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<h2>HL Paper 1</h2><div class="question">
<p>The first term in an arithmetic sequence is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> and the fifth term is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>625</mn></math>.</p>
<p>Find the common difference of the sequence, expressing your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mi>p</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>5</mn></msub><mo>=</mo><mn>4</mn><mo>+</mo><mn>4</mn><mi>d</mi><mo>=</mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>625</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>d</mi><mo>=</mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>625</mn><mo>-</mo><mn>4</mn></math></p>
<p>attempt to write an integer (<em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>) in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>d</mi><mo>=</mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>625</mn><mo>-</mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>16</mn></math></p>
<p>attempt to combine two logs into one <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>d</mi><mo>=</mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mfenced><mfrac><mn>625</mn><mn>16</mn></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mfenced><mfrac><mn>625</mn><mn>16</mn></mfrac></mfenced></math></p>
<p>attempt to use power rule for logs <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><msup><mfenced><mfrac><mn>625</mn><mn>16</mn></mfrac></mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mfenced><mfrac><mn>5</mn><mn>2</mn></mfrac></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[5 marks]<br></strong></em></p>
<p><em><strong><br></strong></em><strong>Note: </strong>Award method marks in any order.<em><strong><br></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the integral <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_1^t {\frac{{ - 1}}{{x + {x^2}}}{\text{ }}} dx">
<munderover>
<mo>∫<!-- ∫ --></mo>
<mn>1</mn>
<mi>t</mi>
</munderover>
<mrow>
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
</mrow>
<mi>d</mi>
<mi>x</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t > 1">
<mi>t</mi>
<mo>></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Very briefly, explain why the value of this integral must be negative.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{ - 1}}{{x + {x^2}}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> in partial fractions.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use parts (a) and (b) to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t < {\text{ln}}\,2">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
<mo><</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>The numerator is negative but the denominator is positive. Thus the integrand is negative and so the value of the integral will be negative. <em><strong>R1AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - 1}}{{x + {x^2}}} = \frac{{ - 1}}{{\left( {1 + x} \right)x}} \equiv \frac{A}{{1 + x}} + \frac{B}{x}">
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>≡</mo>
<mfrac>
<mi>A</mi>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>B</mi>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>M1M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow - 1 \equiv Ax + B(1 + x) \Rightarrow A = 1,\,B = - 1">
<mo stretchy="false">⇒</mo>
<mo>−</mo>
<mn>1</mn>
<mo>≡</mo>
<mi>A</mi>
<mi>x</mi>
<mo>+</mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">⇒</mo>
<mi>A</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>B</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - 1}}{{x + {x^2}}} \equiv \frac{1}{{1 + x}} + \frac{{ - 1}}{x}">
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>≡</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mi>x</mi>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_1^t {\frac{1}{{1 + x}} + \frac{{ - 1}}{x}dx} = \left[ {{\text{ln}}\left( {1 + x} \right) - {\text{ln}}\,x} \right]_1^t = {\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t - {\text{ln}}\,2">
<munderover>
<mo>∫</mo>
<mn>1</mn>
<mi>t</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mi>x</mi>
</mfrac>
<mi>d</mi>
<mi>x</mi>
</mrow>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
<mn>1</mn>
<mi>t</mi>
</msubsup>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> <em><strong>M1A1A1</strong></em></p>
<p>Hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t - {\text{ln}}\,2 < 0 \Rightarrow {\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t < {\text{ln}}\,2">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo><</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
<mo><</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> <em><strong>R1AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Three planes have equations:</p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y + z = 5"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>5</mn> </math></span></p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 3y - z = 4"> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mn>4</mn> </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> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x - 5y + az = b"> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> <mo>+</mo> <mi>a</mi> <mi>z</mi> <mo>=</mo> <mi>b</mi> </math></span></p>
<p>Find the set of values 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> such that the three planes have no points of intersection.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to eliminate a variable (or attempt to find det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A"> <mi>A</mi> </math></span>) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 1&3&{ - 1} \\ 3&{ - 5}&a \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 4 \\ b \end{array}} \right) \to \left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 0&7&{ - 3} \\ 0&{ - 14}&{a + 3} \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 3 \\ {b - 12} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mspace width="thinmathspace"></mspace> </mrow> <mo>|</mo> </mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </mtd> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> <mspace width="thinmathspace"></mspace> </mrow> <mo>|</mo> </mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>b</mi> <mo>−</mo> <mn>12</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> (or det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 14\left( {a - 3} \right)"> <mi>A</mi> <mo>=</mo> <mn>14</mn> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </math></span>)</p>
<p>(or two correct equations in two variables) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \to \left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 0&7&{ - 3} \\ 0&{ 0}&{a - 3} \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 3 \\ {b - 6} \end{array}} \right)"> <mo stretchy="false">→</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mn>0</mn> </mrow> </mtd> <mtd> <mrow> <mi>a</mi> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> <mspace width="thinmathspace"></mspace> </mrow> <mo>|</mo> </mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>b</mi> <mo>−</mo> <mn>6</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> (or solving det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 0"> <mi>A</mi> <mo>=</mo> <mn>0</mn> </math></span>)</p>
<p>(or attempting to reduce to one variable, e.g. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {a - 3} \right)z = b - 6"> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mi>z</mi> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mn>6</mn> </math></span>) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 3{\text{, }}b \ne 6"> <mi>a</mi> <mo>=</mo> <mn>3</mn> <mrow> <mtext>, </mtext> </mrow> <mi>b</mi> <mo>≠</mo> <mn>6</mn> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>. The roots of this equation are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>3</mn></msub></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Im</mi><mfenced><msub><mi>ω</mi><mn>2</mn></msub></mfenced><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Im</mi><mfenced><msub><mi>ω</mi><mn>3</mn></msub></mfenced><mo><</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>3</mn></msub></math> are represented by the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> respectively on an Argand diagram.</p>
</div>
<div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><mtext>i</mtext><msup><mi>z</mi><mn>3</mn></msup><mo>,</mo><mo> </mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></math> is a root of this equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>3</mn></msub></math>, expressing these in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mi>θ</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plot the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> on an Argand diagram.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AC</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using de Moivre’s theorem, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle></mrow></msup></mrow></mfrac></math> is a root of this equation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mi>α</mi></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mfenced><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p><em><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">i</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Candidates who solve the equation correctly can be awarded the above two marks. The working for part (i) may be seen in part (ii).</p>
<p> </p>
<p><em><strong>[2</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>+</mo><mn>2</mn><mi>πk</mi></mrow></mfenced></mrow></msup></math><em> <strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>-</mo><mn>1</mn><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>πk</mi></mrow><mn>6</mn></mfrac></mrow></mfenced></mrow></msup></math><em> <strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></mfenced><mo>⇒</mo><msub><mi>ω</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></mrow></msup></math><em> <strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></mfenced><mo>⇒</mo><msub><mi>ω</mi><mn>3</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></mrow></msup></math><em> <strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></mrow></msup></math> in Cartesian form and translate 1 unit in the positive direction of the real axis<em> <strong>(M1)</strong></em></p>
<p><strong><br>OR</strong></p>
<p>attempt to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>3</mn></msub></math> in Cartesian form<em> <strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p style="padding-left:90px;"><img 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"></p>
<p><strong>Note:</strong> To award <em><strong>A</strong></em> marks, it is not necessary to see <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>1</mn></msub></math>, or the solid lines</p>
<p><em> <strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>1</mn></msub><mo>-</mo><msub><mi>ω</mi><mn>3</mn></msub><mo> </mo><mfenced><mrow><mi>or</mi><mo> </mo><msub><mi>ω</mi><mn>3</mn></msub><mo>-</mo><msub><mi>ω</mi><mn>1</mn></msub></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>1</mn></msub><mo>-</mo><msub><mi>ω</mi><mn>3</mn></msub><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">i</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">i</mi></mrow></mfenced><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi mathvariant="normal">i</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac></math></p>
<p>valid attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi mathvariant="normal">i</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>9</mn><mn>4</mn></mfrac></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AC</mi><mo>=</mo><msqrt><mn>3</mn></msqrt></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi><msup><mi>z</mi><mn>3</mn></msup><mo>⇒</mo><msup><mfenced><mfrac><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow><mi>z</mi></mfrac></mfenced><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow><mi>z</mi></mfrac></mfenced><mn>3</mn></msup><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>α</mi><mo>-</mo><mn>1</mn></mrow><mi>α</mi></mfrac><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> This step to change from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> may occur at any point in MS.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mn>1</mn><mo>=</mo><mi>α</mi><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>α</mi><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfenced><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi><msup><mi>z</mi><mn>3</mn></msup><mo>⇒</mo><msup><mfenced><mfrac><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow><mi>z</mi></mfrac></mfenced><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mi>z</mi></mfrac></mrow></mfenced><mn>3</mn></msup><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mi>z</mi></mfrac><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> This step to change from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> may occur at any point in MS.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>α</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>LHS<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><msup><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mfrac><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi mathvariant="normal">i</mi><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfenced><mn>3</mn></msup></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mi mathvariant="normal">i</mi><mrow><mstyle displaystyle="true"><mfrac><mn>5</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">i</mi><mfenced><mrow><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>5</mn><mn>2</mn></mfrac></mstyle></mrow></mfenced></mrow></mfrac></mrow></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for applying de Moivre’s theorem (may be seen in modulus- argument form.)</p>
<p><br>RHS<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">i</mi><msup><mi>z</mi><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi><msup><mfenced><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi mathvariant="normal">i</mi><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfenced><mn>3</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi><msup><mi>z</mi><mn>3</mn></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi><msup><mi>z</mi><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>z</mi><mo>-</mo><mn>1</mn><mo>=</mo><mi mathvariant="normal">i</mi><msup><mi>z</mi><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">i</mi></mrow></mfenced><msup><mi>z</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>z</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">i</mi></mrow></mfenced><msup><mfenced><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mfenced><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><mn>3</mn><mfenced><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac></mfenced><mo>-</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">i</mi></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfenced><mo>+</mo><mn>3</mn><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfenced><mn>3</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">i</mi></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mn>3</mn><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup><mo>+</mo><mn>3</mn><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> If the candidate does not interpret their conclusion, award <strong>(</strong><em><strong>M1)(A1)A0</strong></em> as appropriate.</p>
<p> </p>
<p><em><strong>[3</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mfenced><mrow><mi>cos</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle></mrow></mfenced></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><mrow><mn>2</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi mathvariant="normal">i</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p>attempt to use conjugate to rationalise <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mo>-</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>2</mn><mi mathvariant="normal">i</mi></mrow><mrow><msup><mfenced><mrow><mn>2</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mo>-</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>2</mn><mi mathvariant="normal">i</mi></mrow><mrow><mn>8</mn><mo>-</mo><mn>4</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mo>-</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac><mi mathvariant="normal">i</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mtext>Re</mtext><mfenced><mi>α</mi></mfenced><mi mathvariant="normal">=</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Their final imaginary part does not have to be correct in order for the final three <em><strong>A</strong></em> marks to be awarded</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mfenced><mrow><mi>cos</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle></mrow></mfenced></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p>attempt to use conjugate to rationalise <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle></mrow></mfenced><mo>-</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfrac><mo>×</mo><mfrac><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow><mrow><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow><mrow><mn>2</mn><mo>-</mo><mn>2</mn><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow><mrow><mn>2</mn><mo>-</mo><mn>2</mn><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mtext>Re</mtext><mfenced><mi>α</mi></mfenced><mi mathvariant="normal">=</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><em><br></em><strong>Note:</strong> Their final imaginary part does not have to be correct in order for the final three <em><strong>A</strong></em> marks to be awarded</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>attempt to multiply through by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></msup></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></msup><mrow><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></msup><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></msup></mrow></mfrac></math><em> <strong>A1</strong></em></p>
<p>attempting to re-write in r-cis form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><mi>cos</mi><mfenced><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mstyle></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mstyle></mrow></mfenced></mrow><mrow><mi>cos</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mstyle><mo>-</mo><mfenced><mrow><mi>cos</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfrac></math><em> <strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac><mo>-</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow><mrow><mn>2</mn><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mstyle></mrow></mfrac></math><em> <strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi mathvariant="normal">i</mi></mrow></mfrac><mi>cot</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">i</mi><mo> </mo><mi>cot</mi><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></mfenced></math></p>
<p><em><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mtext>Re</mtext><mfenced><mi>α</mi></mfenced><mi mathvariant="normal">=</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p>attempt to multiply through by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math><em> <strong>A1</strong></em></p>
<p>attempting to re-write in r-cis form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>-</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow><mrow><mn>2</mn><mo>-</mo><mn>2</mn><mo> </mo><mi>cos</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle></mrow></mfrac></math><em> <strong>A1</strong></em></p>
<p>attempt to re-write in Cartesian form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mstyle><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mi mathvariant="normal">i</mi></mrow><mrow><mn>2</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></mstyle><mrow><mn>2</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo>+</mo><mi mathvariant="normal">i</mi><mfrac><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mrow><mn>2</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></mfenced></math></p>
<p><em><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mtext>Re</mtext><mfenced><mi>α</mi></mfenced><mi mathvariant="normal">=</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Their final imaginary part does not have to be correct in order for the final <em><strong>A</strong></em> mark to be awarded</p>
<p> </p>
<p><em><strong>[6</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = 2\left( {{\text{cos}}\frac{\pi }{3} + {\text{i}}\,{\text{sin}}\frac{\pi }{3}} \right)">
<mi>w</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>These four points form the vertices of a quadrilateral, <em>Q</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup> in modulus-argument form.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch on an Argand diagram the points represented by <em>w</em><sup>0</sup> , <em>w</em><sup>1</sup> , <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the area of the quadrilateral <em>Q</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{21\sqrt 3 }}{2}"> <mfrac> <mrow> <mn>21</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 2\left( {{\text{cos}}\frac{\pi }{n} + {\text{i}}\,{\text{sin}}\frac{\pi }{n}} \right),\,\,n \in {\mathbb{Z}^ + }"> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>. The points represented on an Argand diagram by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^0},\,\,{z^1},\,\,{z^2},\, \ldots \,,\,\,{z^n}"> <mrow> <msup> <mi>z</mi> <mn>0</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mn>1</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> </math></span> form the vertices of a polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_n}"> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></span>.</p>
<p>Show that the area of the polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_n}"> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></span> can be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {{b^n} - 1} \right){\text{sin}}\frac{\pi }{n}"> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>b</mi> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,\,\,b\, \in \mathbb{R}"> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^2} = 4\text{cis}\left( {\frac{{2\pi }}{3}} \right){\text{;}}\,\,{w^3} = 8{\text{cis}}\left( \pi \right)"> <mrow> <msup> <mi>w</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>4</mn> <mtext>cis</mtext> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>;</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>w</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mn>8</mn> <mrow> <mtext>cis</mtext> </mrow> <mrow> <mo>(</mo> <mi>π</mi> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1A1</strong></em></p>
<p><strong>Note:</strong> Accept Euler form.</p>
<p><strong>Note:</strong> <em><strong>M1</strong></em> can be awarded for either both correct moduli or both correct arguments.</p>
<p><strong>Note:</strong> Allow multiplication of correct Cartesian form for <em><strong>M1</strong></em>, final answers must be in modulus-argument form.</p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}ab\,\,{\text{sin}}\,C"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>C</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 1 \times 2 \times {\text{sin}}\frac{\pi }{3} + \frac{1}{2} \times 2 \times 4 \times {\text{sin}}\frac{\pi }{3} + \frac{1}{2} \times 4 \times 8 \times {\text{sin}}\frac{\pi }{3}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>1</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>8</mn> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = \frac{\pi }{3}"> <mi>C</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span>, <em><strong>A1</strong> </em>for correct moduli.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{21\sqrt 3 }}{2}"> <mo>=</mo> <mfrac> <mrow> <mn>21</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong> AG</strong></em></p>
<p><strong>Note:</strong> Other methods of splitting the area may receive full marks.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times {2^0} \times {2^1} \times {\text{sin}}\frac{\pi }{n} + \frac{1}{2} \times {2^1} \times {2^2} \times {\text{sin}}\frac{\pi }{n} + \frac{1}{2} \times {2^2} \times {2^3} \times {\text{sin}}\frac{\pi }{n} + \, \ldots \, + \frac{1}{2} \times {2^{n - 1}} \times {2^n} \times {\text{sin}}\frac{\pi }{n}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>0</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>1</mn> </msup> </mrow> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>1</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>3</mn> </msup> </mrow> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for powers of 2, <em><strong>A1</strong> </em>for any correct expression including both the first and last term.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{sin}}\frac{\pi }{n} \times \left( {{2^0} + {2^2} + {2^4} + \, \ldots \, + {2^{n - 2}}} \right)"> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mn>0</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>2</mn> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>identifying a geometric series with common ratio 2<sup>2</sup>(= 4) <em><strong>(</strong><strong>M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{1 - {2^{2n}}}}{{1 - 4}} \times {\text{sin}}\frac{\pi }{n}"> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> </mrow> </mfrac> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for use of formula for sum of geometric series.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}\left( {{4^n} - 1} \right){\text{sin}}\frac{\pi }{n}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>4</mn> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mo>…</mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>p</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Consider the case where the series is geometric.</p>
</div>
<div class="specification">
<p>Now consider the case where the series is arithmetic with common difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, show that the series is convergent.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of the series is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;">attempt to use a ratio from consecutive terms <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><msup><mi>r</mi><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mfenced><mfrac><mn>1</mn><mrow><mn>3</mn><mi>p</mi></mrow></mfrac></mfenced></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Candidates may use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>p</mi></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mo>…</mo></math> and consider the powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in geometric sequence</p>
<p style="text-align:left;">Award <em><strong>M1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>p</mi><mn>1</mn></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>p</mi></mfrac></math>.</p>
<p style="text-align:left;"><strong><br>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>M0A0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> with no other working seen.</p>
<p style="text-align:left;"> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;">since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>p</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo><</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;">since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>p</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo></math> the geometric series converges. <em><strong>AG</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.<br>Award <em><strong>R0</strong> </em>if both values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> not considered.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mstyle></mrow></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mo>-</mo><mfrac><mn>3</mn><msqrt><mn>3</mn></msqrt></mfrac><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><msqrt><mn>3</mn></msqrt></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>1</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mtext>e</mtext><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">attempt to find a difference from consecutive terms or from <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;">correct equation <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mfenced><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;"><strong><br>Note:</strong> Candidates may use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>p</mi></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mo>…</mo></math> and consider the powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in arithmetic sequence.</p>
<p style="text-align:left;">Award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>-</mo><mn>1</mn><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>-</mo><mi>p</mi></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>p</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">attempt to use arithmetic mean <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>p</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 3</strong></p>
