGPHjnL06FEZNmyY/OlPf1LHg1ke83I2AQqGs+0T1tp9++23Ur9+fRkyZIgULFhQypUrJ5MmTZLvvvtONm/eLAkJCbJr166w1pGF20uAgmEvb61Ki4+PlytXrihhSF3xW7duqUlcjRo1Sn2K+wYT4FoSg40baNMwlBoXFycDBw6UP//5z0ogICAYMUHHJ94ysmXjIxQoZ53u57CqTtYKcl3R6VmsWDElCj169Mgwd7xNnD59Wi5evCiYvFWqVKkMr+UJswnw58Fs+waldZhvERUVpf4FJUNmoi0B9mFoazpWnATsJ0DBsJ85SyQBbQlQMLQ1HStOAvYToGDYz5wlkoC2BCgY2pqOFScB+wlQMOxnzhJJQFsCFAxtTceKk4D9BCgY9jNniSSgLQEKhramY8VJwH4CFAz7mbNEEtCWAAVDW9Ox4iRgPwEKhv3MWSIJaEuAgqGt6VhxErCfAAXDfuYskQS0JUDB0NZ0rDgJ2E+AgmE/c5ZIAtoSoGBoazpWnATsJ0DBsJ85SyQBbQlQMLQ1HStOAvYToGDYz5wlkoC2BCgY2pqOFScB+wlQMOxnzhJJQFsCFAxtTceKk4D9BCgY9jNniSSgLQEKhramY8VJwH4CFAz7mbNEEtCWAAVDW9OlrThin951110SERGR9O/TTz9NeyGPkICfBBhb1U9wTrxt0aJFMmTIEImOjlbVQ2T1pk2bOrGqrJOmBCgYmhoudbVv3rwpmzdvFogGgiczkUAoCPDJCgXVMOQ5fvx4+eijj6RGjRqC7bNnz4ahFizSdAIUDEMsfM8998igQYPk9OnT8sorr0iDBg0kISHBkNaxGU4hQMFwiiUCrMeTTz4p06ZNkx9//FFmzZolx48fl4YNG8q1a9cCzJm3k8BtAhSM2yyM2MqTJ48MHDhQxo0bJ7t27ZLt27cb0S42whkEKBjOsEPQa9G9e3eV56lTp4KeNzN0LwEKhqG2L1SokERGRkrFihUNbSGbFQ4CHFYNB3Ubyvzmm2/k4Ycflvvuuy+ptMuXL8uUKVNS7GMHw7H58uVT11aoUCHpPDdIIDWBCI/H40l9kPt6Edi3b580btxYYmNjpVevXnL48GHZsGGDGl4tWLCgXLlyRdatWydLliyRpUuXZto4XF+/fn1p0qSJPPLII4LRFyYSsAhQMCwSGv/FpK0XX3xRVq9eLcWLF5e+fftKp06dBL8FU6dOlb/85S9JQ6wxMTHSrFkzqVevnhQoUEBat24to0aNktq1awuEZ/369bJx40ZFAxPAWrVqpe5P/qaiMSpWPVACeMNgMotAQkKCZ8KECZ7ixYvj7dFTqlQpz4gRIzx79+5N0dD4+Hh1Pi4uLsXxEydOeObNm+dp2rSpJ0uWLJ6cOXN6YmNjPTt37kxxHXfcRwC/QkwGEfjyyy89RYoUUUIQGRmp/sfPqHkZCUby6w8dOuRp2LChyg/ikVpckl/LbfMJcJQk0Fc0h9yPz5LXX39dmjdvrj4/RowYoWZ99u7dO6Aaog/jX//6l6xYsUKqVq2q+kjwGXPgwIGA8uXNmhIwXxPNb+GmTZs8ZcqUUW8Bffv29Zw/f96rRnvzhpE8o+vXr3vGjx/vwZtL7ty5PePGjfMkJiYmv4TbhhPgJ4nmBl6xYoX6Hxh9FWPGjPGpNb4KhpX5999/74mOjlYC1a5dO+sw/7qAAD9JNH0zRLUx/btt27aCSVrLli2TsWPH2tKaSpUqyaZNm9TQ6/Lly6Vr165q6NaWwllIWAlQMMKK3//C33rrLRkzZowUKVJEPv/8c2nfvr3/mflxJ4Zn4c0L8z4WLlyo5oHA4xeT2QQoGJrZ99atW2q25nPPPSd16tSR3bt3SzhnZ7755ptqwtiWLVukTZs2QtHQ7IHysboUDB+BhfNyjIT06dNHhg8fLnXr1lWjF1FRUeGskuTIkUM+/PBDmTx5snz99ddqejlWyTKZSYCCoZFd+/XrJ3FxcVK+fHlZuXKlYCm7ExJmhA4bNkzNCMXq2Hbt2iXNLHVC/ViH4BGgYASPZUhzghetd955R03p3rp1q2DNh9PS0KFDZcKECXLs2DHlKhBOfJjMIkDB0MCe8+fPVwvJsE4Er/9OFAtgxJsG1qVAODCxC6MnTGYRoGA43J749MBszdKlS6tFYeHus/AGF4Z3n3jiCVXf1157zZtbeI0mBCgYDjYURhx69uwpGBnBW0a5cuUcXNvbVUPfCt6E4FN09OjRqnP29llu6UyAguFQ62FEBHMcEhMTlQ8LeAHXLUHkChcuLI899phs27ZNt+qzvukQoGCkAyXch+Dpu0OHDmrB19/+9jc1zyHcdfKnfEzugsMevCE9/vjjcuTIEX+y4T0OIkDBcJAxrKr89a9/FUy5hqMbvGXonBo1aiRz586VEydOyIMPPihwE8ikLwEKhsNs9+2336qhyWLFigneLkxIiJkyYMAA9YaBSWdM+hKgYDjIdhcuXFBvFei3gKu8MmXKOKh2gVUFM0HxiQIR3LlzZ2CZ8e6wEaBghA192oIHDx6s1mJgkla1atXSXqDxEYycwBEP3pzwtgHHxEz6EaBgOMRmmL2JVZ/wavXSSy85pFbBrUbZsmXV9HF8dr3wwgvBzZy52UKAgmEL5swLwagIVp+WKFFC1q5dK1mzZs38Bo3PIiIbVtnOmTNHeTnXuCmurDoFI8xmh1jACc6OHTvUwjIdZnIGgixbtmyyYMEC5fQH4RA41BoITfvvpWDYzzxFififB0GGHn30UeWEJsVJQ3fgWPi9996T+Ph45UMDk9SY9CBAwQijnTAqAjd7uXPnFretuYBAYrh1z5496vMkjGZg0T4QoGD4ACvYl+J7HkvB8T3vxshikyZNUgGjX331VXrqCvbDFaL8KBghAnunbBHnFB2c+KXt0aPHnS438jz6az744AP55Zdf1FR49OcwOZsABSMM9jlz5oxgzgW8fZsym9NfjPADihGizz77TGbMmOFvNrzPJgIUDJtAJy8G3+4QDXjQio6OTn7KldtwugOnQAgazbUmzn4EKBg22+ejjz5SYQGw5Bu/rkyilsBj0tqvv/4q06ZNIxIHE6Bg2Gycl19+WZWI0RGm2wRatmypnAcjPuz58+dvn+CWowhQMGw0B94u4OsSXqiqVKliY8l6FAUxvXr1qnTu3Flu3LihR6VdVksKho0GHz9+vAoRgL9MaQlUrlxZvWWgA3TVqlVpL+CRsBOgYNhkAsw5+P7778X6JLGpWO2KGTlypGTPnl11gHIGqPPMR8GwwSYII4iRAEQr69Kliw0l6lsElvVjItt3332nhlv1bYmZNadg2GBXdOTBr+WIESNsKE3/Ip566impXbu2mqOC2LFMziFAwQixLfA9jsVl8M+JkQAm7whY4jpx4kTvbuBVthCgYIQQM7xKdevWTXLmzKniiiAyGJN3BDBH5YEHHhCMLB0+fNi7m3hVyAnwCQ4h4nnz5qkZnQiijDCHTL4RePfdd9VK3qlTp/p2I68OGQEKRsjQiuBBh8MYxBpl8p0AIr0hPium0GMqPVP4CVAwQmQDvEofPXpUBVE23YtWiBCqbAcNGqQ6jDG6hI5jpvASoGCEgD+WaWMYFb4rrc67EBTjiizvvfdetfz/iy++kMWLF7uizU5uJAUjBNbBkvVDhw7JkCFDQpC7+7J8/vnnVaNnzZrlvsY7rMUUjCAb5Pjx4+rtonTp0tKqVasg5+7O7OCNDB3H27dvVw6E3UnBGa2mYATZDuigwzLt/v37K/dzQc7etdnNnDlT+cyAsx36zAjfY0DBCCJ7PMgQDDiDgb9OpuARiIyMlD59+ijfn/Pnzw9exszJJwIUDJ9wZX7xM888IydPnpSlS5dK0aJFM7+YZ30mgIhwmM+ChXwMtegzvqDcQMEICkaRffv2qVgbDRo0kCZNmgQpV2aTnAB8oCIcw4kTJ2TNmjXJT3HbJgIUjCCBtpz54i2DKXQEMJGrUqVKjGUSOsSZ5kzByBSPdyfRd4HvakT0ateunXc38Sq/CCDu7NNPPy0bN26UDRs2+JUHb/KfAAXDf3ZJd0IsMDIye/ZsowMpJzU4zBuY9YlocXibY1+GvcagYASBNz5HatasKc2bNw9CbsziTgQwChUbGys//PAD+zLuBCvI5ykYAQLFmhE8uAMGDAgwJ97uCwGLN1YEM9lHgIIRAOuzZ89K3759pWTJkiqwcABZ8VYfCdSqVUvN/vz0009l5cqVPt7Ny/0lQMHwl5yIID7qxYsXpWfPnmoZewBZ8VY/CGA+BvoysNaEDoP9AOjHLRQMP6BZt+B1GB6u3RpM2eIQrr/oy+jUqZMcOXKEYQlsMgIFw0/QeA3eu3ev6ruIiYnxMxfeFigBfBIiYUo+U+gJUDD8YIyhPPi5wC8c/V34ATCIt9SoUUMeeughFf0ds22ZQkuAguEHX3gCR8hDBFSmr04/AAbxFjhWRgDn69evy4svvhjEnJlVegQoGOlRucMx6/UX8TOYwk8AwY+wfmft2rXCOCahtQcFw0e+W7duldWrV6uJQ3CDz+QMApajZbxtMIWOAAXDR7YvvPCCREREyOTJk328k5eHkgACRWEtz7Jly+hgJ4SgKRg+wN22bZsgTmqLFi2EIyM+gLPp0unTp6u1JZgfwxQaAhQMH7guWrRIXY1oZkzOIwAfqhUrVhTLTs6rof41omB4aUMMpX744Ydy9913qzcML2/jZTYTePLJJ9XSd3Z+hgY8BcNLrhCLc+fOydtvvy158uTx8i5eZjcBeBdHLNvevXsz8FEI4FMwvISKIDrly5dnBHYveYXrMrjxg5uBXbt2yTfffBOuahhbLgXDC9Pu379fEHkL7uGYnE/AstPChQudX1nNakjB8MJgY8eOVash4emJyfkEMIqFGbjw3n7jxg3nV1ijGlIw7mAsrISEkxzESeVQ6h1gOeR0jhw5ZPTo0fLzzz/Lm2++6ZBamVENCsYd7IjOTkQNf+KJJ+5wJU87iQA6P9GfMW7cOEFwbKbgEKBg3IEjBAM+LygYdwDlsNPZsmWTjh07yvnz52XdunUOq52+1aFgZGK7jz/+WPbs2SMjR46UP/7xj5lcyVNOJGD5/fz73//uxOppWScKRiZmGzNmjAqojPUjTPoRQNT3hg0bCvx+wpUiU+AEKBgZMPzqq69U+ENMN86bN28GV/Gw0wlghAt9UFxfEhxLUTAy4IjPESTEv2DSl0CjRo3U+hLLnvq2xBk1p2CkYwf8IsFnpzVrMJ1LeEgjAhD9TZs2ybFjxzSqtTOrSsFIxy5wwXf06FGBG/t8+fKlcwUP6UQA60owgYt9UYFbjYKRDkO8vsLBL+KNMOlPIDo6WuAsGBPwTp48qX+DwtgCCkYq+Jjks2bNGuXgF2P5TGYQaN++vWoIo6QFZk8KRip+y5cvlzNnzoj1gKU6zV1NCcDDO9KqVas0bYEzqk3BSGYHdHZOmTJFDaMi1gWTOQTKlSun5mSg85OfJf7blYKRjB0imcGPwsMPP0wnOcm4mLKJoFNXr16V2bNnm9Ik29tBwUiG3Pq+bdeuXbKj3DSFAGKXlC5dWubOnSuXLl0ypVm2toOC8TtuRP+eP3++mnvB/gtbn0HbCsuaNasKnI1l7xgxYfKdAAXjd2affPKJmnvRv39/fo74/hxpc0ebNm1UXa23SW0q7pCKUjB+NwQCE0VGRsqzzz7rENOwGqEgUKVKFalZs6Z8/vnncuHChVAUYXSeFAwRgc9OLDbDykYGVzb6eVeNGz58uPz666/0xuWHqSkYImItTGLfhR9PkIa3tG3bVkqUKCGzZs2SxMREDVsQvipTMJJN5rEm94TPHCzZDgLo/GzdurWKMwNv8EzeE3C9YOzbt0927NghnTt3lqJFi3pPjldqTcD6cWDnp29mdL1gPPfcc4I1I6+99ppv5Hi11gQaN24slStXVuEvL1++rHVb7Ky8qwXj8OHDKkBR/fr1pWTJknZyZ1lhJpAlSxZB52dCQoJ6BsJcHW2Kd7VgrFixQhmKMzu1eV6DWtEOHTpIsWLFxHoOgpq5oZlRMETEmsxjqI3ZrAwIIOAROj+xgpUR0jKAlOqwawXj9OnTsm3bNuX3IioqKhUW7rqFADo/MVWccVi9s7hrBeP9999XhKZNm+YdKV5lJIFmzZqpyXqjRo3inAwvLOxawcB3a926daVs2bJeYOIlphLACBneMk6dOiUbN240tZlBa5crBcOae8G+i6A9R1pnhJmfSPTGdWczulIw4FULy9k5OnLnB8QNV2BORpEiRZKWCLihzf620XWCcfbsWRUFq2rVqhITE+MvN95nEAFMFR86dKiaKo7g20wZE3CdYMAj+JUrV+jkN+NnwpVnBg8erNwbTJgwwZXt97bRrhMM6zuV/RfePiLuuA6+UFq2bKni6cKvK1P6BFwlGAcPHpR169apXnGsI2AigeQEunfvrnbfeeed5Ie5nYyAqwRj+vTpqrMT3qOZSCA1AczJQMQ7rmBNTeb2vmsE4/z587JgwQKpVKmS1KlT5zYBbpHA7wQwVXzQoEFqTgZmATOlJeAawYCTX7hlY99F2oeAR24TQMDm7Nmzc0HabSQptlwjGNaKRLrhS2F/7qQikDdvXmnevDk/S1JxsXZdIRgYRsUbBuZeVKtWzWo7/5JAugQwoe/AgQOyYcOGdM+7+aArBAMjIxANLDBiIoE7EXj00UfVZ8nEiRPvdKnrzrtCMOLi4qRChQoSGxvrOgOzwb4TwDRx+Mn47LPPVKxd33Mw9w7jBePixYvyz3/+k+tGzH2GQ9Iya0GaFYIiJIVomKnxgjFz5ky5fv26mqyloX1Y5TAReOSRRyR37tzs/EzF32jBuHbtmsBBTnR0tNSqVStV07lLAhkTKFSokPTs2VN++OEH2bx5c8YXuuyM0YKxfv165RWacy9c9lQHqbnPP/+86vxkCIrbQI0WjHfffVe1tFevXrdbzC0S8JIAvLG1aNFCrT/avXu3l3eZfZmxgoF4ExhOvf/++zn3wuxnOKStsyb6sfPzN8zGCsaMGTNUZ+dLL70U0geKmZtNgJ2fKe1rpGCgsxMrU++++246yklpb+75SABzMrp27Sr//ve/ZevWrT7ebd7lRgoG5l1g/gU7O817YMPRoiFDhgjc+HHmp4iRgmH5M6CT33D872VemeXKlVMhKeDe8dixY+Y10IcWGScYWDMCw5YqVUoQZJmJBIJBwPrxsX6MgpGnjnkYJxiYqHXu3DmZNGmSwCEKEwkEg0Dnzp2Vk2C3u+8zSjDQ2Ymp4MWLF5eOHTsG4zlhHiSgCBQtWlS6dOkie/fulcOHD7uWilGCgdWFcMVnvT661qpseEgIDB8+XHV+Ll++PCT565CpUYJhGZKjIzo8evrVEZ2ftWvXdnWENGMEA58ja9euVQvNGjZsqN/TyBprQQAzP3fs2CH79+/Xor7BrqQxgvH1119LfHy8YMycnZ3BfkyYn0UAn7uIy4tOdTcmYwQDnyOIKcGFZm58jO1rM+LxVq9eXcXnxQ+U25IRgnH58mVZvHixYOgrf/78brMh22szAfwoYb4P4ty4LRkhGDDchQsXOBXcbU9vmNprue+zQleEqRphKVZ7wbhx44ZMmTJFChcuzJmdYXmE3FdoVFSUGrpHdDS3dX5qLxhwn/bTTz8pRyeIwM1EAnYQgNsEN3Z+ai8Y1twLTtb67X+Ts2fPyltvvSXo12EKHYEaNWqowFhLlixRo3OhK8lZOWstGFZnZ5kyZQTBZ9yeli1bppwd9+/fn4Jhw8PQp08f13V+ai0YVmfnsGHDJFu2bDY8Is4u4vHHH5cOHTo4u5IG1c6NnZ/aCkbyzk54RGL6jQBiaTDZQyB55+fJkyftKTTMpWgrGFZnJz5F8uTJE2aMLN6tBKzOT7f4ydD2Pb5evXoqwAxmdzJ5R+DWrVtqNa91NeauMAVGAJ2f//jHP+Shhx4KLCNN7tZWMNBn8cADD2iC2RnVxNL/YsWKOaMyBtWiZcuWBrUm86ZoKxiZN4tn0yOQM2fOFBHsLXeG6V3LYySQHgEKRnpUDD1WoEABWbp0aVLrMGeDbxxJOLjhBQFtOz29aBsvIQESCDIBCkaQgTI7EjCZAAXDMOvCYzqS9dew5rE5YSZAwQizAYJZPFzHValSRd544w3l3frIkSPBzJ55kYCw09OghwBzAvCPiQRCRYBvGKEiy3xJwEACFAwDjcomkUCoCFAwQkWW+ZKAgQQoGAYalU0igVARoGCEiizzJQEDCVAwDDQqm0QCoSJAwQgVWeZLAgYSoGAYaFQ2iQRSE0Ds4dRewfbt26d8kqa+NrN9CkZmdHiOBAwgsGbNGunYsaMK8YgVyomJidK7d2+pXLmydOvWzacWcqanT7h4MQnoR6BVq1bSrFkzsdwbwB8u/OAiNIev7g0oGPrZnzUmAZ8J5MiRQ0UGXLhwoTz99NPSqFEjn/PADfwk8QsbbyIB/QjUrVtXEN6xTZs2fleeguE3Ot5IAnoRyJs3r6rw7t27/a44BcNvdLyRBPQhgM7OXLlySZYsWeSLL75QFYcXeV8TBcNXYryeBDQkgCiB6LuoWbOmbN26VbXg/fff97klFAyfkfEGEtCDwFdffSX33nuvDB8+XA2pZs2aVbp06SJbtmyRgQMHyv333+9zQzhK4jMy3kACehCoXr269OvXT3VyxsTEqEpDKLJnz67mZfgTBIyCoYftWUsS8JkAOjkHDx6c5j58mvib+EniLzneRwIuJEDBcKHR2WQS8JcABcNfcryPBFxIgILhQqOzySTgLwEKhr/keB8JuJAABcOFRmeTScBfAhQMf8nxPhJwIQEKhguNziaTgL8EKBj+kuN9JOBCAhQMFxqdTSYBfwl4PTW8R48e/pbB+xxK4MqVK6pm8+bNkw0bNji0lqxWIAQGDBggtWrVCiSLFPdGeDweT4ojGexERERkcIaHSYAEnEpg6dKlEhsbG7Tqef1JAl1x0r833nhDQQAQJ9VLp7rEx8crhnFxcWQYwPNdsWJFwS+5E20fTLHAw+K1YARNopgRCZCAtgQoGNqajhUnAfsJaCsY+fPnFzgFyZ07t/3UWCIJJCMQFRUlhQsXTnbE3E2vOz3NReDelsExLALZoA+Do2DufQ58abm2bxi+NJLXkgAJBIcABSM4HJkLCbiCAAXDFWZmI0kgOAQoGMHhyFxIwBUEKBiuMDMbSQLBIUDBCA5H5kICriDg9eIzJ9BAjIXly5eLtWgqdZ2wiKp169apD3OfBIJOYP369TJ69Gg5duxYhnlj2Nq0pI1gzJ8/X06dOiXjxo2TSZMmyYgRI5Qt+vbtKxMnTpQiRYpIlSpVTLMP2+NAAmfOnJH+/fvLkCFDZM+ePXL+/HkVXQzrm27cuJFu8CAHNsO/KmG1qg5pxowZnps3b6qq1q5dW/1NSEjASlvP0aNHdWiC4+oYHx+v+MXFxTmubk6u0MaNGz1ghzRy5EgPnk2kmjVremJjY9W2qf/Rpg8DnyMIVX/t2jX55ZdflDoeOnRI/S1ZsqR/asm7SMAPAg0aNJCiRYuqO7dv3y6FChVS24cPH5Y//OEPfuSozy3aCIaFdNOmTZIvXz61e+LECfEnoKyVF/+SQCAE0JeGCOlYR4Ltn3/+2fjnUSvBuHXrlkyYMEGtf4ChT58+rSJRB2J03ksC/hJYtWqVEgqsx/nvf/+rskFkdJOTVoKxdu1a2bhxo5QtW1bZ5OTJk3Lp0qUk++zcuTNpmxskEGoCkydPVkXcc8896u0CO9bzuHv37lAXH5b8tRklAZ3hw4crSNWqVVN/8b2IV8H27dtLgQIF5NVXXw0LRBbqPgIvv/yy7Nq1S6pWrao+QyyhWLx4seTJk0du3rwp1nNqEh2t3jASExOlePHi0qpVK2WD3r17K4NhbkauXLkkOjraJNuwLQ4mAIFAJzyG+JHw7I0aNUow92L27Nny2GOPObj2/ldNK38YEAw4I46MjExqMca9MWriFgcmSQ0Pwgb9YQQGEc8d3myTJxzDj1eOHDmSHzZmW6tPEhgidcqWLRvFIjUU7ttCILVYoND0jtlSGZsK0eqTxCYmLIYESCADAhSMDMDwMAmQQFoCFIy0THiEBEggAwIUjAzA8DAJkEBaArZ2es6cOVOs9R9pq/Lbkbx588rrr7+edHrOnDly7ty5pH1vNlq0aBHUeJLelMlrzCeA+LP+xKAdO3asMXBsFYxGjRrdcTKL6VPeTaoAAALgSURBVFNrjXly2BBXEtBqHoYrLRTCRnMeRgjhGpo1+zAMNSybRQKhIEDBCAVV5kkChhKgYBhqWDaLBEJBgIIRCqrMkwQMJaCtYKDDDu76mEgg3ASwctVa3h7uuoS6fO0EA+PgcKADL0dY6GN5Dw81KOZPAqkJHDlyRBo3biz58+dX/+rVqycHDx5MfZlR+1oJxsqVK6VTp07StGlTGTp0qBINTPJCLJKMYpUYZS02xjEELl68KO3atVPhLeBMp0mTJrJlyxZ58MEHzRYNndyhd+/e3XPp0qWkKicmJnrKly+vXOV/8sknSce54R0BhhnwjlN6V40ZM8azYsWKFKeeeeYZ9Sy2b98+xXGTdrR6w+jSpYtg6riV4EgHgYyQ4OKdiQTsIhAVFaWCFyUvb/r06XLXXXeJyb5lbZ0anhyuP9t47Uud0JeBdN9996U+xX0SCBmBp556Kk3e8LKFCHx169ZNc86UA1q9YaQHff/+/QKvzfXr10/vNI+RgG0EEhIS5Pjx4yrmqm2F2lyQ1oKBYVV0hC5YsECyZs1qMzoWRwIpCSA4c7du3dQPWMoz5uxp9UmSGvsrr7yiviPr1KmT+hT3ScBWAui3WLJkifznP/+xtVy7C9NWMFavXq1e/95++227mbE8EkhBAN7s4etl8+bNxodK1FIwLly4oNR80aJFKQzHHRIIBwEE0EJMEisiXzjqYFeZ2vVhYMIMvhURLCZ5QmcTYpQwkYBdBNCHhpnGFSpUUP+Sl4tZoCYmrd4wduzYoWbXde3aVT744IMke5w5c0bNrsM3JBMJ2EEAM4s7duwo+fLlU1HP4EoSCZ8nX375pTz77LMSExNjR1VsLUMbwfjxxx/V9Fu8YUycODENpHHjxqU5xgMkEAoC//vf/5TP2AMHDqjsFy5cmKKYggULyooVK1IcM2VHG8HAhBh0KmWUSpQokdEpHieBoBLABK2PP/44wzwRjNnUUIn/B0nHc10ehM4OAAAAAElFTkSuQmCC"></p>
