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<h2>HL Paper 3</h2><div class="specification">
<p>This question will investigate power series, as an extension to the Binomial Theorem for negative and fractional indices.</p>
<p>A power series in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is defined as a function of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> where the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_i} \in \mathbb{R}">
<mrow>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>It can be considered as an infinite polynomial.</p>
</div>
<div class="specification">
<p>This is an example of a power series, but is only a finite power series, since only a finite number of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_i}">
<mrow>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
</math></span> are non-zero.</p>
</div>
<div class="specification">
<p>We will now attempt to generalise further.</p>
<p>Suppose <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span> can be written as the power series <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...">
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^5}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</math></span> using the Binomial Theorem.</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>Consider the power series <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - x + {x^2} - {x^3} + {x^4} - ...">
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span></p>
<p>By considering the ratio of consecutive terms, explain why this series is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 1}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which this equality is true.</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>Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 2}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</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>Repeat this process to find the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 3}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</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>Hence, by recognising the pattern, deduce the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - n}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mi>n</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 class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By substituting <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>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_0}">
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating both sides of the expression and then substituting <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>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_1}">
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Repeat this procedure to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_2}">
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_3}">
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the first four terms in what is called the Extended Binomial Theorem for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{1 + {x^2}}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, using integration, find the power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, giving the first four non-zero terms.</p>
<div class="marks">[4]</div>
<div class="question_part_label">k.</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="1 + 5x + 10{x^2} + 10{x^3} + 5{x^4} + {x^5}">
<mn>1</mn>
<mo>+</mo>
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>It is an infinite GP with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 1{\text{,}}\,\,r = - x">
<mi>a</mi>
<mo>=</mo>
<mn>1</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
</math></span> <em><strong>R1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } = \frac{1}{{1 - \left( { - x} \right)}} = \frac{1}{{1 + x}} = {\left( {1 + x} \right)^{ - 1}}">
<mrow>
<msub>
<mi>S</mi>
<mi mathvariant="normal">∞</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>M1A1AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 1}} = 1 - x + {x^2} - {x^3} + {x^4} - ...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1{\left( {1 + x} \right)^{ - 2}} = - 1 + 2x - 3{x^2} + 4{x^3} - ...">
<mo>−</mo>
<mn>1</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 2}} = 1 - 2x + 3{x^2} - 4{x^3} + ...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2{\left( {1 + x} \right)^{ - 3}} = - 2 + 6x - 12{x^2} + 20{x^3}...">
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mn>12</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>20</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 3}} = 1 - 3x + 6{x^2} - 10{x^3}...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - n}} = 1 - nx + \frac{{n\left( {n + 1} \right)}}{{2{\text{!}}}}{x^2} - \frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{{3{\text{!}}}}{x^3}...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mi>n</mi>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{1^q} = {a_0} \Rightarrow {a_0} = 1">
<mrow>
<msup>
<mn>1</mn>
<mi>q</mi>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q{\left( {1 + x} \right)^{q - 1}} = {a_1} + 2{a_2}x + 3{a_3}{x^2} + ...">
<mi>q</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_1} = q">
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mi>q</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q\left( {q - 1} \right){\left( {1 + x} \right)^{q - 2}} = 1 \times 2{a_2} + 2 \times 3{a_3}x + ...">
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mn>2</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_2} = \frac{{q\left( {q - 1} \right)}}{{2{\text{!}}}}">
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</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="q\left( {q - 1} \right)\left( {q - 2} \right){\left( {1 + x} \right)^{q - 3}} = 1 \times 2 \times 3{a_3} + ...">
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_3} = \frac{{q\left( {q - 1} \right)\left( {q - 2} \right)}}{{3{\text{!}}}}">
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q} = 1 + qx + \frac{{q\left( {q - 1} \right)}}{{2{\text{!}}}}{x^2} + \frac{{q\left( {q - 1} \right)\left( {q - 2} \right)}}{{3{\text{!}}}}{x^3}...">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>q</mi>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{1 + {x^2}}} = 1 - {x^2} + {x^4} - {x^6} + ...">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>6</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x + c = x - \frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} - \frac{{{x^7}}}{7} + ...">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>7</mn>
</msup>
</mrow>
</mrow>
<mn>7</mn>
</mfrac>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>M1A1</strong></em></p>
<p>Putting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0 \Rightarrow c = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x = x - \frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} - \frac{{{x^7}}}{7} + ...">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>7</mn>
</msup>
</mrow>
</mrow>
<mn>7</mn>
</mfrac>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">k.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">k.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to investigate conditions for the existence of complex roots of polynomial equations of degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">3</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">4</mtext></math>.</strong></p>