<p style="text-align:left;">attempt to find difference using <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>3</mn></msub></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>d</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>d</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="⌊" close="⌋"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow></mfenced></math></p>
<p style="text-align:left;">attempt to substitute into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> and equate to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="⌊" close="⌋"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced><mo>=</mo><mo>-</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>3</mn></msup><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">correct working with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="⌊" close="⌋"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>6</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mfrac><mrow><mn>4</mn><mo>-</mo><mi>n</mi></mrow><mn>3</mn></mfrac></mfenced><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;">correct equation without <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>7</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>6</mn></mfrac><mo>=</mo><mo>-</mo><mn>3</mn></math> or equivalent</p>
<p style="text-align:left;"><strong><br>Note:</strong> Award as above if the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mi>p</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mo>…</mo></math> is considered leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>7</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn></math>.</p>
<p style="text-align:left;"><br>attempt to form a quadratic <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>7</mn><mi>n</mi><mo>-</mo><mn>18</mn><mo>=</mo><mn>0</mn></math></p>
<p style="text-align:left;">attempt to solve their quadratic <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>-</mo><mn>9</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>9</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced><mo>=</mo><mo>-</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>3</mn></msup><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">listing the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> terms of the sequence <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>0</mn><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mo>…</mo></math></p>
<p style="text-align:left;">recognizing first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> terms sum to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math><sup>th</sup> term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math><sup>th</sup> term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">sum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math><sup>th</sup> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math><sup>th</sup> term <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>9</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a)(i) was well done with few candidates incorrectly using the value of <em>p</em> to verify rather than to 'show' the given result. In part (a)(ii) most did not consider both values of <em>r</em> and some did know the condition for convergence of a geometric series. Part (a)(iii) was generally well done but some had difficulty in simplifying the surd. Part (b) (i) and (ii) was generally well done. Although many completely correct answers to part b (iii) were noted, weaker candidates often made errors in properties of logarithms or algebraic manipulation leading to an incorrect quadratic equation.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="question">
<p>Determine the roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(z + 2{\text{i}})^3} = 216{\text{i}}">
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}">
<mi>z</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>, giving the answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = a\sqrt 3 + b{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,{\text{ }}b \in \mathbb{Z}">
<mi>a</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="216{\text{i}} = 216\left( {\cos \frac{\pi }{2} + {\text{i}}\sin \frac{\pi }{2}} \right)">
<mn>216</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>=</mo>
<mn>216</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z + 2{\text{i}} = \sqrt[3]{{216}}{\left( {\cos \left( {\frac{\pi }{2} + 2\pi k} \right) = {\text{i}}\sin \left( {\frac{\pi }{2} + 2\pi k} \right)} \right)^{\frac{1}{3}}}">
<mi>z</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>=</mo>
<mroot>
<mrow>
<mn>216</mn>
</mrow>
<mn>3</mn>
</mroot>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z + 2{\text{i}} = 6\left( {\cos \left( {\frac{\pi }{6} + \frac{{2\pi k}}{3}} \right) + {\text{i}}\sin \left( {\frac{\pi }{6} + \frac{{2\pi k}}{3}} \right)} \right)">
<mi>z</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
<mi>k</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
<mi>k</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} + 2{\text{i}} = 6\left( {\cos \frac{\pi }{6} + {\text{i}}\sin \frac{\pi }{6}} \right) = 6\left( {\frac{{\sqrt 3 }}{2} + \frac{{\text{i}}}{2}} \right) = 3\sqrt 3 + 3{\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2} + 2{\text{i}} = 6\left( {\cos \frac{{5\pi }}{6} + {\text{i}}\sin \frac{{5\pi }}{6}} \right) = 6\left( {\frac{{ - \sqrt 3 }}{2} + \frac{{\text{i}}}{2}} \right) = - 3\sqrt 3 + 3{\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mn>6</mn>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_3} + 2{\text{i}} = 6\left( {\cos \frac{{3\pi }}{2} + {\text{i}}\sin \frac{{3\pi }}{2}} \right) = - 6{\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <strong><em>A2</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1A0 </em></strong>for one correct root.</p>
<p> </p>
<p>so roots are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 3\sqrt 3 + {\text{i, }}{z_2} = - 3\sqrt 3 + {\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>3</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mrow>
<mtext>i, </mtext>
</mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_3} = - 8{\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for subtracting 2i from their three roots.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a\sqrt 3 + (b + 2){\text{i}}} \right)^3} = 216{\text{i}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a\sqrt 3 } \right)^3} + 3{\left( {a\sqrt 3 } \right)^2}(b + 2){\text{i}} - 3\left( {a\sqrt 3 } \right){(b + 2)^2} - {\text{i}}{(b + 2)^3} = 216{\text{i}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a\sqrt 3 } \right)^3} - 3\left( {a\sqrt 3 } \right){(b + 2)^2} + {\text{i}}\left( {3{{\left( {a\sqrt 3 } \right)}^2}(b + 2) - {{(b + 2)}^3}} \right) = 216{\text{i}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>216</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a\sqrt 3 } \right)^3} - 3\left( {a\sqrt 3 } \right){(b + 2)^2} = 0">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{\left( {a\sqrt 3 } \right)^2}(b + 2) - {(b + 2)^3} = 216">
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {{a^2} - {{(b + 2)}^2}} \right) = 0">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9{a^2}(b + 2) - {(b + 2)^3} = 216">
<mn>9</mn>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 0">
<mi>a</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^2} = {(b + 2)^2}">
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 0,{\text{ }} - {(b + 2)^3} = 216 \Rightarrow b + 2 = - 6">
<mi>a</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
<mo stretchy="false">⇒</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore b = - 8">
<mo>∴</mo>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>8</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(a,{\text{ }}b) = (0,{\text{ }} - 8)">
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^2} = {(b + 2)^2},{\text{ }}9{(b + 2)^2}(b + 2) - {(b + 2)^3} = 216">
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>9</mn>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8{(b + 2)^3} = 216">
<mn>8</mn>
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(b + 2)^3} = 27">
<mrow>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>27</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b + 2 = 3">
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mn>3</mn>
</math></span></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></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore {a^2} = 9 \Rightarrow a = \pm 3">
<mo>∴</mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mo>±</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore (a,{\text{ }}b) = ( \pm 3,{\text{ }}1)">
<mo>∴</mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>±</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1A1</em></strong></p>
<p>so roots are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 3\sqrt 3 + {\text{i, }}{z_2} = - 3\sqrt 3 + {\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>3</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mrow>
<mtext>i, </mtext>
</mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_3} = - 8{\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(z + 2{\text{i}})^3} - {( - 6{\text{i}})^3} = 0">
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>attempt to factorise: <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {(z + 2{\text{i}}) - ( - 6{\text{i}})} \right)\left( {{{(z + 2{\text{i}})}^2} + (z + 2{\text{i}})( - 6{\text{i}}) + {{( - 6{\text{i}})}^2}} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(z + 8{\text{i}})({z^2} - 2{\text{i}}z - 28) = 0">
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>+</mo>
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>z</mi>
<mo>−</mo>
<mn>28</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z + 8{\text{i}} = 0 \Rightarrow z = - 8{\text{i}}">
<mi>z</mi>
<mo>+</mo>
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^2} - 2{\text{i}}z - 28 = 0 \Rightarrow z = \frac{{2{\text{i}} \pm \sqrt { - 4 - (4 \times 1 \times - 28)} }}{2}">
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>z</mi>
<mo>−</mo>
<mn>28</mn>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>±</mo>
<msqrt>
<mo>−</mo>
<mn>4</mn>
<mo>−</mo>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>×</mo>
<mn>1</mn>
<mo>×</mo>
<mo>−</mo>
<mn>28</mn>
<mo stretchy="false">)</mo>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{{2{\text{i}} \pm \sqrt {108} }}{2}">
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>±</mo>
<msqrt>
<mn>108</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{{2{\text{i}} \pm 6\sqrt 3 }}{2}">
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>±</mo>
<mn>6</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = {\text{i}} \pm 3\sqrt 3 ">
<mi>z</mi>
<mo>=</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>±</mo>
<mn>3</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p>Special Case:</p>
<p><strong>Note: </strong>If a candidate recognises that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt[3]{{216{\text{i}}}} = - 6{\text{i}}">
<mroot>
<mrow>
<mn>216</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mn>3</mn>
</mroot>
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> (anywhere seen), and makes no valid progress in finding three roots, award <strong><em>A1 </em></strong>only.</p>
<p> </p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2x - 4}}{{{x^2} - 1}}{\text{, }} - 1 < x < 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>, </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>For the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span>, then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>show that there are no <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph, showing clearly any asymptotic behaviour.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}"> <mfrac> <mn>3</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The area enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 4"> <mi>y</mi> <mo>=</mo> <mn>4</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,v"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>v</mi> </math></span>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule (or equivalent) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{\left( {{x^2} - 1} \right)\left( 2 \right) - \left( {2x - 4} \right)\left( {2x} \right)}}{{{{\left( {{x^2} - 1} \right)}^2}}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 2{x^2} + 8x - 2}}{{{{\left( {{x^2} - 1} \right)}^2}}}"> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>simplifying numerator (may be seen in part (i)) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {x^2} - 4x + 1 = 0"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </math></span> or equivalent quadratic equation <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>use of quadratic formula</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = \frac{{4 \pm \sqrt {12} }}{2}"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mo>±</mo> <msqrt> <mn>12</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>use of completing the square</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - 2} \right)^2} = 3"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> (since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + \sqrt 3 "> <mn>2</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> is outside the domain) <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not condone verification that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 \Rightarrow f'\left( x \right) = 0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> <mo stretchy="false">⇒</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<p>Do not award the final <em><strong>A1</strong></em> as follow through from part (i).</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(0, 4) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - 4 = 0 \Rightarrow x = 2"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A1</strong></em></p>
<p>outside the domain <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>award <em><strong>A1</strong></em> for concave up curve over correct domain with one minimum point in the first quadrant<br>award <em><strong>A1</strong></em> for approaching <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pm 1"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </math></span> asymptotically</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to combine fractions (using common denominator) <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\left( {x - 1} \right) - \left( {x + 1} \right)}}{{\left( {x + 1} \right)\left( {x - 1} \right)}}"> <mfrac> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{3x - 3 - x - 1}}{{{x^2} - 1}}"> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{2x - 4}}{{{x^2} - 1}}"> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 4 \Rightarrow 2x - 4 = 4{x^2} - 4"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo stretchy="false">⇒</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> or) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p>area under the curve is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int_0^{\frac{1}{2}} {\frac{3}{{x + 1}} - \frac{1}{{x - 1}}{\text{d}}x} "> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mn>3</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><strong>Note:</strong> Ignore absence of, or incorrect limits up to this point.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {3\,{\text{ln}}\,\left| {x + 1} \right| - {\text{ln}}\,\left| {x - 1} \right|} \right]_0^{\frac{1}{2}}"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\,{\text{ln}}\frac{3}{2} - {\text{ln}}\frac{1}{2}\left( { - 0} \right)"> <mo>=</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\frac{{27}}{4}"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>27</mn> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>area is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 - \int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <mn>2</mn> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{1}{2}} {4\,{\text{d}}x} - \int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2 - {\text{ln}}\frac{{27}}{4}"> <mo>=</mo> <mn>2</mn> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>27</mn> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\frac{{4\,{{\text{e}}^2}}}{{27}}"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>27</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { \Rightarrow v = \frac{{4\,{{\text{e}}^2}}}{{27}}} \right)"> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>27</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A farmer has six sheep pens, arranged in a grid with three rows and two columns as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Five sheep called Amber, Brownie, Curly, Daisy and Eden are to be placed in the pens. Each pen is large enough to hold all of the sheep. Amber and Brownie are known to fight.</p>
<p>Find the number of ways of placing the sheep in the pens in each of the following cases:</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Each pen is large enough to contain five sheep. Amber and Brownie must not be placed in the same pen.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Each pen may only contain one sheep. Amber and Brownie must not be placed in pens which share a boundary.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>B has one less pen to select <em><strong>(M1)</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p>A and B can be placed in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>5</mn></math> ways <em><strong>(A1)</strong></em></p>
<p>C, D, E have <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> choices each <em><strong>(A1)</strong></em></p>
<p><strong><br>OR</strong></p>
<p>A (or B), C, D, E have <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> choices each <em><strong>(A1)</strong></em></p>
<p>B (or A) has only <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> choices <em><strong>(A1)</strong></em></p>
<p><strong><br>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>×</mo><msup><mn>6</mn><mn>4</mn></msup><mo> </mo><mfenced><mrow><mo>=</mo><mn>6480</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>total number of ways <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mn>6</mn><mn>5</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>number of ways with Amber and Brownie together <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mn>6</mn><mn>4</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>attempt to subtract (may be seen in words) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>6</mn><mn>5</mn></msup><mo>-</mo><msup><mn>6</mn><mn>4</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>5</mn><mo>×</mo><msup><mn>6</mn><mn>4</mn></msup><mo> </mo><mfenced><mrow><mo>=</mo><mn>6480</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>total number of ways <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo>!</mo><mo>(</mo><mo>=</mo><mn>720</mn><mo>)</mo></math> <em><strong>(A1)</strong></em></p>
<p>number of ways with Amber and Brownie sharing a boundary</p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>4</mn><mo>!</mo><mo>(</mo><mo>=</mo><mn>336</mn><mo>)</mo></math> <em><strong>(A1)</strong></em></p>
<p>attempt to subtract (may be seen in words) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>720</mn><mo>-</mo><mn>336</mn><mo>=</mo><mn>384</mn></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>case 1: number of ways of placing A in corner pen</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></math></p>
<p>Four corners total no of ways is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><mo>(</mo><mn>3</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>12</mn><mo>×</mo><mn>4</mn><mo>!</mo><mo>(</mo><mo>=</mo><mn>288</mn><mo>)</mo></math> <em><strong>(A1)</strong></em></p>
<p>case 2: number of ways of placing A in the middle pen</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></math></p>
<p>two middle pens so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mo>(</mo><mn>2</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>4</mn><mo>×</mo><mn>4</mn><mo>!</mo><mo>(</mo><mo>=</mo><mn>96</mn><mo>)</mo></math> <em><strong>(A1)</strong></em></p>
<p>attempt to add (may be seen in words) <em><strong>(M1)</strong></em></p>
<p>total no of ways <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>288</mn><mo>+</mo><mn>96</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>16</mn><mo>×</mo><mn>4</mn><mo>!</mo><mo>(</mo><mo>=</mo><mn>384</mn><mo>)</mo></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two distinct lines, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, intersect at a point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span>. In addition to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span>, four distinct points are marked out on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and three distinct points on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>. A mathematician decides to join some of these eight points to form polygons.</p>
</div>
<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> has vector equation <em><strong>r</strong></em><sub>1</sub> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 1 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 1 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ<!-- λ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda \in \mathbb{R}">
<mi>λ<!-- λ --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> has vector equation <em><strong>r</strong></em><sub>2</sub> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 1} \\ 0 \\ 2 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}} 5 \\ 6 \\ 2 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>μ<!-- μ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu \in \mathbb{R}">
<mi>μ<!-- μ --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> has coordinates (4, 6, 4).</p>
</div>
<div class="specification">
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> has coordinates (3, 4, 3) and lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> has coordinates (−1, 0, 2) and lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how many sets of four points can be selected which can form the vertices of a quadrilateral.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how many sets of three points can be selected which can form the vertices of a triangle.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> is the point of intersection of the two lines.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> corresponding to the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PA}}} ">
<mover>
<mrow>
<mtext>PA</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PB}}} ">
<mover>
<mrow>
<mtext>PB</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}">
<mrow>
<mtext>C</mtext>
</mrow>
</math></span> be the point on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> with coordinates (1, 0, 1) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{D}}">
<mrow>
<mtext>D</mtext>
</mrow>
</math></span> be the point on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> with parameter <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = - 2">
<mi>μ</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span>.</p>
<p>Find the area of the quadrilateral <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{CDBA}}">
<mrow>
<mtext>CDBA</mtext>
</mrow>
</math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>appreciation that two points distinct from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}"> <mrow> <mtext>P</mtext> </mrow> </math></span> need to be chosen from each line <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{}^4{C_2} \times {}^3{C_2}"> <msup> <mrow> </mrow> <mn>4</mn> </msup> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> <mo>×</mo> <msup> <mrow> </mrow> <mn>3</mn> </msup> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </math></span></p>
<p>=18 <em><strong>A</strong><strong>1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>consider cases for triangles including <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> <strong>or</strong> triangles not including <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 \times 4 + 4 \times {}^3{C_2} + 3 \times {}^4{C_2}">
<mn>3</mn>
<mo>×</mo>
<mn>4</mn>
<mo>+</mo>
<mn>4</mn>
<mo>×</mo>
<msup>
<mrow>
</mrow>
<mn>3</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<msup>
<mrow>
</mrow>
<mn>4</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> <em><strong>(A</strong><strong>1)(</strong></em><em><strong>A</strong><strong>1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for 1st term, <em><strong>A1</strong></em> for 2nd & 3rd term.</p>
<p><strong>OR</strong></p>
<p>consider total number of ways to select 3 points and subtract those with 3 points on the same line <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{}^8{C_3} - {}^5{C_3} - {}^4{C_3}">
<msup>
<mrow>
</mrow>
<mn>8</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>−</mo>
<msup>
<mrow>
</mrow>
<mn>5</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>−</mo>
<msup>
<mrow>
</mrow>
<mn>4</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> <em><strong>(A</strong><strong>1)(</strong></em><em><strong>A</strong><strong>1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for 1st term, <em><strong>A1</strong></em> for 2nd & 3rd term.</p>
<p>56−10−4</p>
<p><strong>THEN</strong></p>
<p>= 42 <em><strong>A</strong><strong>1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substitution of (4, 6, 4) into both equations <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 3">
<mi>λ</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 1">
<mi>μ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>(4, 6, 4) <em><strong>AG</strong></em></p>
<p><strong>METHOD 2</strong></p>
<p>attempting to solve two of the three parametric equations <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 3">
<mi>λ</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 1">
<mi>μ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>check both of the above give (4, 6, 4) <em><strong>M1</strong></em><em><strong>AG</strong></em></p>
<p><strong>Note:</strong> If they have shown the curve intersects for all three coordinates they only need to check (4,6,4) with one of "<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span>" or "<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span>".</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 2">
<mi>λ</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PA}}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ { - 2} \\ { - 1} \end{array}} \right)">
<mover>
<mrow>
<mtext>PA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PB}}} = \left( {\begin{array}{*{20}{c}} { - 5} \\ { - 6} \\ { - 2} \end{array}} \right)">
<mover>
<mrow>
<mtext>PB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if both are given as coordinates.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABP}} = \frac{1}{2}\left| {\overrightarrow {{\text{PB}}} \times \overrightarrow {{\text{PA}}} } \right|">
<mrow>
<mtext>ABP</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>PB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>PA</mtext>
</mrow>
<mo>→</mo>
</mover>
</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="\left( { = \frac{1}{2}\left| {\left( {\begin{array}{*{20}{c}} { - 5} \\ { - 6} \\ { - 2} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} { - 1} \\ { - 2} \\ { - 1} \end{array}} \right)} \right|} \right) = \frac{1}{2}\left| {\left( {\begin{array}{*{20}{c}} 2 \\ { - 3} \\ 4 \end{array}} \right)} \right|">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt {29} }}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PC}}} = 3\overrightarrow {\,{\text{PA}}} ">
<mover>
<mrow>
<mtext>PC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mn>3</mn>
<mover>
<mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>PA</mtext>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PD}}} = 3\overrightarrow {\,{\text{PB}}} ">
<mover>
<mrow>
<mtext>PD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mn>3</mn>
<mover>
<mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>PB</mtext>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> <em><strong>(M1)</strong></em></p>
<p>area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PCD}} = 9 \times ">
<mrow>
<mtext>PCD</mtext>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo>×</mo>
</math></span> area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABP}}">
<mrow>
<mtext>ABP</mtext>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{9\sqrt {29} }}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{D}}">
<mrow>
<mtext>D</mtext>
</mrow>
</math></span> has coordinates (−11, −12, −2) <em><strong>A1</strong></em></p>
<p>area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PCD}} = \frac{1}{2}\left| {\overrightarrow {{\text{PD}}} \times \overrightarrow {{\text{PC}}} } \right| = \frac{1}{2}\left| {\left( {\begin{array}{*{20}{c}} { - 15} \\ { - 18} \\ { - 6} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 6} \\ { - 3} \end{array}} \right)} \right|">
<mrow>