<p>concave down and symmetrical over correct domain <em><strong>A1</strong></em></p>
<p>indication of maximum and minimum values of the function (correct range) <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> = 0 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> = 0 only if consistent with their graph.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant x \leqslant 5"> <mn>1</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>5</mn> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Allow FT from their graph.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 4\,{\text{cos}}\,x + 1"> <mi>y</mi> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 4\,{\text{cos}}\,y + 1"> <mi>x</mi> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{x - 1}}{4} = {\text{cos}}\,y"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow y = {\text{arccos}}\left( {\frac{{x - 1}}{4}} \right)"> <mo stretchy="false">⇒</mo> <mi>y</mi> <mo>=</mo> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {g^{ - 1}}\left( x \right) = {\text{arccos}}\left( {\frac{{x - 1}}{4}} \right)"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>g</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {x^4} - 6{x^2} - 2x + 4">
<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>6</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
</math></span>, <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>.</p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is translated two units to the left to form the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{x^4} + b{x^3} + c{x^2} + dx + e">
<mi>a</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>c</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>d</mi>
<mi>x</mi>
<mo>+</mo>
<mi>e</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="e \in \mathbb{Z}">
<mi>e</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = f\left( {x + 2} \right)\left( { = {{\left( {x + 2} \right)}^4} - 6{{\left( {x + 2} \right)}^2} - 2\left( {x + 2} \right) + 4} \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p>attempt to expand <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{{\left( {x + 2} \right)}^4}}">
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x + 2} \right)^4} = {x^4} + 4\left( {2{x^3}} \right) + 6\left( {{2^2}{x^2}} \right) + 4\left( {{2^3}x} \right) + {2^4}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mn>3</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>4</mn>
</msup>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {x^4} + 8{x^3} + 24{x^2} + 32x + 16">
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>24</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>32</mn>
<mi>x</mi>
<mo>+</mo>
<mn>16</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {x^4} + 8{x^3} + 24{x^2} + 32x + 16 - 6\left( {{x^2} + 4x + 4} \right) - 2x - 4 + 4">
<mi>g</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>8</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>24</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>32</mn>
<mi>x</mi>
<mo>+</mo>
<mn>16</mn>
<mo>−</mo>
<mn>6</mn>
<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>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
<mo>+</mo>
<mn>4</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {x^4} + 8{x^3} + 18{x^2} + 6x - 8">
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>18</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mn>8</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> For correct expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {x - 2} \right) = {x^4} - 8{x^3} + 18{x^2} - 10x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>8</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>18</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>10</mn>
<mi>x</mi>
</math></span> award max <em><strong>M0M1(A1)A0A1</strong></em>.</p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{ax + b}}{{cx + d}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<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></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R},\,\,x \ne - \frac{d}{c}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>d</mi>
<mi>c</mi>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = \frac{{2x - 3}}{{x - 2}},\,\,x \in \mathbb{R},\,\,x \ne 2">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>2</mn>
</math></span></p>
</div>
<div class="question">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right)"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A + \frac{B}{{x - 2}}"> <mi>A</mi> <mo>+</mo> <mfrac> <mi>B</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span> where A, B are constants.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 2 + \frac{1}{{x - 2}}"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span> <em><strong> A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{x - 4}}{{2x - 5}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> </mfrac> </math></span>, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><img 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"></p>
<p>correct shape: two branches in correct quadrants with asymptotic behaviour <em><strong>A1</strong></em></p>
