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<h2>HL Paper 3</h2><div class="specification">
<p>This question investigates the sum of sine and cosine functions</p>
</div>
<div class="specification">
<p>The expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{sin}}\,x + 4\,{\text{cos}}\,x">
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A\,{\text{cos}}(Bx + C) + D">
<mi>A</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>D</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A{\text{,}}\,\,B \in {\mathbb{R}^ + }">
<mi>A</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C{\text{,}}\,\,D \in \mathbb{R}">
<mi>C</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<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=" - \pi < C \leqslant \pi ">
<mo>−<!-- − --></mo>
<mi>π<!-- π --></mi>
<mo><</mo>
<mi>C</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>The expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5\,{\text{sin}}\,x + 12\,{\text{cos}}\,x">
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A\,{\text{cos}}(Bx + C) + D">
<mi>A</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>D</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A{\text{,}}\,\,B \in {\mathbb{R}^ + }">
<mi>A</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C{\text{,}}\,\,D \in \mathbb{R}">
<mi>C</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<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=" - \pi < C \leqslant \pi ">
<mo>−<!-- − --></mo>
<mi>π<!-- π --></mi>
<mo><</mo>
<mi>C</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>In general, the expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\,{\text{sin}}\,x + b\,{\text{cos}}\,x">
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A\,{\text{cos}}(Bx + C) + D">
<mi>A</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>D</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{,}}\,\,b{\text{,}}\,\,A{\text{,}}\,\,B \in {\mathbb{R}^ + }">
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>A</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C{\text{,}}\,\,D \in \mathbb{R}">
<mi>C</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<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=" - \pi < C \leqslant \pi ">
<mo>−<!-- − --></mo>
<mi>π<!-- π --></mi>
<mo><</mo>
<mi>C</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Conjecture an expression, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, for</p>
</div>
<div class="specification">
<p>The expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\,{\text{sin}}\,x + b\,{\text{cos}}\,x">
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> can also be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{a^2} + {b^2}} \left( {\frac{a}{{\sqrt {{a^2} + {b^2}} }}{\text{sin}}\,x + \frac{b}{{\sqrt {{a^2} + {b^2}} }}{\text{cos}}\,x} \right)">
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>a</mi>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mi>b</mi>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{a}{{\sqrt {{a^2} + {b^2}} }} = {\text{sin}}\,\theta ">
<mfrac>
<mi>a</mi>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ<!-- θ --></mi>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 3\,{\text{sin}}\,x + 4\,{\text{cos}}\,x">
<mi>y</mi>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2\pi \leqslant x \leqslant 2\pi ">
<mo>−</mo>
<mn>2</mn>
<mi>π</mi>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<mn>2</mn>
<mi>π</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>Write down the amplitude of this graph</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the period of this graph</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your answers from part (a) to write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span>.</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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\frac{3}{4}">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
</math></span>, giving the answer to 3 significant figures.</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>Comment on your answer to part (c)(i).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 5\,{\text{sin}}\,x + 12\,{\text{cos}}\,x">
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span>.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.iv.</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{b}{{\sqrt {{a^2} + {b^2}} }} = {\text{cos}}\,\theta ">
<mfrac>
<mi>b</mi>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{a}{b} = {\text{tan}}\,\theta ">
<mfrac>
<mi>a</mi>
<mi>b</mi>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence prove your conjectures in part (e).</p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>5 <em><strong>A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi ">
<mn>2</mn>
<mi>π</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 5">
<mi>A</mi>
<mo>=</mo>
<mn>5</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = 1">
<mi>B</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = 0">
<mi>D</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <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>maximum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.644">
<mi>x</mi>
<mo>=</mo>
<mn>0.644</mn>
</math></span> <em><strong>M1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = -0.644">
<mi>C</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.644</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.644 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>it appears that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = - {\text{arctan}}\frac{3}{4}">
<mi>C</mi>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 13">
<mi>A</mi>
<mo>=</mo>
<mn>13</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = 1">
<mi>B</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = 0">
<mi>D</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>maximum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.395">
<mi>x</mi>
<mo>=</mo>
<mn>0.395</mn>
</math></span> <em><strong>M1</strong></em></p>
<p>So C = −0.395 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = - {\text{arctan}}\frac{5}{{12}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mfrac>
<mn>5</mn>
<mrow>
<mn>12</mn>
</mrow>
</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">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \sqrt {{a^2} + {b^2}} ">
<mi>A</mi>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = 1">
<mi>B</mi>
<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">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = - {\text{arctan}}\frac{a}{b}">
<mi>C</mi>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mfrac>
<mi>a</mi>
<mi>b</mi>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = 0">
<mi>D</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong> EITHER</strong></p>
<p>use of a right triangle and Pythgoras’ to show the missing side length is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>OR</strong></p>
<p>Use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^2}\theta + {\text{co}}{{\text{s}}^2}\theta = 1">
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, leading to the required result <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong> EITHER</strong></p>