<mtext>PCD</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>PD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>PC</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>15</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note: <em>A1</em></strong> is for the correct vectors in the correct formula.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{9\sqrt {29} }}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p>area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{CDBA}} = \frac{{9\sqrt {29} }}{2} - \frac{{\sqrt {29} }}{2}">
<mrow>
<mtext>CDBA</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\sqrt {29} ">
<mo>=</mo>
<mn>4</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{D}}">
<mrow>
<mtext>D</mtext>
</mrow>
</math></span> has coordinates (−11, −12, −2) <em><strong>A1</strong></em></p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left| {\overrightarrow {{\text{CB}}} \times \overrightarrow {{\text{CA}}} } \right| + \frac{1}{2}\left| {\overrightarrow {{\text{BC}}} \times \overrightarrow {{\text{BD}}} } \right|">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>CA</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>BD</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for use of correct formula on appropriate non-overlapping triangles.</p>
<p><strong>Note:</strong> Different triangles or vectors could be used.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CB}}} = \left( {\begin{array}{*{20}{c}} { - 2} \\ 0 \\ 1 \end{array}} \right)">
<mover>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CA}}} = \left( {\begin{array}{*{20}{c}} 2 \\ 4 \\ 2 \end{array}} \right)">
<mover>
<mrow>
<mtext>CA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</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="\overrightarrow {{\text{CB}}} \times \overrightarrow {{\text{CA}}} = \left( {\begin{array}{*{20}{c}} { - 4} \\ 6 \\ { - 8} \end{array}} \right)">
<mover>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>CA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</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="\overrightarrow {{\text{BC}}} = \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ { - 1} \end{array}} \right)">
<mover>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{BD}}} = \left( {\begin{array}{*{20}{c}} { - 10} \\ { - 12} \\ { - 4} \end{array}} \right)">
<mover>
<mrow>
<mtext>BD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</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="\overrightarrow {{\text{BC}}} \times \overrightarrow {{\text{BD}}} = \left( {\begin{array}{*{20}{c}} { - 12} \\ {18} \\ { - 24} \end{array}} \right)">
<mover>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>BD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>24</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Other vectors which might be used are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{DA}}} = \left( {\begin{array}{*{20}{c}} {14} \\ {16} \\ {5} \end{array}} \right)">
<mover>
<mrow>
<mtext>DA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>14</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>16</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{BA}}} = \left( {\begin{array}{*{20}{c}} {4} \\ {4} \\ {1} \end{array}} \right)">
<mover>
<mrow>
<mtext>BA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{DC}}} = \left( {\begin{array}{*{20}{c}} {12} \\ {12} \\ {3} \end{array}} \right)">
<mover>
<mrow>
<mtext>DC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p><strong>Note:</strong> Previous <em><strong>A1A1A1A1</strong></em> are all dependent on the first <em><strong>M1</strong></em>.</p>
<p>valid attempt to find a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left| {a \times b} \right|">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong> </em>independent of triangle chosen.</p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times 2 \times \sqrt {29} + \frac{1}{2} \times 6 \times \sqrt {29} ">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>2</mn>
<mo>×</mo>
<msqrt>
<mn>29</mn>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<msqrt>
<mn>29</mn>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\sqrt {29} ">
<mo>=</mo>
<mn>4</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \sqrt {116} + \frac{1}{2}\sqrt {1044} ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msqrt>
<mn>116</mn>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msqrt>
<mn>1044</mn>
</msqrt>
</math></span> or equivalent.</p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Use the method of mathematical induction to prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^n} + 15n - 1">
<mrow>
<msup>
<mn>4</mn>
<mi>n</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> is divisible by 9 for <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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(n)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math></span> be the proposition that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^n} + 15n - 1">
<mrow>
<msup>
<mn>4</mn>
<mi>n</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> is divisible by 9</p>
<p>showing true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><em>ie</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1,{\text{ }}{4^1} + 15 \times 1 - 1 = 18">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mn>4</mn>
<mn>1</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mo>×</mo>
<mn>1</mn>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>18</mn>
</math></span></p>
<p>which is divisible by 9, therefore <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(1)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> is true</p>
<p>assume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(k)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</math></span> is true so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^k} + 15k - 1 = 9A,{\text{ }}(A \in {\mathbb{Z}^ + })">
<mrow>
<msup>
<mn>4</mn>
<mi>k</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>9</mn>
<mi>A</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only award <strong><em>M1 </em></strong>if “truth assumed” or equivalent.</p>
<p> </p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^{k + 1}} + 15(k + 1) - 1">
<mrow>
<msup>
<mn>4</mn>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4 \times {4^k} + 15k + 14">
<mo>=</mo>
<mn>4</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>4</mn>
<mi>k</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>k</mi>
<mo>+</mo>
<mn>14</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4(9A - 15k + 1) + 15k + 14">
<mo>=</mo>
<mn>4</mn>
<mo stretchy="false">(</mo>
<mn>9</mn>
<mi>A</mi>
<mo>−</mo>
<mn>15</mn>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>15</mn>
<mi>k</mi>
<mo>+</mo>
<mn>14</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4 \times 9A - 45k + 18">
<mo>=</mo>
<mn>4</mn>
<mo>×</mo>
<mn>9</mn>
<mi>A</mi>
<mo>−</mo>
<mn>45</mn>
<mi>k</mi>
<mo>+</mo>
<mn>18</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 9(4A - 5k + 2)">
<mo>=</mo>
<mn>9</mn>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mi>A</mi>
<mo>−</mo>
<mn>5</mn>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span> which is divisible by 9 <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>R1 </em></strong>for either the expression or the statement above.</p>
<p> </p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(1)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> is true and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(k)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</math></span> true implies <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(k + 1)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> is true, therefore (by the principle of mathematical induction) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(n)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math></span> is true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only award the final <strong><em>R1 </em></strong>if the 2 <strong><em>M1</em></strong>s have been awarded.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>In the following Argand diagram, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>2</mtext></msub></math> are the vertices of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>1</mtext></msub><msub><mtext>OZ</mtext><mtext>2</mtext></msub></math> described anticlockwise.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub></math> represents the complex number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>α</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></math>. The point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>2</mn></msub></math> represents the complex number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>θ</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></math>.</p>
<p>Angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>θ</mi></math> are measured anticlockwise from the positive direction of the real axis such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>,</mo><mo> </mo><mi>θ</mi><mo><</mo><mn>2</mn><mi>π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math>.</p>
</div>
<div class="specification">
<p>In parts (c), (d) and (e), consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>1</mtext></msub><msub><mtext>OZ</mtext><mtext>2</mtext></msub></math> is an equilateral triangle.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math> be the distinct roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>z</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup></math> is the complex conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub><msub><mtext>OZ</mtext><mn>2</mn></msub></math> is a right-angled triangle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the result from part (c)(ii) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>b</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>z</mi><mo>+</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math>, deduce that only one equilateral triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub><msub><mtext>OZ</mtext><mn>2</mn></msub></math> can be formed from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the roots of this equation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>-i</mtext><mi>θ</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>α</mi></mrow></msup><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>-i</mtext><mi>θ</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced></mrow></msup></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Accept working in modulus-argument form</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup></mrow></mfenced><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub><mo> </mo><mi>cos</mi><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>θ</mi><mo>=</mo><mtext>arcos</mtext><mo> </mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>,</mo><msub><mi>r</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi mathvariant="normal">π</mi></math>) <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub><msub><mtext>OZ</mtext><mn>2</mn></msub></math> is a right-angled triangle <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mfrac><mfenced><mrow><mo>=</mo><mfrac><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub></mfrac><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced></mrow></msup></mrow></mfenced><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub></math>) <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mi>e</mi><mrow><mi>i</mi><mfenced><mrow><mi>θ</mi><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></mfenced></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>θ</mi></mrow></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept working in either modulus-argument form to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></mfenced></math> or in Cartesian form to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mtext>i</mtext></mrow></mfenced></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup></math> <em><strong>M</strong><strong>1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mtext>i</mtext><mo>+</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><msub><mi>z</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> For candidates who work on the LHS and RHS separately to show equality, award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mfenced><mrow><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math>, <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></msup></math>. Accept working in either modulus-argument form or in Cartesian form.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mi>b</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>+</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><mo>=</mo><mn>3</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> into their expression <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>b</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn><mi>b</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> is not clearly recognized, award maximum <em><strong>(A0)A1A1M1A0</strong></em>.</p>
<p> </p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>b</mi><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mi>b</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>2</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>+</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mfenced><mrow><mo>=</mo><mn>3</mn><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> into their expression <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>b</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn><mi>b</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>a</mi></math> is not clearly recognized, award maximum <em><strong>(A0)A1A1M1A0</strong></em>.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>b</mi><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>×</mo><mn>12</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>±</mo><mn>6</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><msup><mi>z</mi><mn>2</mn></msup><mo>±</mo><mn>6</mn><mi>z</mi><mo>+</mo><mn>12</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>6</mn><mo>:</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt><mtext>i</mtext><mo>,</mo><mo> </mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mn>3</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></math> which does not satisfy <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math> <em><strong>R1</strong></em></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>6</mn><mo>:</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mo>-</mo><mn>3</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mtext>i</mtext><mo>,</mo><mo> </mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>-</mo><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>so (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math>), only one equilateral triangle can be formed from point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the two roots of this equation <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The vast majority of candidates scored full marks in parts (a) and (b). If they did not, it was normally due to the lack of rigour in setting out of the answer to a "show that" question. Part (c) was, though, more often than not poorly done. Many candidates could not use the given condition (equilateral triangle) to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math>. Part (d) was well answered by a rather high number of candidates.</p>
<p>Only a handful of students made good progress in (e), not even finding the possible values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following system of equations where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in \mathbb{R}">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + 4y - z = 10">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2y + az = 5">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mi>a</mi>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5x + 12y = 2a">
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>a</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> for which the system of equations does not have a unique solution.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the solution of the system of equations when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 2">
<mi>a</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>an attempt at a valid method<em> eg</em> by inspection or row reduction <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times {R_2} = {R_1} \Rightarrow 2a = - 1">
<mn>2</mn>
<mo>×</mo>
<mrow>
<msub>
<mi>R</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>R</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mn>2</mn>
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a = - \frac{1}{2}">
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using elimination or row reduction to eliminate one variable <em><strong>(M1)</strong></em></p>
<p>correct pair of equations in 2 variables, such as</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left. {\begin{array}{*{20}{c}} {5x + 10y = 25} \\ {5x + 12y = 4} \end{array}} \right\}">
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
<mi>y</mi>
<mo>=</mo>
<mn>25</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
<mi>y</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>}</mo>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> = 0 and one other equation in two variables.</p>
<p> </p>
<p>attempting to solve for these two variables <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 26">
<mi>x</mi>
<mo>=</mo>
<mn>26</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 10.5">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>10.5</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 0">
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> A1A0</strong></em> for only two correct values, and <em><strong>A0A0</strong></em> for only one.</p>
<p><strong>Note:</strong> Award marks in part (b) for equivalent steps seen in part (a).</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A team of four is to be chosen from a group of four boys and four girls.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of different possible teams that could be chosen.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of different possible teams that could be chosen, given that the team must include at least one girl and at least one boy.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 8 \\ 4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{8{\text{!}}}}{{4{\text{!}}4{\text{!}}}} = \frac{{8 \times 7 \times 6 \times 5}}{{4 \times 3 \times 2 \times 1}} = 7 \times 2 \times 5">
<mo>=</mo>
<mfrac>
<mrow>
<mn>8</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<mtext>!</mtext>
</mrow>
<mn>4</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>8</mn>
<mo>×</mo>
<mn>7</mn>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<mn>5</mn>
</mrow>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>3</mn>
<mo>×</mo>
<mn>2</mn>
<mo>×</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>7</mn>
<mo>×</mo>
<mn>2</mn>
<mo>×</mo>
<mn>5</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 70 <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>recognition that they need to count the teams with 0 boys, 1 boy… 4 boys <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \left( {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) + 1">
<mn>1</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 + \left( {4 \times 4} \right) + \left( {6 \times 6} \right) + \left( {4 \times 4} \right) + 1">
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>×</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p>= 70 <em><strong>A1</strong></em></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>recognition that the answer is the total number of teams minus the number of teams with all girls or all boys <em><strong>(M1)</strong></em></p>
<p>70 − 2</p>
<p><strong>OR</strong></p>
<p>recognition that the answer is the total of the number of teams with 1 boy,</p>
<p>2 boys, 3 boys <em><strong>(M1)</strong></em></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) = \left( {4 \times 4} \right) + \left( {6 \times 6} \right) + \left( {4 \times 4} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>×</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong>THEN</strong></p>
<p>= 68 <em><strong>A1</strong></em></p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Find the solution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}x - {\log _2}5 = 2 + {\log _2}3">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>5</mn>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>3</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}x - {\log _2}5 = 2 + {\log _2}3">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>5</mn>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>3</mn>
</math></span></p>
<p>collecting at least two log terms <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}\frac{x}{5} = 2 + {\log _2}3{\text{ or }}{\log _2}\frac{x}{{15}} = 2">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mfrac>
<mi>x</mi>
<mn>5</mn>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>3</mn>
<mrow>
<mtext> or </mtext>
</mrow>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mfrac>
<mi>x</mi>
<mrow>
<mn>15</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
</math></span></p>
<p>obtaining a correct equation without logs <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{5} = 12">
<mfrac>
<mi>x</mi>
<mn>5</mn>
</mfrac>
<mo>=</mo>
<mn>12</mn>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{{15}} = {2^2}">
<mfrac>
<mi>x</mi>
<mrow>
<mn>15</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 60">
<mi>x</mi>
<mo>=</mo>
<mn>60</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <em>S</em> be the sum of the roots found in part (a).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^{24}} = 1"> <mrow> <msup> <mi>z</mi> <mrow> <mn>24</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>1</mn> </math></span> which satisfy the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < {\text{arg}}\left( z \right) < \frac{\pi }{2}"> <mn>0</mn> <mo><</mo> <mrow> <mtext>arg</mtext> </mrow> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, expressing your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{e^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mi>e</mi> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta \in {\mathbb{R}^ + }"> <mi>θ</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Re <em>S</em> = Im <em>S</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>, find the value of cos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sqrt a + \sqrt b }}{c}"> <mfrac> <mrow> <msqrt> <mi>a</mi> </msqrt> <mo>+</mo> <msqrt> <mi>b</mi> </msqrt> </mrow> <mi>c</mi> </mfrac> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> are integers to be determined.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, show that <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left( {1 + \sqrt 2 } \right)\left( {1 + \sqrt 3 } \right)\left( {1 + {\text{i}}} \right)"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {r\left( {{\text{cos}}\,\theta + {\text{i}}\,{\text{sin}}\,\theta } \right)} \right)^{24}} = 1\left( {{\text{cos}}\,0 + {\text{i}}\,{\text{sin}}\,0} \right)"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mn>24</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>0</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>use of De Moivre’s theorem <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^{24}} = 1 \Rightarrow r = 1"> <mrow> <msup> <mi>r</mi> <mrow> <mn>24</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>r</mi> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\theta = 2\pi n \Rightarrow \theta = \frac{{\pi n}}{{12}}"> <mn>24</mn> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mo stretchy="false">⇒</mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mi>n</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {n \in \mathbb{Z}} \right)"> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < {\text{arg}}\left( z \right) < \frac{\pi }{2} \Rightarrow n = "> <mn>0</mn> <mo><</mo> <mrow> <mtext>arg</mtext> </mrow> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo stretchy="false">⇒</mo> <mi>n</mi> <mo>=</mo> </math></span> 1, 2, 3, 4, 5</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = {\text{e}}\frac{{\pi {\text{i}}}}{{12}}"> <mi>z</mi> <mo>=</mo> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{e}}\frac{{2\pi {\text{i}}}}{{12}}"> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{e}}\frac{{3\pi {\text{i}}}}{{12}}"> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{e}}\frac{{4\pi {\text{i}}}}{{12}}"> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{e}}\frac{{5\pi {\text{i}}}}{{12}}"> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> if additional roots are given or if three correct roots are given with no incorrect (or additional) roots.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Re <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{\pi }{{12}} + {\text{cos}}\frac{{2\pi }}{{12}} + {\text{cos}}\frac{{3\pi }}{{12}} + {\text{cos}}\frac{{4\pi }}{{12}} + {\text{cos}}\frac{{5\pi }}{{12}}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span></p>
<p>Im <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{\pi }{{12}} + {\text{sin}}\frac{{2\pi }}{{12}} + {\text{sin}}\frac{{3\pi }}{{12}} + {\text{sin}}\frac{{4\pi }}{{12}} + {\text{sin}}\frac{{5\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for both parts correct.</p>
<p>but <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{5\pi }}{{12}} = {\text{cos}}\frac{\pi }{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{4\pi }}{{12}} = {\text{cos}}\frac{{2\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{3\pi }}{{12}} = {\text{cos}}\frac{{3\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{2\pi }}{{12}} = {\text{cos}}\frac{{4\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{\pi }{{12}} = {\text{cos}}\frac{{5\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p>⇒ Re <em>S</em> = Im <em>S <strong>AG</strong></em></p>
<p><strong>Note:</strong> Accept a geometrical method.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{\pi }{{12}} = {\text{cos}}\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right) = {\text{cos}}\frac{\pi }{4}{\text{cos}}\frac{\pi }{6} + {\text{sin}}\frac{\pi }{4}{\text{sin}}\frac{\pi }{6}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 2 }}{2}\frac{{\sqrt 3 }}{2} + \frac{{\sqrt 2 }}{2}\frac{1}{2}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 6 + \sqrt 2 }}{4}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span><em> <strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{12}} = {\text{cos}}\left( {\frac{\pi }{6} + \frac{\pi }{4}} \right) = {\text{cos}}\frac{\pi }{6}{\text{cos}}\frac{\pi }{4} - {\text{sin}}\frac{\pi }{6}{\text{sin}}\frac{\pi }{4}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>+</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Allow alternative methods <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{12}} = {\text{sin}}\frac{\pi }{{12}} = {\text{sin}}\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 3 }}{2}\frac{{\sqrt 2 }}{2} - \frac{1}{2}\frac{{\sqrt 2 }}{2} = \frac{{\sqrt 6 - \sqrt 2 }}{4}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p>Re <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{\pi }{{12}} + {\text{cos}}\frac{{2\pi }}{{12}} + {\text{cos}}\frac{{3\pi }}{{12}} + {\text{cos}}\frac{{4\pi }}{{12}} + {\text{cos}}\frac{{5\pi }}{{12}}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span></p>
<p>Re <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sqrt 2 + \sqrt 6 }}{4} + \frac{{\sqrt 3 }}{2} + \frac{{\sqrt 2 }}{2} + \frac{1}{2} + \frac{{\sqrt 6 - \sqrt 2 }}{4}"> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> <mo>+</mo> <msqrt> <mn>6</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {\sqrt 6 + 1 + \sqrt 2 + \sqrt 3 } \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>+</mo> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {1 + \sqrt 2 } \right)\left( {1 + \sqrt 3 } \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><em>S</em> = Re(<em>S</em>)(1 + i) since Re <em>S</em> = Im <em>S</em>, <em><strong>R1</strong></em></p>
<p><em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left( {1 + \sqrt 2 } \right)\left( {1 + \sqrt 3 } \right)\left( {1 + {\text{i}}} \right)"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove by mathematical induction that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>n</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo>+</mo><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, determine the Maclaurin series of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced open="[" close="]"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math></p>
<p>LHS: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>d</mo><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>+</mo><mn>2</mn><mi>x</mi><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mfenced><mrow><mo>=</mo><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>RHS: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mn>1</mn></mfenced><mi>x</mi><mo>+</mo><mn>1</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mfenced><mrow><mo>=</mo><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math></p>