<p>crosses at (4, 0) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0{\text{,}}\,\,\frac{4}{5}} \right)"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>asymptotes at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{5}{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{2}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>c</mi></mtd></mtr><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>b</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>c</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math>.</p>
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>≥</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></math>, find an expression for the median of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> be the median</p>
<p><strong><br>EITHER</strong></p>
<p>attempts to find the area of the required triangle <em><strong>M1</strong></em></p>
<p>base is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>and height is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p>area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>×</mo><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><msup><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to integrate the correct function <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>a</mi><mi>m</mi></munderover><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup></mrow></mfenced><mi>a</mi><mi>m</mi></msubsup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><msubsup><mfenced open="[" close="]"><mrow><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mi>a</mi><mi>x</mi></mrow></mfenced><mi>a</mi><mi>m</mi></msubsup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct integration and <em><strong>A1</strong> </em>for correct limits.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>sets up (their) <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>a</mi><mi>m</mi></munderover><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi></math> or area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0A0A0M1A0A0</strong></em> if candidates conclude that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>></mo><mi>c</mi></math> and set up their area or sum of integrals <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>±</mo><msqrt><mfrac><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></msqrt></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>></mo><mi>a</mi></math>, rejects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>-</mo><msqrt><mfrac><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></msqrt></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>+</mo><msqrt><mfrac><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></msqrt></math> <em><strong>A1</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>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{{{x^2} + 3x + 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 2,{\text{ }}x \ne - 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(x + h)^2} + k"> <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 class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Factorize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </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>Sketch the graph 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>, indicating on it the equations of the asymptotes, the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept and the local maximum.</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>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {f(x){\text{d}}x = \ln (p)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the area of the region enclosed between the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis and the lines with equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2 = {\left( {x + \frac{3}{2}} \right)^2} - \frac{1}{4}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2 = (x + 2)(x + 1)"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-08_om_13.58.40.png" alt="M17/5/MATHL/HP1/ENG/TZ1/B11.b/M"></p>
<p><strong><em>A1</em></strong> for the shape</p>
<p><strong><em>A1</em></strong> for the equation <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></p>
<p><strong><em>A1</em></strong> for asymptotes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 2"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span></p>
<p><strong><em>A1</em></strong> for coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \frac{3}{2},{\text{ }} - 4} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong><em>A1</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0,{\text{ }}\frac{1}{2}} \right)"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\frac{1}{{x + 1}} - \frac{1}{{x + 2}}{\text{d}}x} "> <munderover> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </munderover> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\ln (x + 1) - \ln (x + 2)} \right]_0^1"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mn>1</mn> </msubsup> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \ln 2 - \ln 3 - \ln 1 + \ln 2"> <mo>=</mo> <mi>ln</mi> <mo></mo> <mn>2</mn> <mo>−</mo> <mi>ln</mi> <mo></mo> <mn>3</mn> <mo>−</mo> <mi>ln</mi> <mo></mo> <mn>1</mn> <mo>+</mo> <mi>ln</mi> <mo></mo> <mn>2</mn> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \ln \left( {\frac{4}{3}} \right)"> <mo>=</mo> <mi>ln</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore p = \frac{4}{3}"> <mo>∴</mo> <mi>p</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-08_om_14.20.03.png" alt="M17/5/MATHL/HP1/ENG/TZ1/B11.e/M"></p>