<p> <br>The cubic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mi>r</mi><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi></math>, has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><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>q</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Noah believes that if <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>≥</mo><mn>3</mn><mi>q</mi></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> are all real.</p>
</div>
<div class="specification">
<p>Now consider polynomial equations of degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</p>
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>,</mo><mo> </mo><mi>s</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mi>γ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math>.</p>
<p>In a similar way to the cubic equation, it can be shown that:</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi><mo>)</mo></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>β</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>β</mi><mi>δ</mi><mo>+</mo><mi>γ</mi><mi>δ</mi></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mo>(</mo><mi>α</mi><mi>β</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>β</mi><mi>δ</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mi>δ</mi><mo>)</mo></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>α</mi><mi>β</mi><mi>γ</mi><mi>δ</mi></math>.</p>
</div>
<div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>, has one integer root.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>γ</mi></mrow></mfenced></math> show that:</p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mi>α</mi><mi>β</mi><mi>γ</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"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></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>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi></math>, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> cannot all be real.</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>Using the result from part (c), show that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>17</mn></math>, this equation has at least one complex root.</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>By varying the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> in the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, determine the smallest positive integer value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> required to show that Noah is incorrect.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the equation will have at least one real root for all values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state a condition in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> that would imply <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>0</mn></math> has at least one complex root.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your result from part (f)(ii) to show that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math> has at least one complex root.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State what the result in part (f)(ii) tells us when considering this equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the integer root of this equation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn></math> as a product of one linear and one cubic factor, prove that the equation has at least one complex root.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>γ</mi></mrow></mfenced></math> <strong> </strong><em> <strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>α</mi><mi>β</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>γ</mi></mrow></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>β</mi><mi>γ</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mi>r</mi></mrow></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced><mi>x</mi><mo>-</mo><mi>α</mi><mi>β</mi><mi>γ</mi></math> <em><strong>A1</strong></em></p>
<p>comparing coefficients:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced></math> <em><strong>AG</strong> </em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <em><strong>AG</strong> </em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mi>α</mi><mi>β</mi><mi>γ</mi></math> <em><strong>AG</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> For candidates who do not include the <em><strong>AG</strong> </em>lines award full marks.</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"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi><mo>=</mo><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <strong> </strong><em> <strong>(A1)</strong></em></p>
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup></math> <strong> </strong><em> <strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math> or equivalent <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></math> <em><strong>AG</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Accept equivalent working from RHS to LHS.</p>
<p> </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><strong>EITHER</strong></p>
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup></math> <strong> </strong><em> <strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>α</mi><mi>β</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>β</mi><mi>γ</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>α</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>q</mi></math> or equivalent <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi></math> <em><strong>AG</strong> </em></p>
<p><br><strong>OR</strong></p>
<p>attempt to write <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mi>γ</mi></math> <strong> </strong><em> <strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>q</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>α</mi><mi>β</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>β</mi><mi>γ</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>α</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>AG</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Accept equivalent working where LHS and RHS are expanded to identical expressions.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi><mo>⇒</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi><mo><</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>if all roots were real <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup><mo>≥</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Condone strict inequality in the <em><strong>R1</strong> </em>line.<br><strong>Note:</strong> Do not award <em><strong>A0R1</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo></math>roots cannot all be real <em><strong>AG</strong> </em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>7</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>49</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>q</mi><mo>=</mo><mn>51</mn></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi><mo>⇒</mo></math> the equation has at least one complex root <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Allow equivalent comparisons; e.g. checking <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>6</mn><mi>q</mi></math></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of GDC (<em>eg</em> graphs or tables) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>12</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>complex roots appear in conjugate pairs (so if complex roots occur the other root will be real OR all 3 roots will be real).</p>