<p>now assume true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>; i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>k</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>k</mi></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Do not award <em><strong>M1</strong></em> for statements such as "let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>". Subsequent marks can still be awarded.</p>
<p><br>attempt to differentiate the RHS <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mrow><mo>d</mo><msup><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfrac><mo>d</mo><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>k</mi></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>+</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>x</mi><mo>+</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> <em><strong>A1</strong></em></p>
<p>so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> implies true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math> true and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> true <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> true</p>
<p>therefore, true for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math> <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>R1</strong></em> only if three of the previous four marks have been awarded</p>
<p> </p>
<p><em><strong>[7</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>n</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo>+</mo><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>n</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><msub><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><menclose notation="left"><mi>x</mi><mo>=</mo><mn>0</mn></menclose></msub><mo>=</mo><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> may be seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mo>'''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>6</mn><mo>,</mo><mo> </mo><mo> </mo><msup><mi>f</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>12</mn></math></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>x</mi><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mi>f</mi><mo>'''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><msup><mi>f</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mn>0</mn></mfenced><mo>+</mo><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≈</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>'<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>×</mo></math> Maclaurin series of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo> </mo></math>' <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≈</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>≈</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac><mo>≈</mo><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>9</mn></msup><mfenced><mrow><mo>+</mo><mi>higher</mi><mo> </mo><mi>order</mi><mo> </mo><mi>terms</mi></mrow></mfenced></mrow><msup><mi>x</mi><mn>9</mn></msup></mfrac></math></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></mrow></mfenced><mn>3</mn></msup></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>+</mo><mo> </mo><mi>higher</mi><mo> </mo><mi>order</mi><mo> </mo><mi>terms</mi></mrow></mfenced></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced open="[" close="]"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced open="[" close="]"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced><mn>3</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mfenced><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn></mrow><mi>x</mi></mfrac></mfenced><mn>3</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>attempt to use L'Hôpital's rule <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mfenced><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><mn>0</mn></mrow><mn>1</mn></mfrac></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced open="[" close="]"><mrow><munder><mi>lim</mi><mrow><mi mathvariant="normal">x</mi><mo>→</mo><mn>0</mn></mrow></munder><mo> </mo><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup></mrow></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}"> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi "> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>≡</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>k</mi> <mi>π</mi> </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> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the principle of mathematical induction to prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x + \ldots + \sin (2n - 1)x = \frac{{1 - \cos 2nx}}{{2\sin x}},{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}x \ne k\pi "> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>k</mi> <mi>π</mi> </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> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x = \cos x"> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> in the interval <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \pi "> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>π</mi> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4} = \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} - \frac{{\sqrt 2 }}{2} - \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} = \frac{{\sqrt 2 }}{2}"> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for 5 equal terms with \) + \) or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - "> <mo>−</mo> </math></span> signs.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \frac{{1 - (1 - 2{{\sin }^2}x)}}{{2\sin x}}"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{2{{\sin }^2}x}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \sin x"> <mo>≡</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(n):\sin x + \sin 3x + \ldots + \sin (2n - 1)x \equiv \frac{{1 - \cos 2nx}}{{2\sin x}}"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(1):\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>≡</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> which is true (as proved in part (b)) <strong><em>R1</em></strong></p>
<p>assume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(k)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </math></span> true, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x + \ldots + \sin (2k - 1)x \equiv \frac{{1 - \cos 2kx}}{{2\sin x}}"> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Only award <strong><em>M1 </em></strong>if the words “assume” and “true” appear. Do not award <strong><em>M1 </em></strong>for “let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span><em>” </em>only. Subsequent marks are independent of this <strong><em>M1</em></strong><em>.</em></p>
<p> </p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(k + 1)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span>:</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(k + 1):\sin x + \sin 3x + \ldots + \sin (2k - 1)x + \sin (2k + 1)x \equiv \frac{{1 - \cos 2(k + 1)x}}{{2\sin x}}"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="LHS = \sin x + \sin 3x + \ldots + \sin (2k - 1)x + \sin (2k + 1)x"> <mi>L</mi> <mi>H</mi> <mi>S</mi> <mo>=</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos 2kx}}{{2\sin x}} + \sin (2k + 1)x"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos 2kx + 2\sin x\sin (2k + 1)x}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos 2kx + 2\sin x\cos x\sin 2kx + 2{{\sin }^2}x\cos 2kx}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \left( {(1 - 2{{\sin }^2}x)\cos 2kx - \sin 2x\sin 2kx} \right)}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - (\cos 2x\cos 2kx - \sin 2x\sin 2kx)}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos (2kx + 2x)}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos 2(k + 1)x}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p>so if true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span> , then also true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1"> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span></p>
<p>as true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> then true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }"> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept answers using transformation formula for product of sines if steps are shown clearly.</p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>R1 </em></strong>only if candidate is awarded at least 5 marks in the previous steps.</p>
<p> </p>
<p><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x = \cos x \Rightarrow \frac{{1 - \cos 4x}}{{2\sin x}} = \cos x"> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 1 - \cos 4x = 2\sin x\cos x,{\text{ }}(\sin x \ne 0)"> <mo stretchy="false">⇒</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 1 - (1 - 2{\sin ^2}2x) = \sin 2x"> <mo stretchy="false">⇒</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> </mrow> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sin 2x(2\sin 2x - 1) = 0"> <mo stretchy="false">⇒</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sin 2x = 0"> <mo stretchy="false">⇒</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = \frac{1}{2}"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = \pi ,{\text{ }}2x = \frac{\pi }{6}"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = \frac{{5\pi }}{6}"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </math></span></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x = \cos x \Rightarrow 2\sin 2x\cos x = \cos x"> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">⇒</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow (2\sin 2x - 1)\cos x = 0,{\text{ }}(\sin x \ne 0)"> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sin 2x = \frac{1}{2}"> <mo stretchy="false">⇒</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos x = 0"> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = \frac{\pi }{6},{\text{ }}2x = \frac{{5\pi }}{6}"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore x = \frac{\pi }{2},{\text{ }}x = \frac{\pi }{{12}}"> <mo>∴</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{5\pi }}{{12}}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Do not award the final <strong><em>A1 </em></strong>if extra solutions are seen.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Three girls and four boys are seated randomly on a straight bench. Find the probability that the girls sit together and the boys sit together.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>total number of arrangements 7! <strong><em>(A1)</em></strong></p>
<p>number of ways for girls and boys to sit together <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3! \times 4! \times 2">
<mo>=</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>4</mn>
<mo>!</mo>
<mo>×</mo>
<mn>2</mn>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A0 </em></strong>if the 2 is missing.</p>
<p> </p>
<p>probability <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3! \times 4! \times 2}}{{7!}}">
<mfrac>
<mrow>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>4</mn>
<mo>!</mo>
<mo>×</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>7</mn>
<mo>!</mo>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for attempting to write as a probability.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2 \times 3 \times 4! \times 2}}{{7 \times 6 \times 5 \times 4!}}">
<mfrac>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mo>×</mo>
<mn>4</mn>
<mo>!</mo>
<mo>×</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>7</mn>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<mn>5</mn>
<mo>×</mo>
<mn>4</mn>
<mo>!</mo>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{2}{{35}}">
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mn>35</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A0 </em></strong>if not fully simplified.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{7} \times \frac{2}{6} \times \frac{1}{5} + \frac{4}{7} \times \frac{3}{6} \times \frac{2}{5} \times \frac{1}{4}">
<mfrac>
<mn>3</mn>
<mn>7</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>4</mn>
<mn>7</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>3</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>(M1)A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{7} \times \frac{2}{6} \times \frac{1}{5} \times 2">
<mfrac>
<mn>3</mn>
<mn>7</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
<mo>×</mo>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{7} \times \frac{3}{6} \times \frac{2}{5} \times \frac{1}{4} \times 2">
<mfrac>
<mn>4</mn>
<mn>7</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>3</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>×</mo>
<mn>2</mn>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{2}{{35}}">
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mn>35</mn>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A0 </em></strong>if not fully simplified.</p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Prove by mathematical induction that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \ldots + \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} n \\ 3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z},n \geqslant 3">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>,</mo>
<mi>n</mi>
<mo>⩾</mo>
<mn>3</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \ldots + \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} n \\ 3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>show true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 3">
<mi>n</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{LHS}} = \left( {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right) = 1">
<mrow>
<mtext>LHS</mtext>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{RHS}} = \left( {\begin{array}{*{20}{c}} 3 \\ 3 \end{array}} \right) = 1">
<mrow>
<mtext>RHS</mtext>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>hence true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 3">
<mi>n</mi>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k:\left( {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \ldots + \left( {\begin{array}{*{20}{c}} {k - 1} \\ 2 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} k \\ 3 \end{array}} \right)">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>:</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p>consider for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1:\left( {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \ldots + \left( {\begin{array}{*{20}{c}} {k - 1} \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} k \\ 2 \end{array}} \right)">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo>:</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} k \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} k \\ 2 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{k!}}{{(k - 3)!3!}} + \frac{{k!}}{{(k - 2)!2!}}\,\,\,\left( { = \frac{{k!}}{{3!}}\left[ {\frac{1}{{(k - 3)!}} + \frac{3}{{(k - 2)!}}} \right]} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
<mn>3</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
<mn>2</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mn>3</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or any correct expression with a visible common factor <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{k!}}{{3!}}\left[ {\frac{{k - 2 + 3}}{{(k - 2)!}}} \right]">
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mn>3</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mfrac>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
</math></span> or any correct expression with a common denominator <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{k!}}{{3!}}\left[ {\frac{{k + 1}}{{(k - 2)!}}} \right]">
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
<mrow>
<mn>3</mn>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mfrac>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
</math></span></p>
<p> </p>
<p><strong>Note:</strong> At least one of the above three lines or equivalent must be seen.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{(k + 1)!}}{{3!(k - 2)!}}">
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mn>3</mn>
<mo>!</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</math></span> or equivalent <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {k + 1} \\ 3 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>Result is true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 3">
<mi>k</mi>
<mo>=</mo>
<mn>3</mn>
</math></span>. If result is true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> it is true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k + 1">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>. Hence result is true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \geqslant 3">
<mi>k</mi>
<mo>⩾</mo>
<mn>3</mn>
</math></span>. Hence proved by induction. <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> In order to award the <strong><em>R1 </em></strong>at least <strong><em>[5 marks] </em></strong>must have been awarded.</p>
<p> </p>
<p><strong><em>[9 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^4} = - 4">
<mrow>
<msup>
<mi>z</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>4</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}">
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation, giving the solutions in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{i}}b"> <mi>a</mi> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mi>b</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> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt[4]{4}\left( { = \sqrt 2 } \right)"> <mrow> <mo>|</mo> <mi>z</mi> <mo>|</mo> </mrow> <mo>=</mo> <mroot> <mn>4</mn> <mn>4</mn> </mroot> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> </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{arg}}\left( {{z_1}} \right) = \frac{\pi }{4}"> <mrow> <mtext>arg</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p>first solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + {\text{i}}"> <mn>1</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>valid attempt to find all roots (De Moivre or +/− their components) <em><strong>(M1)</strong></em></p>
<p>other solutions are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 + {\text{i}}"> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 - {\text{i}}"> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow> <mtext>i</mtext> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - {\text{i}}"> <mn>1</mn> <mo>−</mo> <mrow> <mtext>i</mtext> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^4} = - 4"> <mrow> <msup> <mi>z</mi> <mn>4</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mn>4</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a + {\text{i}}b} \right)^4} = - 4"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mn>4</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mn>4</mn> </math></span></p>
<p>attempt to expand and equate <strong>both</strong> reals and imaginaries. <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^4} + 4{a^3}b{\text{i}} - 6{a^2}{b^2} - 4a{b^3}{\text{i}} + {b^4} = - 4"> <mrow> <msup> <mi>a</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mn>4</mn> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> </mrow> <mi>b</mi> <mrow> <mtext>i</mtext> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mrow> <msup> <mi>b</mi> <mn>3</mn> </msup> </mrow> <mrow> <mtext>i</mtext> </mrow> <mo>+</mo> <mrow> <msup> <mi>b</mi> <mn>4</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mn>4</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{a^4} - 6{a^4} + {a^4} = - 4 \Rightarrow } \right)a = \pm 1"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>a</mi> <mn>4</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>a</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>a</mi> <mn>4</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mo stretchy="false">⇒</mo> </mrow> <mo>)</mo> </mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </math></span><em><strong> and </strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {4{a^3}b - 4a{b^3} = 0 \Rightarrow } \right)a = \pm b"> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> </mrow> <mi>b</mi> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mrow> <msup> <mi>b</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </mrow> <mo>)</mo> </mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mi>b</mi> </math></span> <em><strong>(A1)</strong></em></p>
<p>first solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + {\text{i}}"> <mn>1</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>valid attempt to find all roots (De Moivre or +/− their components) <em><strong>(M1)</strong></em></p>
<p>other solutions are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 + {\text{i}}"> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 - {\text{i}}"> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow> <mtext>i</mtext> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - {\text{i}}"> <mn>1</mn> <mo>−</mo> <mrow> <mtext>i</mtext> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>complete method to find area of ‘rectangle' <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4"> <mo>=</mo> <mn>4</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>4</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mn>2</mn>
<mi>n</mi>
</msup>
</mrow>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> </math></span> is an odd or even function, justifying your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using mathematical induction, prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mfrac> <mrow> <mi>m</mi> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{Z}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find an expression for the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 1"> <mi>n</mi> <mo>></mo> <mn>1</mn> </math></span>, the equation of the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_n}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y - \pi = 0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>even function <strong><em>A1</em></strong></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos kx = \cos ( - kx)"> <mi>cos</mi> <mo></mo> <mi>k</mi> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <strong>and</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is a product of even functions <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p>even function <strong><em>A1</em></strong></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\cos 2x)(\cos 4x) \ldots = \left( {\cos ( - 2x)} \right)\left( {\cos ( - 4x)} \right) \ldots "> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>…</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>…</mo> </math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>A0R1</em></strong>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider the case <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin 4x}}{{2\sin 2x}} = \frac{{2\sin 2x\cos 2x}}{{2\sin 2x}} = \cos 2x"> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p>hence true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> <strong><em>R1</em></strong></p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>, <em>ie</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\cos 2x)(\cos 4x) \ldots (\cos {2^k}x) = \frac{{\sin {2^{k + 1}}x}}{{{2^k}\sin 2x}}"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>…</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>M1 </em></strong>for “let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>” or “assume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>” or equivalent.</p>
<p> </p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1"> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span>:</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{k + 1}}(x) = {f_k}(x)(\cos {2^{k + 1}}x)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sin {2^{k + 1}}x}}{{{2^k}\sin 2x}}\cos {2^{k + 1}}x"> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{2\sin {2^{k + 1}}x\cos {2^{k + 1}}x}}{{{2^{k + 1}}\sin 2x}}"> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sin {2^{k + 2}}x}}{{{2^{k + 1}}\sin 2x}}"> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> true and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span> true <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow n = k + 1"> <mo stretchy="false">⇒</mo> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span> true. Hence true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }"> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> To obtain the final <strong><em>R1</em></strong>, all the previous <strong><em>M </em></strong>marks must have been awarded.</p>
<p> </p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’ = \frac{{vu' - uv'}}{{{v^2}}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mi>v</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>u</mi> <msup> <mi>v</mi> <mo>′</mo> </msup> </mrow> <mrow> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> (or correct product rule) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}(x) = \frac{{({2^n}\sin 2x)({2^{n + 1}}\cos {2^{n + 1}}x) - (\sin {2^{n + 1}}x)({2^{n + 1}}\cos 2x)}}{{{{({2^n}\sin 2x)}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for correct numerator and <strong><em>A1 </em></strong>for correct denominator.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = \frac{{\left( {{2^n}\sin \frac{\pi }{2}} \right)\left( {{2^{n + 1}}\cos {2^{n + 1}}\frac{\pi }{4}} \right) - \left( {\sin {2^{n + 1}}\frac{\pi }{4}} \right)\left( {{2^{n + 1}}\cos \frac{\pi }{2}} \right)}}{{{{\left( {{2^n}\sin \frac{\pi }{2}} \right)}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = \frac{{({2^n})\left( {{2^{n + 1}}\cos {2^{n + 1}}\frac{\pi }{4}} \right)}}{{{{({2^n})}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\cos {2^{n + 1}}\frac{\pi }{4}{\text{ }}( = 2\cos {2^{n - 1}}\pi )"> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>π</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = 2"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}\left( {\frac{\pi }{4}} \right) = 0"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> This <strong><em>A </em></strong>mark is independent from the previous marks.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2\left( {x - \frac{\pi }{4}} \right)"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y - \pi = 0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Consider integers <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> such that <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></math> is exactly divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>. Prove by contradiction that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> cannot both be odd.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>Assume that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> are both odd. <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> be both odd”.<br><strong>Note:</strong> Subsequent marks after this <em><strong>M1</strong> </em>are independent of this mark and can be awarded.</p>
<p><br>Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>≡</mo><msup><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>+</mo><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mfenced><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow></mfenced><mo>+</mo><mn>2</mn></math> <em><strong>(A1)</strong></em></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow></mfenced></math> is always divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>) but <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> is not divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>. (or equivalent) <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> is not divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>, a contradiction. (or equivalent) <em><strong>R1</strong></em></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> cannot both be odd. <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Award a maximum of <em><strong>M1A0A0(A0)R1R1</strong></em> for considering identical or two consecutive odd numbers for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>Most candidates did not present their proof in a formal manner and merely relied on an algebraic approach rendering the proof incomplete. Very few candidates earned the first mark for making a clear assumption that a and b are both odd. A significant number of candidates only considered consecutive or identical odd numbers. The required reasoning to complete the proof were often poorly expressed or missing altogether. Only a small number of candidates were awarded all the available marks for this question.</p>
</div>
<br><hr><br><div class="specification">
<p>Let the roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^3} = - 3 + \sqrt 3 {\text{i}}">