<p>symmetry about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis <strong><em>M1</em></strong></p>
<p>correct shape <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow <strong><em>FT </em></strong>from part (b).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\int_0^1 {f(x){\text{d}}x} "> <mn>2</mn> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\ln \left( {\frac{4}{3}} \right)"> <mo>=</mo> <mn>2</mn> <mi>ln</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>FT </em></strong>from part (e).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2{x^2} - 5x - 12}}{{x + 2}}{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \ne - 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>5</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>12</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find all the intercepts of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> with both the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> axes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the vertical asymptote.</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>As <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \pm \infty ">
<mi>x</mi>
<mo stretchy="false">→</mo>
<mo>±</mo>
<mi mathvariant="normal">∞</mi>
</math></span> the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> approaches an oblique straight line asymptote.</p>
<p>Divide <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} - 5x - 12">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mi>x</mi>
<mo>−</mo>
<mn>12</mn>
</math></span> by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span> to find the equation of this asymptote.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0 \Rightarrow y = - 6">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
</math></span> intercept on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> axes is (0, −6) <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} - 5x - 12 = 0 \Rightarrow \left( {2x + 3} \right)\left( {x - 4} \right) = 0 \Rightarrow x = \frac{{ - 3}}{2}\,\,{\text{or}}\,\,{\text{4}}">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mi>x</mi>
<mo>−</mo>
<mn>12</mn>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>or</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>4</mtext>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p>intercepts on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> axes are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{{ - 3}}{2}{\text{,}}\,\,0} \right)\,\,{\text{and}}\,\,\left( {4{\text{,}}\,\,0} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>and</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 2">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2x - 9 + \frac{6}{{x + 2}}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>9</mn>
<mo>+</mo>
<mfrac>
<mn>6</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
</math></span> <em><strong> M1A1</strong></em></p>
<p>So equation of asymptote is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2x - 9">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>9</mn>
</math></span> <em><strong> M1A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2{x^3} + 5,{\text{ }} - 2 \leqslant x \leqslant 2">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>2</mn>
</math></span>.</p>
</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">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}(x)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</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 domain and range 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 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=" - 11 \leqslant f(x) \leqslant 21">
<mo>−</mo>
<mn>11</mn>
<mo>⩽</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>⩽</mo>
<mn>21</mn>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>A1 </em></strong>for correct end points, <strong><em>A1 </em></strong>for correct inequalities.</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="{f^{ - 1}}(x) = \sqrt[3]{{\frac{{x - 5}}{2}}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mroot>
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mn>3</mn>
</mroot>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 11 \leqslant x \leqslant 21,{\text{ }} - 2 \leqslant {f^{ - 1}}(x) \leqslant 2">
<mo>−</mo>
<mn>11</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<mn>21</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo>⩽</mo>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>⩽</mo>
<mn>2</mn>
</math></span> <strong><em>A1A1</em></strong></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="question">
<p>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>3</mn></mrow></mfrac><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to use change the base <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mfrac><mrow><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><mn>2</mn></mrow><mn>2</mn></mfrac><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math></p>
<p>attempt to use the power rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mn>2</mn></msqrt><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math></p>
<p>attempt to use product or quotient rule for logs, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>a</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>b</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>a</mi><mi>b</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><mfenced><mrow><mn>4</mn><msqrt><mn>2</mn></msqrt><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math></p>