<p>OR</p>
<p>a cubic curve always crosses the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at at least one point. <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>β</mi><mi>δ</mi><mo>+</mo><mi>γ</mi><mi>δ</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>β</mi><mi>δ</mi><mo>+</mo><mi>γ</mi><mi>δ</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>⇒</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>=</mo></mrow></mfenced><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>2</mn><mi>q</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Allow <em><strong>FT</strong> </em>on their result from part (f)(i).</p>
<p> </p>
<p><em><strong>[1 mark</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo><</mo><mn>6</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo><</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p>hence there is at least one complex root. <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Allow <em><strong>FT</strong> </em>from part (f)(ii) for the <em><strong>R</strong></em> mark provided numerical reasoning is seen.</p>
<p> </p>
<p><em><strong>[1 mark</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>></mo><mn>2</mn><mi>q</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mn>81</mn><mo>></mo><mn>2</mn><mo>×</mo><mn>24</mn></mrow></mfenced></math> (so) nothing can be deduced <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not allow <em><strong>FT</strong> </em>for the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p><em><strong>[1 mark</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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>[1 mark</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to express as a product of a linear and cubic factor <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>34</mn><mi>x</mi><mo>-</mo><mn>12</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each factor. Award at most <em><strong>A1A0</strong></em> if not written as a product.</p>
<p> </p>
<p>since for the cubic, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi><mo> </mo><mfenced><mrow><mn>100</mn><mo><</mo><mn>102</mn></mrow></mfenced></math> <em><strong>R1</strong></em></p>
<p>there is at least one complex root <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">h.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The first part of this question proved to be very accessible, with the majority of candidates expanding their brackets as required, to find the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The first part of this question was usually answered well, though presentation in the second part sometimes left a lot to be desired. The expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>q</mi></math> was expected to be seen more often, as a 'pivot' to reaching the required result. Algebra was often lengthy, but untidily so, sometimes leaving examiners to do some mental tidying up on behalf of the candidate.</p>
<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;">
<p>A good number of candidates recognised the reasoning required in this part of the question and were able to score both marks.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates found applying this specific case to be very straightforward.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates offered incorrect answers in the first part; despite their working suggested utilisation of the GDC, it was clear that many did not appreciate what the question was asking. The second part was usually answered well, with the idea of complex roots occurring in conjugate pairs being put to good use.</p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Some very dubious algebra was seen here, and often no algebra at all. Despite this, a good number of candidates seemed to make the 'leap' to the correct expression <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></math>, perhaps fortuitously so in a number of cases.</p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Of those finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></math> in part f, a surprising number of answers seen employed the test of checking whether <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi></math>.</p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Part i was usually not answered successfully, which may have been due to shortage of time. However, it was pleasing to see a number of candidates reach the end of the paper and successfully factorise the given quartic using a variety of methods. The final part required the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi></math> test. Though correct reasoning was sometimes seen, it was rare for this final mark to be gained.</p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>This question will explore connections between complex numbers and regular polygons.</p>
<p>The diagram below shows a sector of a circle of radius 1, with the angle subtended at the centre <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="O">
<mi>O</mi>
</math></span> being <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha {\text{,}}\,\,0 < \alpha < \frac{\pi }{2}">
<mi>α<!-- α --></mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo><</mo>
<mi>α<!-- α --></mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>. A perpendicular is drawn from point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> to intersect the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q">
<mi>Q</mi>
</math></span>. The tangent to the circle at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> intersects the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAP4AAACJCAYAAAAMsYUoAAAU10lEQVR4Ae1dC3AVZZb+7s07QAiEACKPvECGh4g8JC6PYQQSRNixCGOgZkSdkSBlFbA1hnJLVne1aiW4BVZNyWNVXFTAnbAzxUMSwSiEAUFe4TUIeYFMeAYSIEgSk2yd/6aTvjd9k/vo7tt9+3TVre7++//POf937nf/Z59ra2pqagIfjAAjYCkE7JaqLVeWEWAEBAJMfP4iMAIWRICJb0Gnc5UZASY+fwcYAQsiwMS3oNO5yowAE5+/A4yABRFg4lvQ6VxlRoCJz98BRsCCCDDxLeh0rjIjwMTn7wAjYEEEmPgWdDpXmRFg4vN3gBGwIAJMfAs6navMCDDxg+A7sGjRIthsNqfPli1bgqBmXAWtEGDia4WsznLT0tJAb1jTZ/PmzZg7dy7279+vsxWsziwIMPHN4ikv7MzMzAT9EGzatMmLUpzVSggw8YPU28XFxUFaM66WGggw8dVA0WAyaHxfUlKCefPmGcwyNscoCIQaxRC2wz8E8vPzxeSeJIWIn5SUJN3ymRFwQoBbfCc4zHsjn9yjCT4mvXl9qYflTHw9UGYdjIDBEGDiG8whbA4joAcCTHw9UGYdjIDBELBxXH2DeYTNYQR0QIBbfB1AZhWMgNEQYOIbzSNsDyOgAwJMfB1AZhWMgNEQYOIbzSNsDyOgAwJMfB1AZhWMgNEQYOIbzSMq2kM7+N74aDeu3LqrolQWFQwIMPGDwYsKdait+xnZ6/JxsiQMf/q/4wo5OMnKCDDxg9D7d+7XInt9HpL7dMeDujAcO9+Io+crgrCmXCVfEWDi+4qcQcvdqK7Ba2t3YUTyQ1g4ayxC7HYM7t8FKzadM6jFbFYgEGDiBwJ1jXRevlGN7LV5mDQiES9NHyW0hNiB/3x5pGj5391UpJFmFms2BJj4ZvOYG3tLKm4he10enkkdjHlPjWjJZbfb0IgmZM3qj4Kj93D2YnXLM76wLgJM/CDw/dmL1wXp5z41ArMnDnWqkd0GNDY2Yca4ARiaCPz7J2ednvONNRFg4pvc78cvVIjufdYzYzAzdXCb2ogWv8mRvHz+46i6V4V3PmXytwHKYglMfBM7/LuzP4qW/rXMiZg2ZqBiTQTxGx3M794lCi8+3RdHzt1C3uFrivk50RoIMPFN6ue9RWVY/vEevP3SVEwakeC2FiHNXX0pQ+bk4YiKrMSq/y1B6ZX7UjKfLYYAE9+