<p>use of a right triangle, leading to the required result. <em><strong>M1</strong></em></p>
<p><strong>OR</strong></p>
<p>Use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = \frac{{{\text{sin}}\,\theta }}{{{\text{cos}}\,\theta }}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span>, leading to the required result. <em><strong>M1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\,{\text{sin}}\,x + b\,{\text{cos}}\,x = \sqrt {{a^2} + {b^2}} \left( {{\text{sin}}\,\theta \,{\text{sin}}\,x + {\text{cos}}\,\theta \,{\text{cos}}\,x} \right)">
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\,{\text{sin}}\,x + b\,{\text{cos}}\,x = \sqrt {{a^2} + {b^2}} \left( {{\text{cos}}\left( {x - \theta } \right)} \right)">
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \sqrt {{a^2} + {b^2}} ">
<mi>A</mi>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = 1">
<mi>B</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = 0">
<mi>D</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>And <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = - \theta ">
<mi>C</mi>
<mo>=</mo>
<mo>−</mo>
<mi>θ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = - {\text{arctan}}\frac{a}{b}">
<mi>C</mi>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mfrac>
<mi>a</mi>
<mi>b</mi>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>This question asks you to investigate some properties of the sequence of functions of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right)">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 1 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
<p><strong>Important:</strong> When sketching graphs in this question, you are <strong>not</strong> required to find the coordinates of any axes intercepts or the coordinates of any stationary points unless requested.</p>
</div>
<div class="specification">
<p>For odd values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> > 2, use your graphic display calculator to systematically vary the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>. Hence suggest an expression for odd values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> describing, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, the number of</p>
</div>
<div class="specification">
<p>For even values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> > 2, use your graphic display calculator to systematically vary the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>. Hence suggest an expression for even values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>describing, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, the number of</p>
</div>
<div class="specification">
<p>The sequence of functions, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, defined above can be expressed as a sequence of polynomials of degree <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) = {\text{cos}}\left( {\left( {n + 1} \right)\,{\text{arccos}}\,x} \right)">
<mrow>
<msub>
<mi>f</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the same set of axes, sketch the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_1}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_3}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 1.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>local maximum points;</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>local minimum points;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On a new set of axes, sketch the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_2}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_4}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>4</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 1.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>local maximum points;</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>local minimum points.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}^\prime (x) = 0"> <msup> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mi mathvariant="normal">′</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> and hence show that the stationary points on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_n}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> occur at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{cos}}\frac{{k\pi }}{n}"> <mi>x</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in {\mathbb{Z}^ + }"> <mi>k</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> and 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an appropriate trigonometric identity to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_2}(x) = 2{x^2} - 1"> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an appropriate trigonometric identity to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) - {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) + {f_{n - 1}}(x) = 2x{f_n}\left( x \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }"> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_3}(x)"> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> as a cubic polynomial.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_1}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>A1</strong></em></p>
<p>correct graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_3}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>A1</strong></em></p>
<p><img 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"></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>graphical or tabular evidence that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> has been systematically varied <em><strong>M1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 3, 1 local maximum point and 1 local minimum point</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 5, 2 local maximum points and 2 local minimum points</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 7, 3 local maximum points and 3 local minimum points <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{n - 1}}{2}"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </math></span> local maximum points <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{n - 1}}{2}"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </math></span> local minimum points <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Allow follow through from an incorrect local maximum formula expression.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_2}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>A1</strong></em></p>
<p>correct graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_4}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>4</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>A1</strong></em></p>