<p><strong><br>Note:</strong> The <em><strong>M</strong></em> marks are for attempting to use the relevant log rule and may be applied in any order and at any time during the attempt seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>4</mn><msqrt><mn>2</mn></msqrt><msup><mi>x</mi><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>32</mn><msup><mi>x</mi><mn>6</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>5</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>32</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5</strong></em><em><strong> marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>The quadratic equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 2kx + (k - 1) = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>k</mi>
<mi>x</mi>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> has roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\beta ">
<mi>β</mi>
</math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\alpha ^2} + {\beta ^2} = 4">
<mrow>
<msup>
<mi>α</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>β</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
</math></span>. Without solving the equation, find the possible values of the real number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha + \beta = 2k">
<mi>α</mi>
<mo>+</mo>
<mi>β</mi>
<mo>=</mo>
<mn>2</mn>
<mi>k</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha \beta = k - 1">
<mi>α</mi>
<mi>β</mi>
<mo>=</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(\alpha + \beta )^2} = 4{k^2} \Rightarrow {\alpha ^2} + {\beta ^2} + 2\underbrace {\alpha \beta }_{k - 1} = 4{k^2}">
<mrow>
<mo stretchy="false">(</mo>
<mi>α</mi>
<mo>+</mo>
<mi>β</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>α</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>β</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<munder>
<mrow>
<munder>
<mrow>
<mi>α</mi>
<mi>β</mi>
</mrow>
<mo>⏟</mo>
</munder>
</mrow>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</munder>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\alpha ^2} + {\beta ^2} = 4{k^2} - 2k + 2">
<mrow>
<msup>
<mi>α</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>β</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\alpha ^2} + {\beta ^2} = 4 \Rightarrow 4{k^2} - 2k - 2 = 0">
<mrow>
<msup>
<mi>α</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>β</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mo stretchy="false">⇒</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>attempt to solve quadratic (<strong><em>M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 1,{\text{ }} - \frac{1}{2}">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) - 2{\left( {{\text{ln}}\,2} \right)^2}\left( { = 0} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,x = \frac{{{\text{ln}}\,2 \pm \sqrt {{{\left( {{\text{ln}}\,2} \right)}^2} + 8{{\left( {{\text{ln}}\,2} \right)}^2}} }}{2}">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>±</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{ln}}\,2 \pm 3\,{\text{ln}}\,2}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>±</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{ln}}\,x - 2\,{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x + 2\,{\text{ln}}\,2} \right)\left( { = 0} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,x = 2\,{\text{ln}}\,2">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\text{ln}}\,2">
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = 4">
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong> (M1)A1</strong></em> </p>
<p><strong>Note:</strong> <em><strong>(M1)</strong></em> is for an appropriate use of a log law in either case, dependent on the previous <em><strong>M1</strong></em> being awarded, <strong>A1</strong> for both correct answers.</p>
<p>solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} < x < 4">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>4</mn>
</math></span> <em><strong>A1</strong></em></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 following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, with asymptotes at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math>.</p>
<p style="text-align: center;"><img 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"></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><mtext>arctan </mtext><mi>x</mi></math> to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>p</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi><mo>≡</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo><</mo><mn>1</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan </mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mtext>arctan </mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi mathvariant="normal">+</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <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>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>Using mathematical induction and the result from part (b), prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>n</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[9]</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>EITHER</strong><br>horizontal stretch/scaling with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><br><strong>Note:</strong> Do not allow ‘shrink’ or ‘compression’</p>