EDt99pBgrtxQiJysN44b0a7cG8q6+lPG5yYMwoHcdVv+ZQ3BLmFjtzMQ3mce3HziHNdsOIWfhdIwc2KdD62kd/+eGRqd8vx7/C1Tfv4ouUTb86S+lTs/4xhoIMPFN5Oet+05jc8FJrFw4HUMGxHtkud0ONA/xnfLT7H9EZCWKSu4g/3se7zuBY4EbJr5JnLzp6yLsOPiD6N7TVlxPD/nknrwMEf/EhQpkTIzD+7mlKL/K4305PsF+zcQ3gYc35B0DTeatyEpH3/iuXlksreO7FqIhgCB/SRlefiYBq3i87wpRUN8z8Q3u3rXbDuFE8RXRve8Z28lra0No517zOr5rYSJ+4amLGPVIFPrEReGDv/J43xWjYL1n4hvYs+9vPYCSK7dF9z4mOsInS+02m9i5p1Q4MjxUtPpb953B0jnJOHq+CnuO3lDKymlBhgAT36AOzdlSiBtVNchZkI6IMN//4pAm91wm9Z1qPHvCEOw6dB5VNT9hSUaKWOK7dO0npzx8E3wIMPEN6NO3NxaAAmm88/upsNn8M9Cxju+mrw8gplMknn5iEL76/gKGJ8XgxekDsDqX1/f9Q934pZn4BvKRFCorIjwMy5+frIpl7ib35MKnjR6Ir444yD57Uh/Ex0Zg7bYyeRa+DjIEmPgGcagUKis+thOyMyeoZlVIiB0N7mb3mrUMSeiJLtHhOPT3H0XKkoxkHP77bXzN433V/GA0QUx8A3hEHipr8ewnVbUopJ3JPbki0ep/72j1oyJCsHh2Mt7fWoIfr/N4X45TsFwz8QPsyetV95xCZaltjti557xjV1HFtNEpKDxVjso7jo08I1K64vm0/jzeV0TL/IlM/AD6UITKWpfvFCpLbXNocq+hyf3knqQvKiJMvNorjfUpPWNSH8TFhGPddh7vSzgFy5mJHyBPSqGyZqY+4hQqS21zPJnck3RSq0+z+/KDlvgOnL6Fb47flCfztckRYOIHwIEtobJ+RaGyhmlqgdJrue4UPprUW+wZOPLDP1qyREeGYOmcFNHlr7j5oCWdL8yNABNfZ/9RqKxl6/KxcOZYzHyybagstc1pb+eekq5pY1Lw1RHnVv+xlK6YN6UfVvH6vhJkpkxj4uvoNilU1h+fG4+po1N00UzhtSnYpqcHheY+cPpSm+zPTX4YMdGh+HDHxTbPOMF8CDDxdfKZI1TW7uZQWYk6aQW86eqTUXEx0RjUrwcOnm1L/iVzUrDv5E3sPcHjfd0cqJEiJr5GwMrFtobKSu8wVJa8nBrXShF4OpKbOrQ/Dp5pS/wuUaFYnOFY379SyeP9jnA08nMmvsbe2X7wHNZuP+xxqCy1zXEXgac9PalDiPiOXXyu+UYNisVvJj/M8fpcgTHZPRNfQ4eJUFlfFyEnK93jUFlqmyO6+l6M8Ul/3/gY9OgaLeIAKNmT+au+6BQVio928nhfCR8zpDHxNfIShcraLkJlpYt/rdVITYdivVnHlwtz192X8tB4/5vjN1B4slJK4rOJEGDia+Csj3cdFaGyqKX3NlSW2ua0F4GnPV2C+AoTfFIZmuEn8lOI7mu3a6VkPpsEASa+yo5au+0wikp8D5WlsjmgdfyO3s5T0jnw4TjYbXb88KP7GfzRj8SCXuNd/ecSJRGcZmAEmPgqOkeEyqq4Jcb0vobKUtEcIcrTl3SU9KYO7ac4uy/PSxt7IsPt2LCr7SqAPB9fGwsBJr5K/mgJlZXlX6gslcxpEeNYx/d8A09LQQBjB/fDsQsV8iTFa1ri233kOv52isf7igAZMJGJr4JT3vn0G9TVqxMqSwVznET4OrlHQoYn9cKZ8muorf/ZSabrTWznMBGsc3VuCW5U8XjfFR8j3jPx/fCKFCorPCwUb/xOnVBZfpijWJRafF/G+CQsNMSOYYm9cbrsuqJseeKYwd3wz+Mfwioe78thMew1E99H12gVKstHc9wWC7Xbvdqr7ypoeGIvnC7z7C+2fju1H0JDbPifPB7vu+JotHsmvg8eEaGy1uUhuU83qB0qywdz2i1Ck3vthddut3Bzd/9U6dWOsrU8pyW+XYeu4eCZWy1pfGE8BJj4XvqEQmVlr8vDYwP7YOGsJ7wsrX92fyb3yNrhib1xquwaPAjiIyrXvUsYlsxJFkt8N6t5vK+/xz3TyMT3DCeRSwqVNfHRBLyY/rgXJQOX1Z/JPbI6OjJM7Dw8XeZ5qz9uSHfMSO0l/owzcDVnze0hwMRvDx3ZM71CZclUqnLpaPH9EzUssZdo9b2RQoE66diYz+N9b3DTKy8T3wOkz168Ibr3c3UIleWBOV5l8TYCj5JwmuA7VerZBJ+8PK3v7zx4Dd+d5fG+HBcjXDPxO/CCI1RWnm6hsjowx+vHjgg8XhdzKkDr+ae86OpLhXt0DW8Z79+6Wycl89kACDDx23FCIEJltWOOT4/EOr6nM3NuNMR2jkKnyHBcqbzrJof75NSh3TH9iV68n989RAF5wsR3A/veonIs/1j/UFluzPE52d/JPUlxYu9uKL96W7r16jw/vT8aGprw2W7l4B5eCePMqiDAxFeAkUJlvfdFoXjZZtyQfgo5zJPkyX/neVKbhN6xPhOf5NMS318Lr+D7c779eHhiI+fxHAEmvgtWUqisFVlpGDmwj8tT892KWX0vI/Ao1TKhdzeUXa1SeuRRGv0D79I5yWJLb9W9eo/KcCbtEGDiy7B1DpXVU/bEvJchNu/Ca7urKRHf166+JPOfhsdhyqh40Ms8fAQWASZ+M/5GCZWl9tdBjXV8ssnfrr5Ur5eeHoDaukZs2sPjfQmTQJyZ+AA25B3DvpPlYkwf6FBZan8JfI3A42pHZHgYenXrjEvXfe/uSzIXZyRh674KHPmBx/sSJnqfLU/8tdsO4cSFCkH6nrGd9MZfc33+ROBxNc7R3fef+L27R4L+jJNCdt253/67/q428L06CFia+I5QWbcF6Y0SKksdt7ZKcXT1fYvA0yrFcaXGOF+SOeHROEweGc/x+SVAdD5blvgUKutm9X1B+ojwUJ1h10+dWuv4ZHHNgzpcuuZ/iy/V/vczBqDmQQO2FFyWkvisEwKWJL4UKuvtl6bAZtMJ6QCp8TW8tpK59FYi/ViqeSzJSMYXBZdx7EK1mmJZVgcIWIr4tHP1jY92w8ihsjrwl9eP6b/zGvyJxCHTSP+uU3lHXeI/FCeN94txl8f7MrS1vbQM8R2hsvIQH9sJ2ZkTtEXVQNL9jcAjr0pcTCdU3vlJnqTK9aTHemDC8Di8v5XX91UB1AMhliC+mUJleeAzr7KoObkXFREK+iG5e1/9yDovz0xAdU09vviGx/teOdjHzEFP/OtVNaYKleWjH90WU3Nyj5T0iFG/uy8Zv2R2MjbtuYwTPN6XINHsHNTEd4TKyoOZQmWp7Wm1du5JdsV11aa7T/Ifjo/C0owUrMotxv0HDZJKPmuAQNASXwqV9UzqI5j31AgNoDOHyBCbza/w2q611LLFJ12/HNkDTw6Lw+rcYlfVfK8iAkFJfHmorIyJw1SEy3yi1Ny5R7WPo5n96hpNgciamYDKO3XI/bbjv+/S1JAgFh50xKdQWRT+euHMsZj55OAgdp1nVaOufoOfEXjkmuJionFT5SU9uXzpmrb0bvzqEk6W3JGS+KwiAkFFfAqVtWx9Pl57bjymjk5RhOnChQsYO3YsbDabOB8/flzkmz9/PqZMmYJ79+4pljNrotqTe91jonD77gPN4ejXMwoUrJO6/D/V8nhfbcCDhvh7i8rwbxv24D9enIJJIxIVcdqyZQvS09Px+eefg/73bsGCBfjss8+wY8cO7Ny5EytXrkTnzp0Vy5o1Ua0IPFL9I8JCO/wTTSmvv+enHo/H2F905/V9f4FUKB8UxKdQWSu/2I8VC9LgLlQWtexvvvkmcnNzMXDgQAFFZmYm7HY7Fi9ejNdffx0jR45UgMjcSWJWX4UIPBIK4aEhqP9ZvxZ44awEXL9di617ebwv+UCNs+mJv/3AOazdfhg5HYTKWr16NcaNG9eG3PSDkJiYiKysLDXwNJwMtSLwSBWLCAsB7YLU86Au/4ZdF3GqlMf7auFuauLn7juNzQVF4g27IQPch8oiclNXfs6cOU64ffvttzhx4kRQdvGliqq9jk/vOdTW69fiUz0G9IoG/Rknjfdr6xulqvHZDwRMS3wKlbXzu/OC9Ml9uncIQVJSEkaNGtWSjyb5qIs/Y8aMNr2AlkxBcKFWBB4JCurq1+nY1Zf0Uqy+UYNi+f19CRA/z5oTPyUlRcyg0yw6fWhyzd9DhMoqKseKBdPgaais0tJS7N27V6j+8ssvkZqaimeffRYRERFiJv/DDz+ENMPvr31GKi/W8dWJwyGqJbr69fp29SU8F/06CRWVD/CXwitSEp99REAz4hPRiOjTpk0TM+g0i04fOijd12Pd9sOOUFkL09Ez1rMZeJq0e/fdd/Hqq68K3W+99RY2btwIOpOdCQkJYjafJ/c69goFLQlEiy9ZtnROCv57RznOlPF4X8LEp3OTRkdaWloTfZQOAE0rVqxQetRu2urcvzX9cc2upge19e3m44etCNypqW/K+q/jrQl+XlXXPGia9a+f+inFv+L5h681/SHnWFNdfYN/gixcWrMWPz8/Hy+88ILij9Err7yCgoICxWfuEq0SKstd/X1N7xIdirX/8pivxduUo65+IFt8MmjamJ4YkdIV5y4F12arNmBrmGCjHz215VP3OTk5GYWFhRg/fnwb8Tk5OVi/fj2Kiz17EYNCZRUcL20jhxP0R8CGHrDb4hAaEiLCltHOQFo5oNEbTSS2XjvSKY0GdpROeUU+u73lmkKDURoN/1qv0XztKOMs0+6Q4STTobdVPpVTTnPob7WFIhQ59DfrbB6KSra01k/ZFmedjnrI6yrkwFFHSSbpk/QqpTmeK2NKEYrVOEwRZfKN300GffgwBgLUVDQ0NoLOtDeosZHmb+i6SXbtSKc0alkoD+UV+RobW64bmstS+9N6TfIdc0LO8klOs14nmQ69znlb0xw6SX+znTJbpHpQnlb9jS3XrTId9rfW1WFLa52k+jnrFTLh0NsqvxU/pTS5La6YDurXCcuf9/8dFE2Jf/mycjSV8vJy0Gw/H+ZEgFqk0BDNRonmBMVkVmviPVozp67+vn37FOFYs2aN2/G/YgFOZAQYAVUR0KzF/+STTzBhwgSxVJadnd1iNLX0aWlpoH3yfDACjEBgENBkck+qijTJJ93TmWb0P/jgA3kSXzMCjIDOCGhKfJ3rwuoYAUbAQwQ0GeN7qJuzMQKMQIAQYOIHCHhWywgEEgEmfiDRZ92MQIAQYOIHCHhWywgEEgEmfiDR91O3Fq88+2mS5sUXLVoktvfSFl/ps3//fs31BpsCJr4JParVK89mgYL2gUiveW/evFnsF2Hye+c9Jr53eBkiN7V69OV33Q+Rl5cn7KOXoKxy0EYw2iV64MABq1RZlXoy8VWBUV8har/yrK/1rM0ICDDxjeAFL2ygbj4dffv2VSxF0YQ8fd1ZUYDJEql3U1JSgoyMDJNZHlhzNdurH9hqsfZgRoB6PFL4NurmaxBSIpjhE3XjFt+kLrbyK8/yyT1q7WnOgw/vEGDie4dXwHPzK8/OLqBZfXrNWxoCOT/lO3cIMPHdIWPgdHrlmb7srrP3tK5PXV8rvfIszeq/9957BvaY8Uxj4hvPJx1aRHEMqYu7bNmylk0s0piX0tX474IOjTBQBvrzU/oh5MNzBPi1XM+xMk1OGvMSEehHgIYGfDACrggw8V0R4XtGwAIIcFffAk7mKjICrggw8V0R4XtGwAIIMPEt4GSuIiPgigAT3xURvmcELIAAE98CTuYqMgKuCDDxXRHhe0bAAggw8S3gZK4iI+CKABPfFRG+ZwQsgAAT3wJO5ioyAq4I/D8BC8J3uiqNsgAAAABJRU5ErkJggg=="></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the area of two triangles and the area of the sector show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\alpha \,{\text{sin}}\,\alpha < \alpha < \frac{{{\text{sin}}\,\alpha }}{{{\text{cos}}\,\alpha }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo><</mo> <mi>α</mi> <mo><</mo> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> </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>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{\alpha \to 0} \frac{\alpha }{{{\text{sin}}\,\alpha }} = 1"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mi>α</mi> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> </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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^n} = 1{\text{,}}\,\,z \in \mathbb{C}{\text{,}}\,\,n \in \mathbb{N}{\text{,}}\,\,n \geqslant 5"> <mrow> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>z</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>⩾</mo> <mn>5</mn> </math></span>. Working in modulus/argument form find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> solutions to this equation.