<p><img src="data:image/png;base64,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"></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>graphical or tabular evidence that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> has been systematically varied <em><strong>M1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 2, 0 local maximum point and 1 local minimum point</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 4, 1 local maximum points and 2 local minimum points</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> = 6, 2 local maximum points and 3 local minimum points <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{n - 2}}{2}"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </math></span> local maximum points <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{n}{2}"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </math></span> local minimum points <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\left( x \right)} \right)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}^\prime (x) = \frac{{n\,{\text{sin}}\left( {n\,{\text{arccos}}\left( x \right)} \right)}}{{\sqrt {1 - {x^2}} }}"> <msup> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mi mathvariant="normal">′</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to use the chain rule.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}^\prime (x) = 0 \Rightarrow n\,{\text{sin}}\left( {n\,{\text{arccos}}\left( x \right)} \right) = 0"> <msup> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mi mathvariant="normal">′</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n\,{\text{arccos}}\left( x \right) = k\pi \,\,\,\left( {k \in {\mathbb{Z}^ + }} \right)"> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> <mi>π</mi> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>leading to</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{cos}}\frac{{k\pi }}{n}"> <mi>x</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in {\mathbb{Z}^ + }"> <mi>k</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> and 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>) <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_2}(x) = {\text{cos}}\left( {2\,{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2{\left( {{\text{cos}}\left( {{\text{arccos}}\,x} \right)} \right)^2} - 1"> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>M1</strong></em></p>
<p>stating that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{cos}}\left( {{\text{arccos}}\,x} \right)} \right) = x"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_2}(x) = 2{x^2} - 1"> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) = {\text{cos}}\left( {\left( {n + 1} \right)\,{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{cos}}\left( {n\,{\text{arccos}}\,x + {\text{arccos}}\,x} \right)"> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>use of cos(<em>A</em> + <em>B</em>) = cos <em>A </em>cos <em>B</em> − sin <em>A </em>sin <em>B</em> leading to <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) - {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)"> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n - 1}}(x) = {\text{cos}}\left( {\left( {n - 1} \right)\,{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) + {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)"> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) + {f_{n - 1}}(x) = 2\,{\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2x{f_n}\left( x \right)"> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_3}(x) = 2x{f_2}\left( x \right) - {f_1}(x)"> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2x\left( {2{x^2} - 1} \right) - x"> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4{x^3} - 3x"> <mo>=</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to examine various polygons for which the numerical value of the area is the same as the numerical value of the perimeter. For example, a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> rectangle has an area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math> and a perimeter of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math>.</strong></p>
<p> </p>
<p>For each polygon in this question, let the numerical value of its area be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and let the numerical value of its perimeter be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
</div>
<div class="specification">
<p>An <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon can be divided into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> congruent isosceles triangles. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> be the length of each of the two equal sides of one such isosceles triangle and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> be the length of the third side. The included angle between the two equal sides has magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></math>.</p>
<p>Part of such an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Consider a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
</div>
<div class="specification">
<p>The Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>5</mn></msup></mrow><mn>15</mn></mfrac><mo>+</mo><mo>…</mo></math></p>
</div>
<div class="specification">
<p>Consider a right-angled triangle with side lengths <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≥</mo><mi>b</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the side length, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>></mo><mn>0</mn></math>, of a square such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, an expression for the area, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub></math>, of one of these isosceles triangles.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the results from parts (b) and (c) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Interpret your answer to part (e)(i) geometrically.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>8</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>4</mn></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the result of part (f) or otherwise, determine the three side lengths of the only two right-angled triangles for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo> </mo><mi>P</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the area and perimeter of these two right-angled triangles.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><msup><mi>s</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>4</mn><mi>s</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><msup><mi>s</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mi>s</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mrow><mi>s</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>s</mi><mo>=</mo><mn>4</mn><mfenced><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1M1A0</strong></em> if both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mn>0</mn></math> are stated as final answers.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mrow><mn>2</mn><mo> </mo></mrow></msup><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1 </strong></em>for a correct alternative form expressed in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> only.</p>