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>+</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>On an Argand diagram, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span> are represented by the points U, V and W respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + \sqrt 3 {\text{i}}"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> <mrow> <mtext>i</mtext> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{{\text{e}}^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r > 0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \pi < \theta \leqslant \pi "> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo>⩽</mo> <mi>π</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> expressing your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{{\text{e}}^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r > 0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \pi < \theta \leqslant \pi "> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo>⩽</mo> <mi>π</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle UVW.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the sum of the roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span>, show that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{7\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}} = 0"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to find modulus <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 2\sqrt 3 \left( { = \sqrt {12} } \right)"> <mi>r</mi> <mo>=</mo> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <msqrt> <mn>12</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to find argument in the correct quadrant <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \pi + {\text{arctan}}\left( { - \frac{{\sqrt 3 }}{3}} \right)"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>+</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{5\pi }}{6}"> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + \sqrt 3 {\text{i}} = \sqrt {12} {{\text{e}}^{\frac{{5\pi {\text{i}}}}{6}}}\left( { = 2\sqrt 3 {{\text{e}}^{\frac{{5\pi {\text{i}}}}{6}}}} \right)"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <msqrt> <mn>12</mn> </msqrt> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></span></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find a root using de Moivre’s theorem <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}{{\text{e}}^{\frac{{5\pi {\text{i}}}}{{18}}}}"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to find further two roots by adding and subtracting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> to the argument <em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}{{\text{e}}^{ - \frac{{7\pi {\text{i}}}}{{18}}}}"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}{{\text{e}}^{\frac{{17\pi {\text{i}}}}{{18}}}}"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Ignore labels for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> at this stage.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em><br>attempting to find the total area of (congruent) triangles UOV, VOW and UOW <em><strong>M1</strong></em></p>
<p>Area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\left( {\frac{1}{2}} \right)\left( {{{12}^{\frac{1}{6}}}} \right)\left( {{{12}^{\frac{1}{6}}}} \right){\text{sin}}\frac{{2\pi }}{3}"> <mo>=</mo> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{{12}^{\frac{1}{6}}}} \right)\left( {{{12}^{\frac{1}{6}}}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> and <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{2\pi }}{3}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\sqrt 3 }}{4}\left( {{{12}^{\frac{1}{3}}}} \right)"> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>UV<sup>2</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {{{12}^{\frac{1}{6}}}} \right)^2} + {\left( {{{12}^{\frac{1}{6}}}} \right)^2} - 2\left( {{{12}^{\frac{1}{6}}}} \right)\left( {{{12}^{\frac{1}{6}}}} \right){\text{cos}}\frac{{2\pi }}{3}"> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>UV <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt 3 \left( {{{12}^{\frac{1}{6}}}} \right)"> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>attempting to find the area of UVW using Area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{1}{2}} "> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></span> × UV × VW × sin <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span> for example <em><strong>M1</strong></em></p>
<p>Area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {\sqrt 3 \times {{12}^{\frac{1}{6}}}} \right)\left( {\sqrt 3 \times {{12}^{\frac{1}{6}}}} \right){\text{sin}}\frac{\pi }{3}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>×</mo> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>×</mo> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\sqrt 3 }}{4}\left( {{{12}^{\frac{1}{3}}}} \right)"> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> = 0 <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}\left( {{\text{cos}}\left( { - \frac{{7\pi }}{{18}}} \right) + {\text{i}}\,{\text{sin}}\left( { - \frac{{7\pi }}{{18}}} \right) + {\text{cos}}\frac{{5\pi }}{{18}} + {\text{i}}\,{\text{sin}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}} + {\text{i}}\,{\text{sin}}\frac{{17\pi }}{{18}}} \right) = 0"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>A1</strong></em></p>
<p>consideration of real parts <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{12^{\frac{1}{6}}}\left( {{\text{cos}}\left( { - \frac{{7\pi }}{{18}}} \right) + {\text{cos}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}}} \right) = 0"> <mrow> <msup> <mn>12</mn> <mrow> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( { - \frac{{7\pi }}{{18}}} \right) = {\text{cos}}\frac{{17\pi }}{{18}}"> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> </math></span> explicitly stated <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{7\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}} = 0"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>9</mn></msubsup><mfenced><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></mfenced><mo>d</mo><mi>x</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mo>∫</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>5</mn><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>10</mn><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p>substituting limits into their integrated function and subtracting <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mn>9</mn></mfenced><mo>-</mo><mn>10</mn><msup><mfenced><mn>9</mn></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>-</mo><mfenced><mrow><mn>3</mn><mfenced><mn>1</mn></mfenced><mo>-</mo><mn>10</mn><msup><mfenced><mn>1</mn></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>-</mo><mn>10</mn><mo>×</mo><mn>3</mn><mo>-</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>10</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>A mixed response was noted for this question. Candidates who simplified the algebraic fraction before integrating were far more successful in gaining full marks in this question. Many candidates used other valid approaches such as integration by substitution and integration by parts with varying degrees of success. A small number of candidates substituted the limits without integrating.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the three planes</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle><mo>:</mo><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>4</mn></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle><mo>:</mo><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>5</mn></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle><mo>:</mo><mo>-</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>32</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the three planes do not intersect.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> lies on both <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>, the line of intersection of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">attempt to eliminate a variable <em><strong>M1</strong></em></p>
<p style="text-align:left;">obtain a pair of equations in two variables</p>
<p style="text-align:left;"><br><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>3</mn></math> and <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>44</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>7</mn></math> and <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>40</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>3</mn></math> and <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mo>-</mo><mfrac><mn>79</mn><mn>5</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;">the two lines are parallel (<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>≠</mo><mn>44</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>7</mn><mo>≠</mo><mn>40</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>≠</mo><mo>-</mo><mfrac><mn>79</mn><mn>5</mn></mfrac></math>) <em><strong>R1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> There are other possible pairs of equations in two variables.<br>To obtain the final <em><strong>R1</strong></em>, at least the initial <em><strong>M1</strong> </em>must have been awarded.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">hence the three planes do not intersect <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">vector product of the two normals <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>A0</strong></em> if “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo></math>” is missing. Subsequent marks may still be awarded.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">Attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>+</mo><mi>λ</mi><mo>,</mo><mo>-</mo><mn>2</mn><mo>+</mo><mn>5</mn><mi>λ</mi><mo>,</mo><mn>3</mn><mi>λ</mi></mrow></mfenced></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mo>+</mo><mn>5</mn><mi>λ</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mi>λ</mi></mrow></mfenced><mo>=</mo><mn>32</mn></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>15</mn><mo>=</mo><mn>32</mn></math>, a contradiction <em><strong>R1</strong></em></p>
<p style="text-align:left;">hence the three planes do not intersect <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 3</strong></p>
<p style="text-align:left;">attempt to eliminate a variable <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>=</mo><mn>6</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>=</mo><mn>100</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>94</mn></math>, a contradiction <em><strong>R1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Accept other equivalent alternatives. Accept other valid methods.<br>To obtain the final <em><strong>R1</strong></em>, at least the initial <em><strong>M1</strong> </em>must have been awarded.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">hence the three planes do not intersect <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder><mo>:</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>0</mn><mo>=</mo><mn>4</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder><mo>:</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>0</mn><mo>=</mo><mn>5</mn></mstyle></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">attempt to find the vector product of the two normals <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>×</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo></math>” is missing.<br>Accept any multiple of the direction vector.<br>Working for (b)(ii) may be seen in part (a) Method 2. In this case penalize lack of “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo></math>” only once.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">attempt to eliminate a variable from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>3</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>y</mi><mo>-</mo><mn>5</mn><mi>z</mi><mo>=</mo><mo>-</mo><mn>6</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>7</mn></math></p>
<p style="text-align:left;">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math></p>
<p style="text-align:left;">substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>3</mn></math> to obtain</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>+</mo><mn>3</mn><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><mi>t</mi><mo>-</mo><mn>7</mn></math> (for all three variables in parametric form) <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo></math>” is missing.<br>Accept any multiple of the direction vector. Accept other position vectors which satisfy both the planes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle></math> .</p>
<p style="text-align:left;"> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">the line connecting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math></p>
<p style="text-align:left;">attempt to substitute position and direction vector to form <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">s</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mn>9</mn><mi>t</mi><mo>,</mo><mo>-</mo><mn>2</mn><mo>+</mo><mn>3</mn><mi>t</mi><mo>,</mo><mo>-</mo><mn>2</mn><mi>t</mi></mrow></mfenced></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mn>9</mn><mi>t</mi></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mo>+</mo><mn>3</mn><mi>t</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></mfenced><mo>=</mo><mn>32</mn></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>94</mn><mi>t</mi><mo>=</mo><mn>47</mn><mo>⇒</mo><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">attempt to find distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> and their point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="|" close="|"><mrow><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>-</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><msup><mfenced><mrow><mo>-</mo><mn>9</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msqrt><mn>94</mn></msqrt><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">unit normal vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced></mrow><msqrt><mn>81</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>4</mn></msqrt></mfrac></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><msqrt><mn>94</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>4</mn></munder></mstyle></math> be the plane parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> and passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, <br>then the normal vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>4</mn></munder></mstyle></math> is given by</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mo>-</mo><mn>15</mn></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">unit normal vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>4</mn></munder></mstyle></math> is given by</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced></mrow><msqrt><mn>81</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>4</mn></msqrt></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>15</mn></mrow><msqrt><mn>94</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">distance between the planes is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><msqrt><mn>94</mn></msqrt></mfrac><mo>-</mo><mfrac><mrow><mo>-</mo><mn>15</mn></mrow><msqrt><mn>94</mn></msqrt></mfrac></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>47</mn><msqrt><mn>94</mn></msqrt></mfrac><mfenced><mrow><mo>=</mo><mfrac><msqrt><mn>94</mn></msqrt><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) was well attempted using a variety of approaches. Most candidates were able to gain marks for part (a) through attempts to eliminate a variable with many subsequently making algebraic errors. Part (b)(i) was well done. For part (b)(ii) few successful attempts were noted, many candidates failed to use an appropriate notation "<em>r</em> =" while giving the vector equation of a line. Part (c) proved to be challenging for most candidates with very few correct answers seen. Many candidates did not attempt part (c).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Prove by contradiction that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has no integer roots.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1 (rearranging the equation)</strong></p>
<p>assume there exists some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>M1</strong></em> for equivalent statements such as ‘assume that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> is an integer root of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>’. Condone the use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> throughout the proof.</p>
<p>Award <em><strong>M1</strong></em> for an assumption involving <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><mi>α</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math>.</p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>for statements such as “let’s consider the equation has integer roots…” ,“let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> be a root of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>…”</p>
<p><strong>Note:</strong> Subsequent marks after this <em><strong>M1</strong> </em>are independent of this <em><strong>M1</strong> </em>and can be awarded.</p>
<p> </p>
<p>attempts to rearrange their equation into a suitable form <em><strong>M1</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>⇒</mo><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi></math> is even <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> which is not even and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> cannot be an integer <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Accept ‘<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> which gives a contradiction’.</p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><mn>2</mn><mfenced><mrow><mo>-</mo><msup><mi>α</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>α</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>⇒</mo><mfenced><mrow><mo>-</mo><msup><mi>α</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>α</mi></mrow></mfenced><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>1</mn></math> is even which is not true and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> cannot be an integer <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Accept ‘<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>1</mn></math> is even which gives a contradiction’.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mo>-</mo><msup><mi>α</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>α</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>⇒</mo><mfenced><mrow><mo>-</mo><msup><mi>α</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>α</mi></mrow></mfenced><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>α</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>α</mi></math> is is not an integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> cannot be an integer <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Accept ‘ <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>α</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>α</mi></math> is not an integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></math> which gives a contradiction’.</p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>⇒</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfrac><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></math> is not an integer and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> cannot be an integer <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></math> is not an integer which gives a contradiction’.</p>
<p><strong><br>THEN</strong></p>
<p>so the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has no integer roots <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>assume there exists some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>M1</strong></em> for equivalent statements such as ‘assume that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> is an integer root of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>’. Condone the use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> throughout the proof. Award <em><strong>M1</strong></em> for an assumption involving <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><mi>α</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math> and award subsequent marks based on this.</p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>for statements such as “let’s consider the equation has integer roots…” ,“let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> be a root of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>α</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>α</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>…”</p>
<p><strong>Note:</strong> Subsequent marks after this <em><strong>M1</strong> </em>are independent of this <em><strong>M1</strong> </em>and can be awarded.</p>
<p> </p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></math> (and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>α</mi></mfenced><mo>=</mo><mn>0</mn></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mo>></mo><mn>0</mn></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo> </mo><mo>⇒</mo><mi>f</mi></math> is a (strictly) increasing function <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mo>-</mo><mn>7</mn></math> <em><strong>R1</strong></em></p>
<p>thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> has only one real root between <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math>, which gives a contradiction</p>
<p>(or therefore, contradicting the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>α</mi></mfenced><mo>=</mo><mn>0</mn></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>), <em><strong>R1</strong></em></p>
<p>so the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has no integer roots <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Consider the quartic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>4</mn></msup><mo>+</mo><mn>4</mn><msup><mi>z</mi><mn>3</mn></msup><mo>+</mo><mn>8</mn><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mn>80</mn><mi>z</mi><mo>+</mo><mn>400</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p>Two of the roots of this equation are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>+</mo><mi>a</mi><mtext>i</mtext></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>other two roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>-</mo><mi>b</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>-</mo><mi>a</mi><mtext>i</mtext></math> <em><strong>A1</strong></em></p>
<p>sum of roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>4</mn></math> and product of roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>400</mn></math> <em><strong>A1</strong></em></p>
<p>attempt to set sum of four roots equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> OR<br>attempt to set product of four roots equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>400</mn></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mtext>i</mtext><mo>+</mo><mi>a</mi><mo>-</mo><mi>b</mi><mtext>i</mtext><mo>+</mo><mi>b</mi><mo>+</mo><mi>a</mi><mtext>i</mtext><mo>+</mo><mi>b</mi><mo>−</mo><mi>a</mi><mi>i</mi><mo>=</mo><mo>−</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>a</mi><mo>+</mo><mn>2</mn><mi>b</mi><mo>=</mo><mo>−</mo><mn>4</mn><mo>(</mo><mo>⇒</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mtext>i</mtext><mo>)</mo><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mtext>i</mtext><mo>)</mo><mo> </mo><mo>(</mo><mi>b</mi><mo>+</mo><mi>a</mi><mtext>i</mtext><mo>)</mo><mo>(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mtext>i</mtext><mo>)</mo><mo>=</mo><mn>400</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>400</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>20</mn></math></p>
<p>attempt to solve simultaneous equations <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>4</mn></math> <em><strong> A1A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>other two roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>-</mo><mi>b</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>-</mo><mi>a</mi><mtext>i</mtext></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mtext>i</mtext></mrow></mfenced></mrow></mfenced><mfenced><mrow><mi>z</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>b</mi><mtext>i</mtext></mrow></mfenced></mrow></mfenced><mfenced><mrow><mi>z</mi><mo>-</mo><mfenced><mrow><mi>b</mi><mo>+</mo><mi>a</mi><mtext>i</mtext></mrow></mfenced></mrow></mfenced><mfenced><mrow><mi>z</mi><mo>-</mo><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi><mtext>i</mtext></mrow></mfenced></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>b</mi><mi>z</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>Attempt to equate coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>3</mn></msup></math> and constant with the given quartic equation <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mn>2</mn><mi>b</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>400</mn></math> <em><strong>A1</strong></em></p>
<p>attempt to solve simultaneous equations <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>4</mn></math> <em><strong> A1A1</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></msqrt></mfrac><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><mi>x</mi></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>≠</mo><mn>0</mn></math>.</p>
<p>The binomial expansion of this expression, in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, as far as the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>b</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the restriction which must be placed on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for this expansion to be valid.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to expand binomial with negative fractional power <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></msqrt></mfrac><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mrow><mi>a</mi><mi>x</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>8</mn></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>1</mn><mo>-</mo><mi>x</mi></msqrt><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>8</mn></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></msqrt></mfrac><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><mi>x</mi></msqrt><mo>=</mo><mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfenced><mfrac><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mn>8</mn></mfrac></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math></p>
<p>attempt to equate coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mo>:</mo><mo> </mo><mo> </mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow><mn>2</mn></mfrac><mo>=</mo><mn>4</mn><mi>b</mi><mo>;</mo><mo> </mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>:</mo><mo> </mo><mfrac><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mn>8</mn></mfrac><mo>=</mo><mi>b</mi></math></p>
<p>attempt to solve simultaneously <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>x</mi></mfenced><mo><</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1</strong></em><em><strong> mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 1 - \cos 2\theta - {\text{i}}\sin 2\theta ,{\text{ }}z \in \mathbb{C},{\text{ }}0 \leqslant \theta \leqslant \pi ">
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>θ<!-- θ --></mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ ">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msup>
<mn>0</mn>
<mo>∘</mo>
</msup>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<msup>
<mn>180</mn>
<mo>∘</mo>
</msup>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the modulus and argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>. Express each answer in its simplest form.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cube roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in modulus-argument form.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin (x + 60^\circ ) = \cos (x + 30^\circ )">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2(\sin x\cos 60^\circ + \cos x\sin 60^\circ ) = \cos x\cos 30^\circ - \sin x\sin 30^\circ ">
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin x \times \frac{1}{2} + 2\cos x \times \frac{{\sqrt 3 }}{2} = \cos x \times \frac{{\sqrt 3 }}{2} - \sin x \times \frac{1}{2}">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{3}{2}\sin x = - \frac{{\sqrt 3 }}{2}\cos x">
<mo stretchy="false">⇒</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \tan x = - \frac{1}{{\sqrt 3 }}">
<mo stretchy="false">⇒</mo>
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = 150^\circ ">
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<msup>
<mn>150</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>choosing two appropriate angles, for example 60° and 45° <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ ">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
</math></span> and</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos 105^\circ = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ ">
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }} + \frac{1}{2} \times \frac{1}{{\sqrt 2 }} + \frac{1}{2} \times \frac{1}{{\sqrt 2 }} - \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{\sqrt 2 }}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>OR</strong></p>
<p>attempt to square the expression <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(\sin 105^\circ + \cos 105^\circ )^2} = {\sin ^2}105^\circ + 2\sin 105^\circ \cos 105^\circ + {\cos ^2}105^\circ ">
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mrow>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(\sin 105^\circ + \cos 105^\circ )^2} = 1 + \sin 210^\circ ">
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>210</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = (1 - \cos 2\theta ) - {\text{i}}\sin 2\theta ">
<mi>z</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt {{{(1 - \cos 2\theta )}^2} + {{(\sin 2\theta )}^2}} ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt {1 - 2\cos 2\theta + {{\cos }^2}2\theta + {{\sin }^2}2\theta } ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mn>2</mn>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mn>2</mn>