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Represent these <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> solutions on an Argand diagram. Let their positions be denoted by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{\text{,}}\,\,{P_1}{\text{,}}\,\,{P_2}{\text{,}}\, \ldots {P_{n - 1}}"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mrow> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></span> placed in order in an anticlockwise direction round the circle, starting on the positive <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. Show the positions of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{\text{,}}\,\,{P_1}{\text{,}}\,\,{P_2}"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_{n - 1}}"> <mrow> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the length of the line segment <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{P_1}"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\,{\text{sin}}\frac{\pi }{n}"> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the total length of the perimeter of the regular <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> sided polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{P_1}{P_2} \ldots {P_{n - 1}}{P_0}"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> <mo>…</mo> <mrow> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using part (b) find the limit of this perimeter as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \to \infty "> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total area of this <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> sided polygon.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using part (b) find the limit of this area as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \to \infty "> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">i.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="OPQ = \frac{1}{2}{\text{cos}}\,\alpha \,{\text{sin}}\,\alpha "> <mi>O</mi> <mi>P</mi> <mi>Q</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </math></span> <em><strong>A1</strong></em></p>
<p>Area sector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{1^2}\alpha "> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mn>1</mn> <mn>2</mn> </msup> </mrow> <mi>α</mi> </math></span> <em><strong>A1</strong></em></p>
<p>Area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="OPR = \frac{1}{2}1\,{\text{tan}}\,\alpha "> <mi>O</mi> <mi>P</mi> <mi>R</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </math></span> <em><strong>A1</strong></em></p>
<p>So looking at the diagram <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{\text{cos}}\,\alpha \,{\text{sin}}\,\alpha < \frac{1}{2}\alpha < \frac{1}{2}\frac{{{\text{sin}}\,\alpha }}{{{\text{cos}}\,\alpha }}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>α</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{cos}}\,\alpha \,{\text{sin}}\,\alpha < \alpha < \frac{{{\text{sin}}\,\alpha }}{{{\text{cos}}\,\alpha }}"> <mo stretchy="false">⇒</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo><</mo> <mi>α</mi> <mo><</mo> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\alpha < \frac{\alpha }{{{\text{sin}}\,\alpha }} < \frac{1}{{{\text{cos}}\,\alpha }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo><</mo> <mfrac> <mi>α</mi> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> <mo><</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> </math></span> and as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha \to 0{\text{,}}\,\,{\text{cos}}\,\alpha \to 1"> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo stretchy="false">→</mo> <mn>1</mn> </math></span> we have <em><strong>M1R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{\alpha \to 0} \frac{\alpha }{{{\text{sin}}\,\alpha }} = 1"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mi>α</mi> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">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( {r\,cis\,\theta } \right)^n} = 1\,cis\,0 \Rightarrow {r^n}cis\,n\,\theta = 1\,cis\,\theta "> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mi>i</mi> <mi>s</mi> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mi>i</mi> <mi>s</mi> <mspace width="thinmathspace"></mspace> <mn>0</mn> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>r</mi> <mi>n</mi> </msup> </mrow> <mi>c</mi> <mi>i</mi> <mi>s</mi> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mi>i</mi> <mi>s</mi> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span> <em><strong>M1A1</strong></em><em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^n} = 1 \Rightarrow r = 1"> <mrow> <msup> <mi>r</mi> <mi>n</mi> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>r</mi> <mo>=</mo> <mn>1</mn> </math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n\theta = 0 + 2\pi k{\text{,}}\,\,k \in \mathbb{Z}"> <mi>n</mi> <mi>θ</mi> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{{2\pi k}}{n}{\text{,}}\,\,0 \leqslant k \leqslant n - 1"> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>0</mn> <mo>⩽</mo> <mi>k</mi> <mo>⩽</mo> <mi>n</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="z = cis\frac{{2\pi k}}{n}{\text{,}}\,\,0 \leqslant k \leqslant n - 1"> <mi>z</mi> <mo>=</mo> <mi>c</mi> <mi>i</mi> <mi>s</mi> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>0</mn> <mo>⩽</mo> <mi>k</mi> <mo>⩽</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Bisecting the triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="O{P_0}{P_1}"> <mi>O</mi> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> </math></span> to form two right angle triangles <em><strong>M1</strong></em></p>
<p><img src="data:image/png;base64,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"> </p>
<p>Length of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{P_1} = 2t"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mn>2</mn> <mi>t</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {\text{sin}}\left( {\frac{{\frac{{2\pi }}{n}}}{2}} \right)"> <mi>t</mi> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong> </strong><em><strong> M1A1A1</strong></em></p>
<p>So length is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\,{\text{sin}}\frac{\pi }{n}"> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> <em><strong>A</strong><strong>G</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Length of perimeter is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2n\,{\text{sin}}\frac{\pi }{n}"> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2n\,{\text{sin}}\frac{\pi }{n} = 2\pi \frac{n}{\pi }{\text{sin}}\frac{\pi }{n} \to 2\pi "> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mfrac> <mi>n</mi> <mi>π</mi> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo stretchy="false">→</mo> <mn>2</mn> <mi>π</mi> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \to \infty "> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="O{P_0}{P_1} = \frac{1}{2}1 \times 1\,{\text{sin}}\frac{{2\pi }}{n}"> <mi>O</mi> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mn>1</mn> <mo>×</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </math></span> so total area is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{n}{2}{\text{sin}}\frac{{2\pi }}{n}"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </math></span>. <em><strong>M1A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{n}{2}{\text{sin}}\frac{{2\pi }}{n} = \pi \frac{n}{{2\pi }}{\text{sin}}\frac{{2\pi }}{n} \to \pi "> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> <mo>=</mo> <mi>π</mi> <mfrac> <mi>n</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> <mo stretchy="false">→</mo> <mi>π</mi> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \to \infty "> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">i.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">i.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will be exploring the strategies required to solve a system of linear differential equations.</strong></p>