<p> For example, using Pythagoras’ theorem, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mrow><mn>2</mn><mo> </mo></mrow></msup><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mn>2</mn><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mtext>opp</mtext><mtext>hyp</mtext></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mstyle displaystyle="true"><mfrac><mi>y</mi><mn>2</mn></mfrac></mstyle><mi>x</mi></mfrac><mo>=</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>uses Pythagoras’ theorem <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mi>y</mi><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mi>y</mi><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>uses the cosine rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p>uses the sine rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>y</mi><mrow><mi>sin</mi><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></mstyle></mrow></mfrac><mo>=</mo><mfrac><mi>x</mi><mrow><mi>sin</mi><mstyle displaystyle="true"><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mstyle></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><mi>n</mi><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mi>n</mi><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for equating correct expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo> </mo><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math> (seen anywhere in part (d) or in part (b)) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math></p>
<p>attempts to either factorise or divide their expression <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac><mo>,</mo><mo> </mo><mfenced><mrow><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>≠</mo><mn>0</mn></mrow></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> (or equivalent) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>n</mi><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mi>n</mi><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mfenced><mrow><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Other approaches are possible. For example, award<em><strong> A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>tan</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math> and <em><strong>M1</strong></em> for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
<p><strong><br>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> (or equivalent) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>n</mi><msub><mi>A</mi><mi>T</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mn>2</mn></msup><mfenced><mrow><mi>sin</mi><mfrac><mstyle displaystyle="true"><mn>2</mn><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mn>2</mn></msup><mfenced><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><msup><mfenced><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mfenced><mn>3</mn></msup></mstyle><mn>3</mn></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><mn>2</mn><msup><mfenced><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mfenced><mn>5</mn></msup></mstyle><mn>15</mn></mfrac><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mn>4</mn><mi>n</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>+</mo><mfrac><msup><mi mathvariant="normal">π</mi><mn>3</mn></msup><mrow><mn>3</mn><msup><mi>n</mi><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi mathvariant="normal">π</mi><mn>5</mn></msup></mrow><mrow><mn>15</mn><msup><mi>n</mi><mn>5</mn></msup></mrow></mfrac><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></mrow></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mfenced><mrow><mi mathvariant="normal">π</mi><mo>+</mo><mfrac><msup><mi mathvariant="normal">π</mi><mn>3</mn></msup><mrow><mn>3</mn><msup><mi>n</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi mathvariant="normal">π</mi><mn>5</mn></msup></mrow><mrow><mn>15</mn><msup><mi>n</mi><mn>4</mn></msup></mrow></mfrac><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mo>=</mo><mn>4</mn><mi mathvariant="normal">π</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>M1A1A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder></math> is not stated anywhere.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><mi>P</mi><mo>→</mo><mn>4</mn><mi mathvariant="normal">π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>→</mo><mn>4</mn><mi mathvariant="normal">π</mi></math>)</p>
<p>the polygon becomes a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong></em> for alternative responses such as:<br>the polygon becomes a circle of area <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">π</mi></math> OR<br>the polygon becomes a circle of perimeter <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">π</mi></math> OR<br>the polygon becomes a circle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi mathvariant="normal">π</mi></math>.<br>Award <em><strong>R0</strong></em> for polygon becomes a circle.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> <em><strong>(A1)(A1)</strong></em></p>
<p>equates their expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for isolating <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>2</mn><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math>. This step may be seen later.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></mrow></mfenced><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>+</mo><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo>-</mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to expand their RHS of either <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mo>…</mo></math>.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>a</mi><mi>b</mi><mo>≠</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mo>-</mo><mn>4</mn><mi>a</mi><mo>=</mo><mn>4</mn><mi>b</mi><mo>-</mo><mn>8</mn></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo>-</mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mi>b</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>2</mn><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>a</mi><mfenced><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>b</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>8</mn><mi>b</mi><mo>-</mo><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>b</mi></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>b</mi></mrow></mfrac></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>b</mi><mo>-</mo><mn>8</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>b</mi><mo>-</mo><mn>16</mn><mo>+</mo><mn>8</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>8</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>4</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>A1A1M1M1M0A0A0</strong></em> for attempting to verify.<br>For example, verifying that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mfrac><mn>16</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mi>b</mi><mo>+</mo><mn>4</mn></math> gains <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> marks.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using an appropriate method <em><strong>(M1)</strong></em></p>
<p><em>eg</em> substituting values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> or using divisibility properties</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>12</mn><mo>,</mo><mo> </mo><mn>13</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>6</mn><mo>,</mo><mo> </mo><mn>8</mn><mo>,</mo><mo> </mo><mn>10</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> for either one set of three correct side lengths or two sets of two correct side lengths.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>30</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>24</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>A1FT</strong></em>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br>