<mi>θ</mi>
</msqrt>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt 2 \sqrt {(1 - \cos 2\theta )} ">
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<msqrt>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {2(2{{\sin }^2}\theta )} ">
<mo>=</mo>
<msqrt>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta ">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \alpha ">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>α</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \alpha = - \frac{{\sin 2\theta }}{{1 - \cos 2\theta }}">
<mi>tan</mi>
<mo></mo>
<mi>α</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 2\sin \theta \cos \theta }}{{2{{\sin }^2}\theta }}">
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \cot \theta ">
<mo>=</mo>
<mo>−</mo>
<mi>cot</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \alpha = - \arctan \left( {\tan \left( {\frac{\pi }{2} - \theta } \right)} \right)">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>α</mi>
<mo>=</mo>
<mo>−</mo>
<mi>arctan</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>tan</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \theta - \frac{\pi }{2}">
<mo>=</mo>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = (1 - \cos 2\theta ) - {\text{i}}\sin 2\theta ">
<mi>z</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2{\sin ^2}\theta - 2{\text{i}}\sin \theta \cos \theta ">
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta (\sin \theta - {\text{i}}\cos \theta )">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 2{\text{i}}\sin \theta (\cos \theta + {\text{i}}\sin \theta )">
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta \left( {\cos \left( {\theta - \frac{\pi }{2}} \right) + {\text{i}}\sin \left( {\theta - \frac{\pi }{2}} \right)} \right)">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = 2\sin \theta ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \theta - \frac{\pi }{2}">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to apply De Moivre’s theorem <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(1 - \cos 2\theta - {\text{i}}\sin 2\theta )^{\frac{1}{3}}} = {2^{\frac{1}{3}}}{(\sin \theta )^{\frac{1}{3}}}\left[ {\cos \left( {\frac{{\theta - \frac{\pi }{2} + 2n\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{\theta - \frac{\pi }{2} + 2n\pi }}{3}} \right)} \right]">
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <strong><em>A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>A1 </em></strong>for modulus, <strong><em>A1 </em></strong>for dividing argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> by 3 and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2n\pi ">
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</math></span>.</p>
<p> </p>
<p>Hence cube roots are the above expression when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = - 1,{\text{ }}0,{\text{ }}1">
<mi>n</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
</math></span>. Equivalent forms are acceptable. <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the binomial theorem to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>4</mn></msup></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mi mathvariant="normal">i</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> are expressed in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use de Moivre’s theorem and the result from part (a) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cot</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cot</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cot</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the identity from part (b) to show that the quadratic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce a quadratic equation with integer coefficients, having roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p style="text-align:left;">uses the binomial theorem on <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>4</mn></msup></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mmultiscripts><mi>C</mi><mn>0</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mfenced><mrow><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>3</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>3</mn></msup><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>4</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>4</mn></msup><mo> </mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mfenced><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mfenced><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">(using de Moivre’s theorem with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math> gives) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> <strong>(A1)</strong></p>
<p style="text-align:left;">equates both the real and imaginary parts of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mfenced><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow></mfenced></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></math></p>
<p style="text-align:left;">recognizes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow><mrow><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow></mfrac></math> <strong>(A1)</strong></p>
<p style="text-align:left;">substitutes for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow><mrow><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow></mfrac></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow></mfrac></math></p>
<p style="text-align:left;">divides the numerator and denominator by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></math> to obtain</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mstyle displaystyle="true"><mfrac><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow><mrow><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow><mrow><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfrac></mstyle></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cot</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cot</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cot</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn></math> and putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> in the numerator of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cot</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cot</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cot</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <strong>M1</strong></p>
<p style="text-align:left;">attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo>…</mo><mo> </mo><mfenced><mrow><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi mathvariant="normal">π</mi><mo>,</mo><mo> </mo><mi mathvariant="normal">n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mo>…</mo></mrow></mfenced></math> <strong>(A1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Do not award the final <strong>A1</strong> if solutions other than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> are listed.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">finding the roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn><mo> </mo><mfenced><mrow><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></mrow></mfenced></math> corresponds to finding the roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <strong>R1</strong></p>
<p style="text-align:left;">so the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> as roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math> <strong>A1</strong></p>
<p style="text-align:left;">since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>></mo><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac><mo>,</mo><mo> </mo><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> has the smaller value of the two roots <strong>R1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>R1</strong> for an alternative convincing valid reason.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac><mo>=</mo><mn>3</mn><mo>-2</mo><msqrt><mn>2</mn></msqrt></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math></p>
<p style="text-align:left;">uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>6</mn><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Use the principle of mathematical induction to prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + \, \ldots \, + n{\left( {\frac{1}{2}} \right)^{n - 1}} = 4 - \frac{{n + 2}}{{{2^{n - 1}}}}">
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<mo>…</mo>
<mspace width="thinmathspace"></mspace>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>, where <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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{LHS}} = 1\,{\text{;}}\,\,{\text{RHS}} = 4 - \frac{3}{{{2^0}}} = 4 - 3 = 1">
<mrow>
<mtext>LHS</mtext>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>;</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>RHS</mtext>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mn>0</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mn>3</mn>
<mo>=</mo>
<mn>1</mn>
</math></span> <em> <strong>M1</strong></em></p>
<p>hence true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Assumption of truth must be present. Following marks are not dependent on the first two <em><strong>M1</strong> </em>marks.</p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + \, \ldots \, + k{\left( {\frac{1}{2}} \right)^{k - 1}} = 4 - \frac{{k + 2}}{{{2^{k - 1}}}}">
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<mo>…</mo>
<mspace width="thinmathspace"></mspace>
<mo>+</mo>
<mi>k</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + \, \ldots \, + k{\left( {\frac{1}{2}} \right)^{k - 1}} + \left( {k + 1} \right){\left( {\frac{1}{2}} \right)^k}">
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<mo>…</mo>
<mspace width="thinmathspace"></mspace>
<mo>+</mo>
<mi>k</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>k</mi>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4 - \frac{{k + 2}}{{{2^{k - 1}}}} + \left( {k + 1} \right){\left( {\frac{1}{2}} \right)^k}">
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>k</mi>
</msup>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>finding a common denominator for the two fractions <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4 - \frac{{2\left( {k + 2} \right)}}{{{2^k}}} + \frac{{k + 1}}{{{2^k}}}">
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mi>k</mi>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mi>k</mi>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4 - \frac{{2\left( {k + 2} \right) - \left( {k + 1} \right)}}{{{2^k}}} = 4 - \frac{{k + 3}}{{{2^k}}}\left( { = 4 - \frac{{\left( {k + 1} \right) + 2}}{{{2^{\left( {k + 1} \right) - 1}}}}} \right)">
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mi>k</mi>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mi>k</mi>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>hence if true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span> then also true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, as true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, so true (for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>) <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award the final <em><strong>R1</strong> </em>only if the first four marks have been awarded.</p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the complex numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i, </mtext>
</mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = \frac{{{z_1}}}{{{z_2}}}">
<mi>w</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form write down the modulus of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>;</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form write down the argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the smallest positive integer value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^n}">
<mrow>
<msup>
<mi>w</mi>
<mi>n</mi>
</msup>
</mrow>
</math></span> is a real number.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>cis</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<mtext>cis</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0 </strong></em>for correct moduli and arguments found, but not written in mod-arg form.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| w \right| = \sqrt 2 ">
<mrow>
<mo>|</mo>
<mi>w</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>cis</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<mtext>cis</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0 </strong></em>for correct moduli and arguments found, but not written in mod-arg form.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg w = \frac{\pi }{{12}}">
<mi>arg</mi>
<mo></mo>
<mi>w</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Notes:</strong> Allow <em><strong>FT </strong></em>from incorrect answers for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \left( {\frac{{\pi n}}{{12}}} \right) = 0">
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>n</mi>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg ({w^n}) = \pi ">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>w</mi>
<mi>n</mi>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>π</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{n\pi }}{{12}} = \pi ">
<mfrac>
<mrow>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mi>π</mi>
</math></span></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore n = 12">
<mo>∴</mo>
<mi>n</mi>
<mo>=</mo>
<mn>12</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = x\,{{\text{e}}^{2x}}"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}"> <mi>x</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>. The <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{n^{{\text{th}}}}"> <mrow> <msup> <mi>n</mi> <mrow> <mrow> <mtext>th</mtext> </mrow> </mrow> </msup> </mrow> </math></span> derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> is denoted by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( n \right)}}\left( x \right)"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<p> </p>
<p>Prove, by mathematical induction, that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( n \right)}}\left( x \right) = \left( {{2^n}x + n{2^{n - 1}}} \right){{\text{e}}^{2x}}"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }"> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = {{\text{e}}^{2x}} + 2x{{\text{e}}^{2x}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> This must be obtained from the candidate differentiating <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {{2^1}x + 1 \times {2^{1 - 1}}} \right){{\text{e}}^{2x}}"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mn>1</mn> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>(hence true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span>)</p>
<p> </p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>: <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( k \right)}}\left( x \right) = \left( {{2^k}x + k{2^{k - 1}}} \right){{\text{e}}^{2x}}"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> if truth is assumed. Do not allow “let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>”.</p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1"> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span>:</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( {k + 1} \right)}}\left( x \right) = \frac{{\text{d}}}{{{\text{d}}x}}\left( {\left( {{2^k}x + k{2^{k - 1}}} \right){{\text{e}}^{2x}}} \right)"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mtext>d</mtext> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>attempt to differentiate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( k \right)}}\left( x \right)"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <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^{\left( {k + 1} \right)}}\left( x \right) = {2^k}{{\text{e}}^{2x}} + 2\left( {{2^k}x + k{2^{k - 1}}} \right){{\text{e}}^{2x}}"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( {k + 1} \right)}}\left( x \right) = \left( {{2^k} + {2^{k + 1}}x + k{2^k}} \right){{\text{e}}^{2x}}"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( {k + 1} \right)}}\left( x \right) = \left( {{2^{k + 1}}x + \left( {k + 1} \right){2^k}} \right){{\text{e}}^{2x}}"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {{2^{k + 1}}x + \left( {k + 1} \right){2^{\left( {k + 1} \right) - 1}}} \right){{\text{e}}^{2x}}"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span></p>
<p>True for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span> true implies true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1"> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span>.</p>
<p>Therefore the statement is true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n\left( { \in {\mathbb{Z}^ + }} \right)"> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Do not award final <em><strong>R1</strong></em> if the two previous <em><strong>M1s</strong></em> are not awarded. Allow full marks for candidates who use the base case <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{4x - 5}}{{{x^2} - 3x + 2}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 1{\text{,}}\,x \ne 2">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>1</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> in partial fractions.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use part (a) to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is always decreasing.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use part (a) to find the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_{ - 1}^0 {f(x)dx} ">
<munderover>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>0</mn>
</munderover>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>d</mi>
<mi>x</mi>
</mrow>
</math></span>, giving the answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,q">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Q}">
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{4x - 5}}{{\left( {x - 1} \right)\left( {x - 2} \right)}} \equiv \frac{A}{{x - 1}} + \frac{B}{{x - 2}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>≡</mo>
<mfrac>
<mi>A</mi>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>B</mi>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
</math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 4x - 5 \equiv A\left( {x - 2} \right) + B\left( {x - 1} \right)">
<mo stretchy="false">⇒</mo>
<mn>4</mn>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
<mo>≡</mo>
<mi>A</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>B</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1 \Rightarrow A = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>A</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 \Rightarrow B = 3">
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">⇒</mo>
<mi>B</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <em><strong>A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{1}{{x - 1}} + \frac{3}{{x - 2}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
</math></span></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = - {\left( {x - 1} \right)^{ - 2}} - 3{\left( {x - 2} \right)^{ - 2}}">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p>This is always negative so function is always decreasing. <em><strong>R1AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_{ - 1}^0 {\frac{1}{{x - 1}} + \frac{3}{{x - 2}}{\text{ }}} dx = \left[ {{\text{ln}}\left| {x - 1} \right| + 3\,{\text{ln}}\left| {x - 2} \right|} \right]_{ - 1}^0">
<munderover>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>0</mn>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
</mrow>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>0</mn>
</msubsup>
</math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {3\,{\text{ln}}\,2} \right) - \left( {\,{\text{ln}}\,2 + 3\,{\text{ln}}\,3} \right) = 2\,{\text{ln}}\,2 - 3\,{\text{ln}}\,3 = \,{\text{ln}}\frac{4}{{27}}">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mfrac>
<mn>4</mn>
<mrow>
<mn>27</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>b</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mn>2</mn><mi>b</mi><mtext>i</mtext></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</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>Hence, given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arg</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>2</mn><mi>b</mi></mrow></mfenced><mtext>i</mtext></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mtext>i</mtext><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>+</mo><mtext>i</mtext><mfenced><mrow><mo>-</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>+</mo><mtext>i</mtext><mfenced><mrow><mo>-</mo><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> and A1 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>b</mi><mtext>i</mtext><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup><mtext>i</mtext></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arg</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mo>-</mo><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>EITHER</strong><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mrow><mo>-</mo><mi>b</mi></mrow></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>≠</mo><mn>0</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>) <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) was generally well done with many completely correct answers seen. Part (b) proved to be challenging with many candidates incorrectly equating the ratio of their imaginary and real parts to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>4</mn></mfrac></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi>π</mi><mn>4</mn></mfrac></math>. Stronger candidates realized that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></math>, it forms an isosceles right-angled triangle and equated the real and imaginary parts to obtain the value of <em>b</em> .</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.<br>Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_14.39.35.png" alt="N17/5/MATHL/HP1/ENG/TZ0/10"></p>
<p>Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.<br>Chloe wins if <strong>no</strong> matches occur; otherwise Selena wins.</p>
</div>
<div class="specification">
<p>Chloe and Selena repeat their game so that they play a total of 50 times.<br>Suppose the discrete random variable <em>X </em>represents the number of times Chloe wins.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability that Chloe wins the game is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{8}">
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the mean of <em>X</em>.</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>Determine the variance of <em>X</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>number of possible “deals” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4! = 24">
<mo>=</mo>
<mn>4</mn>
<mo>!</mo>
<mo>=</mo>
<mn>24</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>consider ways of achieving “no matches” (Chloe winning):</p>
<p>Selena could deal B, C, D (<em>ie</em>, 3 possibilities)</p>
<p>as her first card <strong><em>R1</em></strong></p>
<p>for each of these matches, there are only 3 possible combinations for the remaining 3 cards <strong><em>R1</em></strong></p>
<p>so no. ways achieving no matches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3 \times 3 = 9">
<mo>=</mo>
<mn>3</mn>
<mo>×</mo>
<mn>3</mn>
<mo>=</mo>
<mn>9</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p>so probability Chloe wins <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{9}{{23}} = \frac{3}{8}">
<mo>=</mo>
<mfrac>
<mn>9</mn>
<mrow>
<mn>23</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span> <strong><em>A1AG</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>number of possible “deals” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4! = 24">
<mo>=</mo>
<mn>4</mn>
<mo>!</mo>
<mo>=</mo>
<mn>24</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>consider ways of achieving a match (Selena winning)</p>
<p>Selena card A can match with Chloe card A<em>, </em>giving 6 possibilities for this happening <strong><em>R1</em></strong></p>
<p>if Selena deals B as her first card, there are only 3 possible combinations for the remaining 3 cards. Similarly for dealing C and dealing D <strong><em>R1</em></strong></p>
<p>so no. ways achieving one match is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6 + 3 + 3 + 3 = 15">
<mo>=</mo>
<mn>6</mn>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mn>3</mn>
<mo>=</mo>
<mn>15</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p>so probability Chloe wins <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 - \frac{{15}}{{24}} = \frac{3}{8}">
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>15</mn>
</mrow>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span> <strong><em>A1AG</em></strong></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>systematic attempt to find number of outcomes where Chloe wins (no matches)</p>
<p>(using tree diag. or otherwise) <strong><em>M1</em></strong></p>
<p>9 found <strong><em>A1</em></strong></p>
<p>each has probability <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{4} \times \frac{1}{3} \times \frac{1}{2} \times 1">
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>1</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{24}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p>their 9 multiplied by their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{24}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{3}{8}">
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{B}}\left( {50,{\text{ }}\frac{3}{8}} \right)">
<mi>X</mi>
<mo>∼</mo>
<mrow>
<mtext>B</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>50</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = np = 50 \times \frac{3}{8} = \frac{{150}}{8}{\text{ }}\left( { = \frac{{75}}{4}} \right){\text{ }}( = 18.75)">
<mi>μ</mi>
<mo>=</mo>
<mi>n</mi>
<mi>p</mi>
<mo>=</mo>
<mn>50</mn>
<mo>×</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>150</mn>
</mrow>
<mn>8</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>75</mn>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>18.75</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></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="{\sigma ^2} = np(1 - p) = 50 \times \frac{3}{8} \times \frac{5}{8} = \frac{{750}}{{64}}{\text{ }}\left( { = \frac{{375}}{{32}}} \right){\text{ }}( = 11.7)">
<mrow>
<msup>
<mi>σ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>n</mi>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>50</mn>
<mo>×</mo>
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>5</mn>
<mn>8</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>750</mn>
</mrow>
<mrow>
<mn>64</mn>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>375</mn>
</mrow>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>11.7</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi><mo> </mo><mo>\</mo><mo> </mo><mfenced open="{" close="}"><mi>k</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>≠</mo><mn>5</mn></math>. </p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the vertical asymptote on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the horizontal asymptote on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an algebraic method to determine whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is a self-inverse function.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region bounded by the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>7</mn></math> is rotated through <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. Find the volume of the solid generated, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>)</mo><mo> </mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>k</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mfenced><mstyle displaystyle="true"><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></mstyle></mfenced><mo>-</mo><mn>5</mn></mrow><mrow><mfenced><mstyle displaystyle="true"><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></mstyle></mfenced><mo>-</mo><mi>k</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>k</mi><mfenced><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo>-</mo><mn>5</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced></mrow><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn><mo>-</mo><mi>k</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><mn>5</mn><mi>k</mi><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>k</mi></mrow><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn><mo>-</mo><mi>k</mi><mi>x</mi><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><mn>5</mn><mi>x</mi></mrow><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>x</mi><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfenced></mrow><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi></math> , (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is self-inverse) <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> The statement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>x</mi></math> could be seen anywhere in the candidate’s working to award <em><strong>R1</strong></em>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Interchanging <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> can be done at any stage.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfenced><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>-</mo><mi>x</mi><mi>k</mi><mo>=</mo><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>-</mo><mi>k</mi><mi>y</mi><mo>=</mo><mi>x</mi><mi>k</mi><mo>-</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math> (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is self-inverse) <em><strong>R1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img style="display: block;margin-left:auto;margin-right:auto;" 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"></p>