<p> </p>
<p>Consider the system of linear differential equations of the form:</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>y</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>y</mi></math>,</p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>,</mo><mo> </mo><mi>t</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is a parameter.</p>
<p>First consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>4</mn></math>.</p>
</div>
<div class="specification">
<p>From previous cases, we might conjecture that a solution to this differential equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is a constant.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation in part (a)(ii) to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></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>By substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</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>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the two values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></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>Let the two values found in part (c)(ii) be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math>.</p>
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math> is a solution to the differential equation in (c)(i),where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> is a constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>=</mo><mi>y</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mi>y</mi></mfrac><mo>=</mo><mo>∫</mo><mo>d</mo><mtext>t</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>y</mi><mo>=</mo><mi>t</mi><mo>+</mo><mi>c</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mfenced open="|" close="|"><mi>y</mi></mfenced><mo>=</mo><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>y</mi></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>rearranging to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>y</mi><mo>=</mo><mn>0</mn></math> AND multiplying by integrating factor <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mi>A</mi></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></p>
<p> </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>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> into differential equation in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>integrating factor (IF) is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>∫</mo><mo>-</mo><mn>1</mn><mo>d</mo><mi>t</mi></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>A</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>A</mi><mi>t</mi><mo>+</mo><mi>D</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfenced><mrow><mo>-</mo><mi>A</mi><mi>t</mi><mo>+</mo><mi>D</mi></mrow></mfenced><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The first constant must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, and the second can be any constant for the final <em><strong>A1</strong></em> to be awarded. Accept a change of constant applied at the end.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></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><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mfenced><mrow><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>Y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>Y</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>Y</mi></mrow><mi>Y</mi></mfrac><mo>=</mo><mo>∫</mo><mn>2</mn><mo>d</mo><mi>t</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced open="|" close="|"><mi>Y</mi></mfenced><mo>=</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi>c</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>Y</mi><mo>=</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>∫</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mtext> </mtext><mo>d</mo><mi>t</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> The first constant must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, and the second can be any constant for the final <em><strong>A1</strong></em> to be awarded. Accept a change of constant applied at the end.</p>
<p> </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><strong>METHOD 1</strong></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> and their (iii) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></math> <em><strong>M1(M1)</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> A1</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> AG</strong></em></p>
<p><strong>Note:</strong> Follow through from incorrect part (iii) cannot be awarded if it does not lead to the <em><strong>AG</strong></em>.</p>
<p><br><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>∫</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mtext> </mtext><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>+</mo><mi>D</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi><mo>+</mo><mi>D</mi><msup><mtext>e</mtext><mi>t</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>⇒</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>=</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi><mo>-</mo><mi>D</mi><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi><mo>⇒</mo><mi>D</mi><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mi>y</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> seen anywhere <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>attempt to eliminate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>4</mn><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mrow><mi>y</mi><mo>-</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>3</mn><mi>y</mi></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>rewriting LHS in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mfenced><mrow><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>-</mo><mn>3</mn><mi>y</mi></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> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>F</mi><mi>λ</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>F</mi><msup><mi>λ</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup></math><em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><msup><mi>λ</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mi>F</mi><mi>λ</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>=</mo><mn>0</mn></math><em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>≠</mo><mn>0</mn></math>)<em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math> (either order)<em><strong> A1</strong></em></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"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced></math><em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>6</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>2</mn><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></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> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>F</mi><msub><mi>λ</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msub><mi>λ</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mstyle displaystyle="true"><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mstyle></mfrac><mo>=</mo><mi>F</mi><msup><msub><mi>λ</mi><mn>1</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><msub><mi>λ</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mi>F</mi><msup><msub><mi>λ</mi><mn>1</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><msub><mi>λ</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>F</mi><msub><mi>λ</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msub><mi>λ</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></mrow></mfenced></math><em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mfenced><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mfenced><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn></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> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.