<p><br>followed by a horizontal translation/shift <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> units to the left <em><strong>A2</strong></em></p>
<p><br><strong>Note:</strong> Do not allow ‘move’</p>
<p><br><em><strong>OR</strong></em></p>
<p>horizontal translation/shift <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit to the left</p>
<p>followed by horizontal stretch/scaling with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A2</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>vertical translation/shift up by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> (or translation through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em><br>(may be seen anywhere)</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>let <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mi>p</mi></math></strong> and <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi></math> <em>M1</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mtext>tan</mtext><mo> </mo><mi>α</mi></math> </strong>and <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mtext>tan</mtext><mo> </mo><mi>β</mi></math> <em>(A1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></math> <em>A1</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>β</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></mfenced></math> <em>A1</em></strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>p</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi><mo>≡</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo><</mo><mn>1</mn></math>. <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mfenced><mn>1</mn></mfenced></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfenced><mrow><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mn>2</mn><mi>x</mi><mo>+1</mo><mo>-</mo><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mstyle displaystyle="true"><mn>1</mn><mo>+</mo><mfrac><mrow><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+1</mo></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>x</mi></mstyle><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan 1</mtext></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>tan </mtext><mfenced><mrow><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">=</mi><mi>tan</mi><mo> </mo><mfenced><mrow><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi mathvariant="normal">+</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHS</mtext><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mrow></mfrac><mfenced><mrow><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>n</mi></mfenced></math> be the proposition that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>n</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math></p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>1</mn></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mo>=</mo><mtext>RHS</mtext></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math> is true <strong><em>R1</em></strong></p>
<p>assume <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> is true, ie. <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi> </mi><mfenced><mrow><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>”.<br><strong>Note:</strong> Subsequent marks after this <em><strong>M1</strong></em> are independent of this mark and can be awarded.</p>
<p> </p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>+</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mstyle><mrow><mn>1</mn><mo>-</mo><mfenced><mstyle displaystyle="true"><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfenced><mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mstyle></mfenced></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mi>k</mi></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct numerator, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math> factored. Denominator does not need to be simplified</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn><msup><mi>k</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for denominator correctly expanded. Numerator does not need to be simplified. These two <em><strong>A</strong></em> marks may be awarded in any order</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> The word ‘arctan’ must be present to be able to award the last three A marks</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> is true whenever <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> is true and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math> is true, so</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>n</mi></mfenced></math> is true for for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award the final <em><strong>R1</strong></em> mark provided at least four of the previous marks have been awarded.<br><strong>Note:</strong> To award the final <em><strong>R1</strong></em>, the truth of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> must be mentioned. ‘<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>’ is insufficient to award the mark.</p>