<p>attempt to draw both branches of a rectangular hyperbola <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>5</mn><mn>7</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong>EITHER</strong></p>
<p>attempt to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mi>q</mi><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mn>4</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup></math> and divide out <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo>+</mo><mfrac><mrow><mn>24</mn><mi>x</mi><mo>-</mo><mn>56</mn></mrow><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo>+</mo><mfrac><mn>24</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>16</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>5</mn><mn>7</mn></munderover><mfenced><mrow><mn>9</mn><mo>+</mo><mfrac><mn>24</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>16</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></mrow></mfenced><mo> </mo><mtext>d</mtext><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mn>9</mn><mi>x</mi><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mfrac><mn>16</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mrow></mfenced><mn>5</mn><mn>7</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced open="⌊" close="⌋"><mrow><mfenced><mrow><mn>63</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>4</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>45</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>-</mo><mn>8</mn></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>22</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>5</mn><mn>7</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>⇒</mo><mfrac><mrow><mtext>d</mtext><mi>u</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn><mo>=</mo><mn>3</mn><mfenced><mrow><mi>u</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mn>5</mn><mo>=</mo><mn>3</mn><mi>u</mi><mo>+</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>2</mn><mn>4</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>u</mi><mo>+</mo><mn>4</mn></mrow><mi>u</mi></mfrac></mfenced><mn>2</mn></msup><mtext>d</mtext><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>2</mn><mn>4</mn></msubsup><mn>9</mn><mo>+</mo><mfrac><mn>16</mn><msup><mi>u</mi><mn mathvariant="italic">2</mn></msup></mfrac><mo>+</mo><mfrac><mn>24</mn><mi>u</mi></mfrac><mo> </mo><mtext>d</mtext><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mn>9</mn><mi>u</mi><mo>-</mo><mfrac><mn>16</mn><mi>u</mi></mfrac><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>u</mi></mrow></mfenced><mn>2</mn><mn>4</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Ignore absence of or incorrect limits seen up to this point.</p>
<p><em><strong><br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>22</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math><em><strong> A1<br></strong></em></p>
<p><em><strong><br></strong></em><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\omega ">
<mi>ω<!-- ω --></mi>
</math></span> be one of the non-real solutions of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^3} = 1">
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the complex numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 1 - 3{\text{i}}">
<mi>p</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = x + (2x + 1){\text{i}}">
<mi>q</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of</p>
<p>(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \omega + {\omega ^2}">
<mn>1</mn>
<mo>+</mo>
<mi>ω</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>;</p>
<p>(ii) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \omega {\text{*}} + {(\omega {\text{*}})^2}">
<mn>1</mn>
<mo>+</mo>
<mi>ω</mi>
<mrow>
<mtext>*</mtext>
</mrow>
<mo>+</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>ω</mi>
<mrow>
<mtext>*</mtext>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\omega - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13">
<mo stretchy="false">(</mo>
<mi>ω</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>ω</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>13</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> that satisfy the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| p \right| = \left| q \right|">
<mrow>
<mo>|</mo>
<mi>p</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mi>q</mi>
<mo>|</mo>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2}">
<mi>Re</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>8</mn>
<mo><</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>Im</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mi>q</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(i) <strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \omega + {\omega ^2} = \frac{{1 - {\omega ^3}}}{{1 - \omega }} = 0">
<mn>1</mn>
<mo>+</mo>
<mi>ω</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>ω</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\omega \ne 1">
<mi>ω</mi>
<mo>≠</mo>
<mn>1</mn>
</math></span> <strong><em>R1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>solutions of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - {\omega ^3} = 0">
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\omega = 1,{\text{ }}\omega {\text{ = }}\frac{{ - 1 \pm \sqrt 3 {\text{i}}}}{2}">
<mi>ω</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>ω</mi>
<mrow>
<mtext> = </mtext>
</mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
<mo>±</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p>verification that the sum of these roots is 0 <strong><em>R1</em></strong></p>
<p>(ii) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \omega {\text{*}} + {(\omega {\text{*}})^2} = 0">
<mn>1</mn>
<mo>+</mo>
<mi>ω</mi>
<mrow>
<mtext>*</mtext>
</mrow>
<mo>+</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>ω</mi>
<mrow>
<mtext>*</mtext>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\omega - 3{\omega ^2})({\omega ^2} - 3\omega ) = - 3{\omega ^4} + 10{\omega ^3} - 3{\omega ^2}">
<mo stretchy="false">(</mo>
<mi>ω</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>ω</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 3{\omega ^2}({\omega ^2} + \omega + 1) + 13{\omega ^3}">
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>ω</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>13</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 3{\omega ^2} \times 0 + 13 \times 1">
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>0</mn>
<mo>+</mo>
<mn>13</mn>
<mo>×</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 3\omega + 10 - 3{\omega ^2} = - 3({\omega ^2} + \omega + 1) + 13">
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mi>ω</mi>
<mo>+</mo>
<mn>10</mn>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>ω</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>13</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 3 \times 0 + 13">
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo>×</mo>
<mn>0</mn>
<mo>+</mo>
<mn>13</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>substitution by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\omega = \frac{{ - 1 \pm \sqrt 3 {\text{i}}}}{2}">
<mi>ω</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
<mo>±</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> in any form <strong><em>M1</em></strong></p>
<p>numerical values of each term seen <strong><em>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 13">
<mo>=</mo>
<mn>13</mn>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| p \right| = \left| q \right| \Rightarrow \sqrt {{1^2} + {3^2}} = \sqrt {{x^2} + {{(2x + 1)}^2}} ">
<mrow>
<mo>|</mo>
<mi>p</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mi>q</mi>
<mo>|</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<msqrt>
<mrow>
<msup>
<mn>1</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>3</mn>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5{x^2} + 4x - 9 = 0">
<mn>5</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>x</mi>
<mo>−</mo>
<mn>9</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(5x + 9)(x - 1) = 0">
<mo stretchy="false">(</mo>
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1,{\text{ }}x = - \frac{9}{5}">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>9</mn>
<mn>5</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="pq = (1 - 3{\text{i}})\left( {x + (2x + 1){\text{i}}} \right) = (7x + 3) + (1 - x){\text{i}}">
<mi>p</mi>
<mi>q</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>7</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2} \Rightarrow (7x + 3) + 8 < {(1 - x)^2}">
<mi>Re</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>8</mn>
<mo><</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>Im</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mi>q</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">⇒</mo>
<mo stretchy="false">(</mo>
<mn>7</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>8</mn>
<mo><</mo>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {x^2} - 9x - 10 > 0">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mi>x</mi>
<mo>−</mo>
<mn>10</mn>
<mo>></mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow (x + 1)(x - 10) > 0">
<mo stretchy="false">⇒</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < - 1,{\text{ }}x > 10">
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>></mo>
<mn>10</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>An arithmetic sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}{\text{, }}{u_2}{\text{, }}{u_3} \ldots ">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>…<!-- … --></mo>
</math></span> has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 1">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> and common difference <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \ne 0">
<mi>d</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2}{\text{, }}{u_3}">
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6}">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
</math></span> are the first three terms of a geometric sequence</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_N} = - 15">
<mrow>
<msub>
<mi>u</mi>
<mi>N</mi>
</msub>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>15</mn>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^N {{u_r}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>r</mi>
</msub>
</mrow>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} = {u_1} + (n - 1)d">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>d</mi>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(1 + 2d)^2} = (1 + d)(1 + 5d)">
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>d</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>d</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mn>5</mn>
<mi>d</mi>
<mo stretchy="false">)</mo>
</math></span> (or equivalent) <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - 2">
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + (N - 1) \times - 2 = - 15">
<mn>1</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>N</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mo>−</mo>
<mn>2</mn>
<mo>=</mo>
<mo>−</mo>
<mn>15</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N = 9">
<mi>N</mi>
<mo>=</mo>
<mn>9</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^9 {{u_r}} = \frac{9}{2}(2 + 8 \times - 2)">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mn>9</mn>
</munderover>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>r</mi>
</msub>
</mrow>
</mrow>
<mo>=</mo>
<mfrac>
<mn>9</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>+</mo>
<mn>8</mn>
<mo>×</mo>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 63">
<mo>=</mo>
<mo>−</mo>
<mn>63</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^x} + {2^{x + 2}} = 3">
<mrow>
<msup>
<mn>4</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to form a quadratic in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^x}">
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{({2^x})^2} + 4 \bullet {2^x} - 3 = 0">
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mo>∙</mo>
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^x} = \frac{{ - 4 \pm \sqrt {16 + 12} }}{2}{\text{ }}\left( { = - 2 \pm \sqrt 7 } \right)">
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>4</mn>
<mo>±</mo>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>12</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>±</mo>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^x} = - 2 + \sqrt 7 {\text{ }}\left( {{\text{as }} - 2 - \sqrt 7 < 0} \right)">
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<msqrt>
<mn>7</mn>
</msqrt>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>as </mtext>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>7</mn>
</msqrt>
<mo><</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\log _2}\left( { - 2 + \sqrt 7 } \right){\text{ }}\left( {x = \frac{{\ln \left( { - 2 + \sqrt 7 } \right)}}{{\ln 2}}} \right)">
<mi>x</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<mn>2</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>R0 A1 </em></strong>if final answer is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\log _2}\left( { - 2 + \sqrt 7 } \right)">
<mi>x</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Solve the simultaneous equations</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}6x = 1 + 2\,{\text{lo}}{{\text{g}}_2}y">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mn>6</mn>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mi>y</mi>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + {\text{lo}}{{\text{g}}_6}x = {\text{lo}}{{\text{g}}_6}\left( {15y - 25} \right)">
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>15</mn>
<mi>y</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>use of at least one “log rule” applied correctly for the first equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}6x = {\text{lo}}{{\text{g}}_2}2 + 2\,{\text{lo}}{{\text{g}}_2}y">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mn>6</mn>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mn>2</mn>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mi>y</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{lo}}{{\text{g}}_2}2 + \,{\text{lo}}{{\text{g}}_2}{y^2}">
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mn>2</mn>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{lo}}{{\text{g}}_2}\left( {2{y^2}} \right)">
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 6x = 2{y^2}">
<mo stretchy="false">⇒</mo>
<mn>6</mn>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>use of at least one “log rule” applied correctly for the second equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_6}\left( {15y - 25} \right) = 1 + {\text{lo}}{{\text{g}}_6}x">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>15</mn>
<mi>y</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{lo}}{{\text{g}}_6}6 + {\text{lo}}{{\text{g}}_6}x">
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mn>6</mn>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{lo}}{{\text{g}}_6}6x">
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mn>6</mn>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 15y - 25 = 6x">
<mo stretchy="false">⇒</mo>
<mn>15</mn>
<mi>y</mi>
<mo>−</mo>
<mn>25</mn>
<mo>=</mo>
<mn>6</mn>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to eliminate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> (or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>) from their two equations <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{y^2} = 15y - 25">
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>15</mn>
<mi>y</mi>
<mo>−</mo>
<mn>25</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{y^2} - 15y + 25 = 0">
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>15</mn>
<mi>y</mi>
<mo>+</mo>
<mn>25</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {2y - 5} \right)\left( {y - 5} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{25}}{{12}}{\text{,}}\,\,y = \frac{5}{2}{\text{,}}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>25</mn>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{25}}{3}{\text{,}}\,\,y = 5">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>25</mn>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> values do not have to be “paired” to gain either of the final two<em><strong> A</strong></em> marks.</p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}(x + 3) + {\log _2}(x - 3) = 4">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}(x + 3) + {\log _2}(x - 3) = 4">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}({x^2} - 9) = 4">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 9 = {2^4}{\text{ }}( = 16)">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>4</mn>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>16</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = 25">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>25</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pm 5">
<mi>x</mi>
<mo>=</mo>
<mo>±</mo>
<mn>5</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 5">
<mi>x</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The 1st, 4th and 8th terms of an arithmetic sequence, with common difference <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \ne 0">
<mi>d</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>, are the first three terms of a geometric sequence, with common ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>. Given that the 1st term of both sequences is 9 find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>;</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>EITHER</strong></p>
<p>the first three terms of the geometric sequence are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9">
<mn>9</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9r">
<mn>9</mn>
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9{r^2}">
<mn>9</mn>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 + 3d = 9r( \Rightarrow 3 + d = 3r)">
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>d</mi>
<mo>=</mo>
<mn>9</mn>
<mi>r</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">⇒</mo>
<mn>3</mn>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>3</mn>
<mi>r</mi>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 + 7d = 9{r^2}">
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
<mo>=</mo>
<mn>9</mn>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve simultaneously <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 + 7d = 9{\left( {\frac{{3 + d}}{3}} \right)^2}">
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
<mo>=</mo>
<mn>9</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>3</mn>
<mo>+</mo>
<mi>d</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><strong>OR</strong></p>
<p>the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{1}}^{{\text{st}}}}">
<mrow>
<msup>
<mrow>
<mtext>1</mtext>
</mrow>
<mrow>
<mrow>
<mtext>st</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{4}}^{{\text{th}}}}">
<mrow>
<msup>
<mrow>
<mtext>4</mtext>
</mrow>
<mrow>
<mrow>
<mtext>th</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{8}}^{{\text{th}}}}">
<mrow>
<msup>
<mrow>
<mtext>8</mtext>
</mrow>
<mrow>
<mrow>
<mtext>th</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> terms of the arithmetic sequence are</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9,{\text{ }}9 + 3d,{\text{ }}9 + 7d">
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>d</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{9 + 7d}}{{9 + 3d}} = \frac{{9 + 3d}}{9}">
<mfrac>
<mrow>
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
</mrow>
<mrow>
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>d</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>d</mi>
</mrow>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve <strong><em>(M1)</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 1">
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \frac{4}{3}">
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept answers where a candidate obtains <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> by finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> first. The first two marks in either method for part (a) are awarded for the same ideas and the third mark is awarded for attempting to solve an equation in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) - 2{\left( {{\text{ln}}\,2} \right)^2}\left( { = 0} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,x = \frac{{{\text{ln}}\,2 \pm \sqrt {{{\left( {{\text{ln}}\,2} \right)}^2} + 8{{\left( {{\text{ln}}\,2} \right)}^2}} }}{2}">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>±</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{ln}}\,2 \pm 3\,{\text{ln}}\,2}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>±</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{ln}}\,x - 2\,{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x + 2\,{\text{ln}}\,2} \right)\left( { = 0} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,x = 2\,{\text{ln}}\,2">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\text{ln}}\,2">
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = 4">
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong> (M1)A1</strong></em> </p>
<p><strong>Note:</strong> <em><strong>(M1)</strong></em> is for an appropriate use of a log law in either case, dependent on the previous <em><strong>M1</strong></em> being awarded, <strong>A1</strong> for both correct answers.</p>
<p>solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} < x < 4">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>4</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>In the following Argand diagram the point A represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 + 4{\text{i}}"> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mrow> <mtext>i</mtext> </mrow> </math></span> and the point B represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + 0{\text{i}}"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>0</mn> <mrow> <mtext>i</mtext> </mrow> </math></span>. The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_06.11.20.png" alt="M17/5/MATHL/HP1/ENG/TZ2/05"></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>C represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2{\text{i}}"> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <mtext>i</mtext> </mrow> </math></span> <strong><em>A2</em></strong></p>
<p>D represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 + 2{\text{i}}"> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow> <mtext>i</mtext> </mrow> </math></span> <strong><em>A2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r,\,x \in {\mathbb{R}^ + }">
<mi>r</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{{{\text{lo}}{{\text{g}}_r}\,{r^2}}}\left( { = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{{{\text{2}}\,{\text{lo}}{{\text{g}}_r}\,r}}} \right)">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{{{\text{lo}}{{\text{g}}_x}\,{r^2}}}">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>x</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{2\,{\text{lo}}{{\text{g}}_x}\,r}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>x</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{{1 - {x^2}}}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 < x < 1">
<mo>−</mo>
<mn>1</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>1</mn>
</math></span>. Use partial fractions to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f\left( x \right)} {\text{ }}dx">
<mo>∫</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mi>d</mi>
<mi>x</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{1 - {x^2}}} = \frac{1}{{\left( {1 - x} \right)\left( {1 + x} \right)}} \equiv \frac{A}{{1 - x}} + \frac{B}{{1 + x}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>≡</mo>
<mfrac>
<mi>A</mi>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>B</mi>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> <em><strong>M1M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 1 \equiv A\left( {1 + x} \right) + B\left( {1 - x} \right) \Rightarrow A = B = \frac{1}{2}">
<mo stretchy="false">⇒</mo>
<mn>1</mn>
<mo>≡</mo>
<mi>A</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>B</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mi>A</mi>
<mo>=</mo>
<mi>B</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong> M1A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{\tfrac{1}{2}}}{{1 - x}}} + \frac{{\tfrac{1}{2}}}{{1 + x}}dx = \frac{{ - 1}}{2}\ln \left( {1 - x} \right) + \frac{1}{2}\ln \left( {1 + x} \right) + c">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \ln k\sqrt {\frac{{1 + x}}{{1 - x}}} } \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>k</mi>
<msqrt>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
</mfrac>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[8 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>A</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>B</mi><mo>≡</mo><mi>sin</mi><mfenced><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow></mfenced><mo>-</mo><mi>sin</mi><mfenced><mrow><mi>A</mi><mo>-</mo><mi>B</mi></mrow></mfenced></math>. (Do <strong>not</strong> prove this identity.)</p>
<p>Using mathematical induction and the above identity, prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mi mathvariant="normal">Σ</mi><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>cos</mi><mfenced><mrow><mn>2</mn><mi>r</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle><mstyle displaystyle="true"><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mi>n</mi></mfenced></math> be the proposition that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mi mathvariant="normal">Σ</mi><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>cos</mi><mfenced><mrow><mn>2</mn><mi>r</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle><mstyle displaystyle="true"><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math></p>
<p>considering <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mn>1</mn></mfenced></math>:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>LHS</mi><mo>=</mo><mi>cos</mi><mfenced><mn>1</mn></mfenced><mi>θ</mi><mo>=</mo><mi>cos</mi><mo> </mo><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>RHS</mi><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo> </mo><mi>cos</mi><mi>θ</mi></mrow><mstyle displaystyle="true"><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mstyle></mfrac><mo>=</mo><mi>cos</mi><mi>θ</mi><mo>=</mo><mi>LHS</mi></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mn>1</mn></mfenced></math> is true <strong>R1</strong></p>
<p>assume <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mi>k</mi></mfenced></math> is true, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mi mathvariant="normal">Σ</mi><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mi>cos</mi><mfenced><mrow><mn>2</mn><mi>r</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>k</mi></mstyle><mstyle displaystyle="true"><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac><mo> </mo><mfenced><mrow><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></mrow></mfenced></math> <strong>M1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M0</strong> for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>”.</p>
<p><strong>Note:</strong> Subsequent marks after this <strong>M1</strong> are independent of this mark and can be awarded.</p>
<p> </p>
<p>considering <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mi mathvariant="normal">Σ</mi><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></munderover><mi>cos</mi><mfenced><mrow><mn>2</mn><mi>r</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>θ</mi><mo>=</mo><munderover><mi mathvariant="normal">Σ</mi><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mi>cos</mi><mfenced><mrow><mn>2</mn><mi>r</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>θ</mi><mo>+</mo><mi>cos</mi><mfenced><mrow><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced><mi>θ</mi></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>k</mi></mstyle><mstyle displaystyle="true"><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac><mo> </mo><mo>+</mo><mi>cos</mi><mfenced><mrow><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced><mi>θ</mi></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>k</mi></mstyle><mstyle displaystyle="true"><mi>θ</mi><mo>+</mo><mn>2</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>θ</mi></mrow></mfenced><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac><mo> </mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>k</mi></mstyle><mstyle displaystyle="true"><mi>θ</mi><mo>+</mo><mi>sin</mi><mfenced><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>θ</mi><mo>+</mo><mi>θ</mi></mrow></mfenced><mo>-</mo><mo> </mo><mi>sin</mi><mfenced><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>θ</mi><mo>-</mo><mi>θ</mi></mrow></mfenced></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac><mo> </mo></math> <strong>M1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>A</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>B</mi><mo>=</mo><mi>sin</mi><mfenced><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow></mfenced><mo>-</mo><mi>sin</mi><mfenced><mrow><mi>A</mi><mo>-</mo><mi>B</mi></mrow></mfenced></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfenced><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>θ</mi></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>k</mi></mstyle><mstyle displaystyle="true"><mi>θ</mi><mo>+</mo><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mi>θ</mi><mo>-</mo><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>k</mi><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac><mo> </mo></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mn>2</mn><mstyle displaystyle="true"><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac><mo> </mo></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> is true whenever <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mi>k</mi></mfenced></math> is true, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mn>1</mn></mfenced></math> is true, so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi><mfenced><mi>n</mi></mfenced></math> is true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math> <strong>R1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award the final <strong>R1</strong> mark provided at least five of the previous marks have been awarded.</p>