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.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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iv.</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">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>A <strong>Gaussian integer</strong> is a complex number, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</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>. In this question, you are asked to investigate certain divisibility properties of Gaussian integers.</p>
</div>
<div class="specification">
<p>Consider two Gaussian integers, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>3</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mn>1</mn><mo>-</mo><mn>2</mn><mtext>i</mtext></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>=</mo><mi>α</mi><mi>β</mi></math> for some Gaussian integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math>.</p>
</div>
<div class="specification">
<p>Now consider two Gaussian integers, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>3</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>=</mo><mn>11</mn><mo>+</mo><mn>2</mn><mtext>i</mtext></math>.</p>
</div>
<div class="specification">
<p>The norm of a complex number <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>z</mi></mfenced></math>, is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><msup><mfenced open="|" close="|"><mi>z</mi></mfenced><mn>2</mn></msup></math>. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>2</mn><mo>+</mo><mn>3</mn><mtext>i</mtext></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mtext>i</mtext></mrow></mfenced><mo>=</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>=</mo><mn>13</mn></math>.</p>
</div>
<div class="specification">
<p>A <strong>Gaussian prime</strong> is a Gaussian integer, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>, that <strong>cannot</strong> be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>α</mi><mi>β</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> are Gaussian integers with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>,</mo><mo> </mo><mi>N</mi><mfenced><mi>β</mi></mfenced><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>The positive integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> is a prime number, however it is not a Gaussian prime.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> be Gaussian integers.</p>
</div>
<div class="specification">
<p>The result from part (h) provides a way of determining whether a Gaussian integer is a Gaussian prime.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>γ</mi><mi>α</mi></mfrac></math> is a Gaussian integer.</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>On an Argand diagram, plot and label all Gaussian integers that have a norm less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</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>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing the positive integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></math> as a product of two Gaussian integers each of norm <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is not a Gaussian prime.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> is not a Gaussian prime.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down another prime number of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></math> that is not a Gaussian prime and express it as a product of two Gaussian integers.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mo>=</mo><mi>N</mi><mfenced><mi>α</mi></mfenced><mi>N</mi><mfenced><mi>β</mi></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">h.</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"><mn>1</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></math> is a Gaussian prime.</p>
<div class="marks">[3]</div>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use proof by contradiction to prove that a prime number, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, that is not of the form <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 a Gaussian prime.</p>
<div class="marks">[6]</div>
<div class="question_part_label">j.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mtext>i</mtext></mrow></mfenced><mo>=</mo><mn>11</mn><mo>-</mo><mn>2</mn><mtext>i</mtext></math> <em><strong>(M1)</strong><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>γ</mi><mi>α</mi></mfrac><mo>=</mo><mfrac><mn>41</mn><mn>25</mn></mfrac><mo>-</mo><mfrac><mn>38</mn><mn>25</mn></mfrac><mtext>i</mtext></math> <em><strong>(M1)</strong><strong>A1</strong></em> </p>
<p>(Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfrac><mi>γ</mi><mi>α</mi></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>41</mn><mn>25</mn></mfrac></mrow></mfenced></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Im</mtext><mfrac><mi>γ</mi><mi>α</mi></mfrac><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>38</mn><mn>25</mn></mfrac></mrow></mfenced></math> are not integers)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>γ</mi><mi>α</mi></mfrac></math> is not a Gaussian integer <em><strong> R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award<em><strong> R1</strong></em> for correct conclusion from their answer.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>±</mo><mtext>i</mtext><mo>,</mo><mo> </mo><mn>0</mn></math> plotted and labelled <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>±</mo><mtext>i</mtext><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>±</mo><mtext>i</mtext></math> plotted and labelled <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if extra points to the above are plotted and labelled.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>z</mi></mfenced><mo>=</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> (and as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><msup><mfenced open="|" close="|"><mi>z</mi></mfenced><mn>2</mn></msup></math>) <em><strong>A1</strong></em></p>