<p> </p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the binomial theorem to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>4</mn></msup></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mi mathvariant="normal">i</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> are expressed in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</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>Use de Moivre’s theorem and the result from part (a) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cot</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cot</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cot</mi><mo> </mo><mi>θ</mi></mrow></mfrac></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>Use the identity from part (b) to show that the quadratic equation <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>1</mn><mo>=</mo><mn>0</mn></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce a quadratic equation with integer coefficients, having roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></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 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 style="text-align:left;">uses the binomial theorem on <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>4</mn></msup></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mmultiscripts><mi>C</mi><mn>0</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mfenced><mrow><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>3</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>3</mn></msup><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>4</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>4</mn></msup><mo> </mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mfenced><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mfenced><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">(using de Moivre’s theorem with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math> gives) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> <strong>(A1)</strong></p>
<p style="text-align:left;">equates both the real and imaginary parts of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mfenced><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow></mfenced></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></math></p>
<p style="text-align:left;">recognizes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow><mrow><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow></mfrac></math> <strong>(A1)</strong></p>
<p style="text-align:left;">substitutes for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow><mrow><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow></mfrac></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow></mfrac></math></p>
<p style="text-align:left;">divides the numerator and denominator by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></math> to obtain</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mstyle displaystyle="true"><mfrac><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow><mrow><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow><mrow><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfrac></mstyle></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cot</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cot</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cot</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn></math> and putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> in the numerator of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cot</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cot</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cot</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math> gives <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>1</mn><mo>=</mo><mn>0</mn></math> <strong>M1</strong></p>
<p style="text-align:left;">attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo>…</mo><mo> </mo><mfenced><mrow><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi mathvariant="normal">π</mi><mo>,</mo><mo> </mo><mi mathvariant="normal">n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mo>…</mo></mrow></mfenced></math> <strong>(A1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Do not award the final <strong>A1</strong> if solutions other than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> are listed.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">finding the roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn><mo> </mo><mfenced><mrow><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></mrow></mfenced></math> corresponds to finding the roots of <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>1</mn><mo>=</mo><mn>0</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <strong>R1</strong></p>
<p style="text-align:left;">so the equation <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>1</mn><mo>=</mo><mn>0</mn></math> as roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts to solve <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>1</mn><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math> <strong>A1</strong></p>
<p style="text-align:left;">since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>></mo><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac><mo>,</mo><mo> </mo><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> has the smaller value of the two roots <strong>R1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>R1</strong> for an alternative convincing valid reason.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac><mo>=</mo><mn>3</mn><mo>-2</mo><msqrt><mn>2</mn></msqrt></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math></p>
<p style="text-align:left;">uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><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>1</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>6</mn><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
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