<p> </p>
<p><strong>[8 marks]</strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>z</mi></mrow><mrow><mn>3</mn><mo>-</mo><mi>z</mi><mo>*</mo></mrow></mfrac><mo>=</mo><mtext>i</mtext></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mtext>i</mtext><mi>y</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mtext>i</mtext><mi>y</mi></math> <strong>and </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>*</mo><mo>=</mo><mi>x</mi><mo>-</mo><mtext>i</mtext><mi>y</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mtext>i</mtext><mi>y</mi></mrow></mfenced></mrow><mrow><mn>3</mn><mo>-</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mtext>i</mtext><mi>y</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mtext>i</mtext></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mtext>i</mtext><mi>y</mi><mo>=</mo><mo>-</mo><mi>y</mi><mo>+</mo><mtext>i</mtext><mfenced><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfenced></math> </p>
<p>equate real and imaginary: <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi></math> AND <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mo>=</mo><mn>3</mn><mo>-</mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> If they multiply top and bottom by the conjugate, the equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>x</mi><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>y</mi><mo>-</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>=</mo><msup><mfenced><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></math> may be seen. Allow for <em><strong>A1</strong></em>.</p>
<p><strong><br></strong>solving simultaneously:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>2</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>z</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>+</mo><mn>2</mn><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><strong><br></strong><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, with asymptotes at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe a sequence of transformations that transforms the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan </mtext><mi>x</mi></math> to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>p</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi><mo>≡</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo><</mo><mn>1</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan </mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mtext>arctan </mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi mathvariant="normal">+</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using mathematical induction and the result from part (b), prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>n</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong><br>horizontal stretch/scaling with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><br><strong>Note:</strong> Do not allow ‘shrink’ or ‘compression’</p>
<p><br>followed by a horizontal translation/shift <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> units to the left <em><strong>A2</strong></em></p>
<p><br><strong>Note:</strong> Do not allow ‘move’</p>
<p><br><em><strong>OR</strong></em></p>
<p>horizontal translation/shift <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit to the left</p>
<p>followed by horizontal stretch/scaling with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A2</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>vertical translation/shift up by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> (or translation through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em><br>(may be seen anywhere)</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mi>p</mi></math></strong> and <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi></math> <em>M1</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mtext>tan</mtext><mo> </mo><mi>α</mi></math> </strong>and <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mtext>tan</mtext><mo> </mo><mi>β</mi></math> <em>(A1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></math> <em>A1</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>β</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></mfenced></math> <em>A1</em></strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>p</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi><mo>≡</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo><</mo><mn>1</mn></math>. <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mfenced><mn>1</mn></mfenced></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfenced><mrow><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mn>2</mn><mi>x</mi><mo>+1</mo><mo>-</mo><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mstyle displaystyle="true"><mn>1</mn><mo>+</mo><mfrac><mrow><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+1</mo></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>x</mi></mstyle><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan 1</mtext></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>tan </mtext><mfenced><mrow><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">=</mi><mi>tan</mi><mo> </mo><mfenced><mrow><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi mathvariant="normal">+</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHS</mtext><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mrow></mfrac><mfenced><mrow><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>n</mi></mfenced></math> be the proposition that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>n</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math></p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>1</mn></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mo>=</mo><mtext>RHS</mtext></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math> is true <strong><em>R1</em></strong></p>
<p>assume <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> is true, ie. <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi> </mi><mfenced><mrow><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>”.<br><strong>Note:</strong> Subsequent marks after this <em><strong>M1</strong></em> are independent of this mark and can be awarded.</p>
<p> </p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>+</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mstyle><mrow><mn>1</mn><mo>-</mo><mfenced><mstyle displaystyle="true"><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfenced><mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mstyle></mfenced></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mi>k</mi></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct numerator, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math> factored. Denominator does not need to be simplified</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn><msup><mi>k</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for denominator correctly expanded. Numerator does not need to be simplified. These two <em><strong>A</strong></em> marks may be awarded in any order</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> The word ‘arctan’ must be present to be able to award the last three A marks</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> is true whenever <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> is true and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math> is true, so</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>n</mi></mfenced></math> is true for for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award the final <em><strong>R1</strong></em> mark provided at least four of the previous marks have been awarded.<br><strong>Note:</strong> To award the final <em><strong>R1</strong></em>, the truth of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> must be mentioned. ‘<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>’ is insufficient to award the mark.</p>
<p> </p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Use mathematical induction to prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^n {r\left( {r{\text{!}}} \right)} = \left( {n + 1} \right){\text{!}} - 1">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>r</mi>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, for <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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1\left( {1{\text{!}}} \right) = 1">
<mn>1</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\text{!}} - 1 = 1">
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>1</mn>
</math></span> therefore true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> There must be evidence that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> has been substituted into both expressions, or an expression such LHS=RHS=1 is used. “therefore true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>” or an equivalent statement must be seen.</p>
<p> </p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span>, (so that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^k {r\left( {r{\text{!}}} \right)} = \left( {k + 1} \right){\text{!}} - 1">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>k</mi>
</munderover>
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>r</mi>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>) <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Assumption of truth must be present.</p>
<p> </p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^{k + 1} {r\left( {r{\text{!}}} \right)} = \sum\limits_{r = 1}^k {r\left( {r{\text{!}}} \right)} + \left( {k + 1} \right)\left( {k + 1} \right){\text{!}}">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</munderover>
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>r</mi>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>k</mi>
</munderover>
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>r</mi>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ = }}\left( {k + 1} \right){\text{!}} - 1 + \left( {k + 1} \right)\left( {k + 1} \right){\text{!}}">
<mrow>
<mtext> = </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ = }}\left( {k + 2} \right)\left( {k + 1} \right){\text{!}} - 1">
<mrow>
<mtext> = </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong></em> is for factorising <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {k + 1} \right){\text{!}}">
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
</math></span></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ = }}\left( {k + 2} \right){\text{!}} - 1">
<mrow>
<mtext> = </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\left( {k + 1} \right) + 1} \right){\text{!}} - 1">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p>so if true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span>, then also true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, and as true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> then true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { \in {\mathbb{Z}^ + }} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Only award final <em><strong>R1</strong> </em>if all three method marks have been awarded.<br>Award <em><strong>R0</strong> </em>if the proof is developed from both LHS and RHS.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^4} + a{z^3} + b{z^2} + cz + d = 0"> <mrow> <msup> <mi>z</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mi>a</mi> <mrow> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>b</mi> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>c</mi> <mi>z</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}"> <mi>d</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}"> <mi>z</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> </math></span>.</p>
<p>Two of the roots of the equation are log<sub>2</sub>6 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\sqrt 3 "> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </math></span> and the sum of all the roots is 3 + log<sub>2</sub>3.</p>
<p>Show that 6<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d"> <mi>d</mi> </math></span> + 12 = 0.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - i\sqrt 3 "> <mo>−</mo> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </math></span> is a root <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 + {\text{lo}}{{\text{g}}_2}3 - {\text{lo}}{{\text{g}}_2}6\left( { = 3 + {\text{lo}}{{\text{g}}_2}\frac{1}{2} = 3 - 1 = 2} \right)"> <mn>3</mn> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo>−</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span> is a root <em><strong>(A1)</strong></em></p>
<p>sum of roots: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - a = 3 + {\text{lo}}{{\text{g}}_2}3 \Rightarrow a = - 3 - {\text{lo}}{{\text{g}}_2}3"> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award M1 for use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - a"> <mo>−</mo> <mi>a</mi> </math></span> is equal to the sum of the roots, do not award if minus is missing.</p>
<p><strong>Note:</strong> If expanding the factored form of the equation, award <em><strong>M1</strong> </em>for equating <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> to the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^3}"> <mrow> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> </math></span>.</p>
<p> </p>
<p>product of roots: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( { - 1} \right)^4}d"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mn>4</mn> </msup> </mrow> <mi>d</mi> </math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\left( {{\text{lo}}{{\text{g}}_2}6} \right)\left( {i\sqrt 3 } \right)\left( { - i\sqrt 3 } \right)"> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6\,{\text{lo}}{{\text{g}}_2}6"> <mo>=</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - 6\,{\text{lo}}{{\text{g}}_2}6"> <mi>d</mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> </math></span></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6a + d + 12 = - 18 - 6\,{\text{lo}}{{\text{g}}_2}3 + 6\,{\text{lo}}{{\text{g}}_2}6 + 12"> <mn>6</mn> <mi>a</mi> <mo>+</mo> <mi>d</mi> <mo>+</mo> <mn>12</mn> <mo>=</mo> <mo>−</mo> <mn>18</mn> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>6</mn> <mo>+</mo> <mn>12</mn> </math></span></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 6 + 6\,{\text{lo}}{{\text{g}}_2}2 = 0"> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>2</mn> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1A1AG</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong> </em>is for a correct use of one of the log laws.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 6 - 6\,{\text{lo}}{{\text{g}}_2}3 + 6\,{\text{lo}}{{\text{g}}_2}3 + 6\,{\text{lo}}{{\text{g}}_2}2 = 0"> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>2</mn> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1A1AG</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong> </em>is for a correct use of one of the log laws.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider two events <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="A">
<mi>A</mi>
</math></span> defined in the same sample space.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = \frac{4}{9},{\text{ P}}(B|A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B|A') = \frac{1}{6}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span>;</p>
<p>(ii) hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A \cap B) + {\text{P}}(A' \cap B) - {\text{P}}(A \cap B)"> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A' \cap B)"> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A|B) \times {\text{P}}(B)"> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + \left( {1 - {\text{P}}(A|B)} \right) \times {\text{P}}(B)"> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A'|B) \times {\text{P}}(B)"> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A' \cap B)"> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = {\text{P}}(B|A'){\text{P}}(A')">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{9} = {\text{P}}(A) + \frac{1}{6}\left( {1 - {\text{P}}(A)} \right)">
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 = 18{\text{P}}(A) + 3\left( {1 - {\text{P}}(A)} \right)">
<mn>8</mn>
<mo>=</mo>
<mn>18</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p>(ii) <strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = {\text{P}}(A \cap B) + {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(B|A){\text{P}}(A) + {\text{P}}(B|A'){\text{P}}(A')">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3} \times \frac{1}{3} + \frac{1}{6} \times \frac{2}{3} = \frac{2}{9}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cap B) = {\text{P}}(B|A){\text{P}}(A) \Rightarrow {\text{P}}(A \cap B) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = {\text{P}}(A \cup B) + {\text{P}}(A \cap B) - {\text{P}}(A)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = \frac{4}{9} + \frac{1}{9} - \frac{1}{3} = \frac{2}{9}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{ax + b}}{{cx + d}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mrow>
<mi>c</mi>
<mi>x</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
</mfrac>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R},\,\,x \ne - \frac{d}{c}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>d</mi>
<mi>c</mi>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = \frac{{2x - 3}}{{x - 2}},\,\,x \in \mathbb{R},\,\,x \ne 2">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>2</mn>
</math></span></p>
</div>
<div class="question">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right)"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A + \frac{B}{{x - 2}}"> <mi>A</mi> <mo>+</mo> <mfrac> <mi>B</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span> where A, B are constants.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 2 + \frac{1}{{x - 2}}"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span> <em><strong> A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>3</mn></mrow></mfrac><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to use change the base <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mfrac><mrow><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><mn>2</mn></mrow><mn>2</mn></mfrac><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math></p>
<p>attempt to use the power rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mn>2</mn></msqrt><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math></p>
<p>attempt to use product or quotient rule for logs, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>a</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>b</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>a</mi><mi>b</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><mfenced><mrow><mn>4</mn><msqrt><mn>2</mn></msqrt><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math></p>
<p><strong><br>Note:</strong> The <em><strong>M</strong></em> marks are for attempting to use the relevant log rule and may be applied in any order and at any time during the attempt seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>4</mn><msqrt><mn>2</mn></msqrt><msup><mi>x</mi><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>32</mn><msup><mi>x</mi><mn>6</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>5</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>32</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5</strong></em><em><strong> marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Consider the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></mfenced><mi>n</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>. Determine all possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> for which the expansion has a non-zero constant term.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>EITHER</strong></p>
<p>attempt to obtain the general term of the expansion</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mmultiscripts><mi>C</mi><mi>r</mi><mprescripts></mprescripts><mi>n</mi></mmultiscripts><msup><mfenced><mrow><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow></msup><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></mfenced><mi>r</mi></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mmultiscripts><mi>C</mi><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow><mprescripts></mprescripts><mi>n</mi></mmultiscripts><msup><mfenced><mrow><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mi>r</mi></msup><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>recognize power of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> starts at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>n</mi></math> and goes down by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> each time <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>recognizing the constant term when the power of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is zero (or equivalent) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>n</mi></mrow><mn>4</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>r</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>n</mi><mo>-</mo><mn>4</mn><mi>r</mi><mo>=</mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>r</mi><mo>-</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mfenced><mrow><mi>r</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>9</mn><mo>,</mo><mo>…</mo></mrow></mfenced></math> or one correct value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn><mi>k</mi><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is a (positive) multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>12</mn><mo>,</mo><mo>…</mo></math><br>Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>12</mn></math></p>
<p><strong>Note:</strong> Award full marks for a correct answer using trial and error approach<br>showing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>12</mn><mo>,</mo><mo>…</mo></math> and for recognizing that this pattern continues.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>There was a mixed response to this question. Candidates who used a trial and error approach were more successful in obtaining completely correct answers than those who tried to solve algebraically by finding the general term to form an equation relating n and r . Poor explanations were often noted in the trial and error approach. Candidates often failed to make progress after obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>r</mi></math> in the algebraic approach. Some candidates did not attempt this question.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mi>n</mi></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>n</mi></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>2</mn></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mo>,</mo><mo> </mo><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>×</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
<p>It is given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> term in the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use the chain rule <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1A0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt></mfrac></math> or equivalent seen</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>''</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac><mo>=</mo></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mn>1</mn></msup><mfrac><mrow><mn>1</mn><mo>!</mo></mrow><mrow><mn>0</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mn>2</mn></mrow></msup></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R0</strong></em> for not starting at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>. Award subsequent marks as appropriate.</p>
<p> </p>
<p>assume true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>, (so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mi>k</mi></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi></mrow></msup></math>) <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>M1</strong></em> for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>” or “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> is true”. Subsequent marks can still be awarded.</p>
<p> </p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHS</mtext><mo>=</mo><msup><mi>f</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>d</mi><mfenced><mrow><msup><mi>f</mi><mfenced><mi>k</mi></mfenced></msup><mfenced><mi>x</mi></mfenced></mrow></mfenced></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi></mrow></mfenced><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext><mo>=</mo><msup><mi>f</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for leading coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><strong>Note:</strong> The following <em><strong>A</strong></em> marks can be awarded in any order.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>k</mi></mrow><mn>2</mn></mfrac></mfenced><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for isolating <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> correctly.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for multiplying top and bottom by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for leading coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></msup><mo>=</mo><mtext>RHS</mtext></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>since true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>, and true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> if true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>, the statement is true for all, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>2</mn></math> by mathematical induction <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>To obtain the final <em><strong>R1</strong></em>, at least four of the previous marks must have been awarded.</p>
<p> </p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math></p>
<p>using product rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><mfenced><mrow><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mrow><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>x</mi><mi>h</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mo>…</mo></math></p>
<p>equating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> coefficient to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math> A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to apply binomial theorem for rational exponents <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mfenced><mfenced><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></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>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>m</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>attempt to set equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> and solve <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>m</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </strong></em>or <em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>x</mi><mi>h</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mo>…</mo></math></p>
<p>equating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> coefficient to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p>using product rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mfenced><mi>x</mi></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>2</mn><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mfenced><mi>x</mi></mfenced></math><em><strong> A1</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mi>g</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mn>2</mn><mi>g</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>g</mi><mfenced><mn>0</mn></mfenced><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>×</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>m</mi><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mn>1</mn><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </strong></em>or <em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> A1</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br>