<p>then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</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>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup><mo>=</mo><mfenced><mrow><mi>c</mi><mo>+</mo><mi>d</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>d</mi><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>c</mi><mo>+</mo><mi>d</mi><mtext>i</mtext></mrow></mfenced><mo>=</mo><mi>N</mi><mfenced><mrow><mi>c</mi><mo>-</mo><mi>d</mi><mtext>i</mtext></mrow></mfenced><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>c</mi><mo>+</mo><mi>d</mi><mtext>i</mtext></mrow></mfenced><mo>,</mo><mo> </mo><mi>N</mi><mfenced><mrow><mi>c</mi><mo>-</mo><mi>d</mi><mtext>i</mtext></mrow></mfenced><mo>></mo><mn>1</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi></math> are positive) <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></math> is not a Gaussian prime, by definition <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mo>=</mo><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mtext>i</mtext></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mtext>i</mtext></mrow></mfenced><mo>=</mo><mi>N</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mtext>i</mtext></mrow></mfenced><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> is not a Gaussian prime <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>For example, </em><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mfenced><mrow><mo>=</mo><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mtext>i</mtext></mrow></mfenced></math> <em><strong>(M1)A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mi>m</mi><mo>+</mo><mi>n</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi><mtext>i</mtext></math></p>
<p>LHS:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mi>β</mi><mo>=</mo><mfenced><mrow><mi>m</mi><mi>p</mi><mo>-</mo><mi>n</mi><mi>q</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mi>m</mi><mi>q</mi><mo>+</mo><mi>n</mi><mi>p</mi></mrow></mfenced><mtext>i</mtext></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mo>=</mo><msup><mfenced><mrow><mi>m</mi><mi>p</mi><mo>-</mo><mi>n</mi><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>m</mi><mi>q</mi><mo>+</mo><mi>n</mi><mi>p</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>m</mi><mi>p</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>m</mi><mi>n</mi><mi>p</mi><mi>q</mi><mo>+</mo><msup><mfenced><mrow><mi>n</mi><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>m</mi><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>m</mi><mi>n</mi><mi>p</mi><mi>q</mi><mo>+</mo><msup><mfenced><mrow><mi>n</mi><mi>p</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>m</mi><mi>p</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>n</mi><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>m</mi><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>n</mi><mi>p</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p>RHS:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mi>N</mi><mfenced><mi>β</mi></mfenced><mo>=</mo><mfenced><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>m</mi><mi>p</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>m</mi><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>n</mi><mi>p</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>n</mi><mi>q</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p>LHS = RHS and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mo>=</mo><mi>N</mi><mfenced><mi>α</mi></mfenced><mi>N</mi><mfenced><mi>β</mi></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mi>m</mi><mo>+</mo><mi>n</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi><mtext>i</mtext></math></p>
<p>LHS</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>2</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><mi>m</mi><mo>+</mo><mi>n</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mi>m</mi><mo>-</mo><mi>n</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mi>p</mi><mo>-</mo><mi>q</mi><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>m</mi><mo>+</mo><mi>n</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mi>m</mi><mo>-</mo><mi>n</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mi>p</mi><mo>-</mo><mi>q</mi><mtext>i</mtext></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mfenced><mrow><mi>m</mi><mi>p</mi><mo>-</mo><mi>n</mi><mi>q</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mi>m</mi><mi>q</mi><mo>+</mo><mi>n</mi><mi>p</mi></mrow></mfenced><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mfenced><mrow><mi>m</mi><mi>p</mi><mo>-</mo><mi>n</mi><mi>q</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>m</mi><mi>q</mi><mo>+</mo><mi>n</mi><mi>p</mi></mrow></mfenced><mtext>i</mtext></mrow></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>m</mi><mi>p</mi><mo>-</mo><mi>n</mi><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>m</mi><mi>q</mi><mo>+</mo><mi>n</mi><mi>p</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfenced><mrow><mfenced><mrow><mi>m</mi><mi>p</mi><mo>-</mo><mi>n</mi><mi>q</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mi>m</mi><mi>q</mi><mo>+</mo><mi>n</mi><mi>p</mi></mrow></mfenced><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>N</mi><mfenced><mi>α</mi></mfenced><mi>N</mi><mfenced><mi>β</mi></mfenced></math> (= RHS) <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></mrow></mfenced><mo>=</mo><mn>17</mn></math> which is a prime (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">ℤ</mi></math>) <em><strong>R1</strong></em></p>
<p>if <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>4</mn><mtext>i</mtext><mo>=</mo><mi>α</mi><mi>β</mi></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn><mo>=</mo><mi>N</mi><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mo>=</mo><mi>N</mi><mfenced><mi>α</mi></mfenced><mi>N</mi><mfenced><mi>β</mi></mfenced></math> <em><strong>R1</strong></em></p>
<p>we cannot have <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>,</mo><mo> </mo><mi>N</mi><mfenced><mi>β</mi></mfenced><mo>></mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong></em> for stating that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></math> is not the product of Gaussian integers of smaller norm because no such norms divide <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn></math></p>
<p> </p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></math> is a Gaussian prime <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Assume <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> is not a Gaussian prime</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>p</mi><mo>=</mo><mi>α</mi><mi>β</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> are Gaussian integers and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>,</mo><mo> </mo><mi>N</mi><mfenced><mi>β</mi></mfenced><mo>></mo><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>N</mi><mfenced><mi>p</mi></mfenced><mo>=</mo><mi>N</mi><mfenced><mi>α</mi></mfenced><mi>N</mi><mfenced><mi>β</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mi>N</mi><mfenced><mi>α</mi></mfenced><mi>N</mi><mfenced><mi>β</mi></mfenced></math> <em><strong>A1</strong></em></p>
<p>It cannot be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>N</mi><mfenced><mi>β</mi></mfenced><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></math> from definition of Gaussian prime <em><strong>R1</strong></em></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>=</mo><mi>p</mi><mo>,</mo><mo> </mo><mi>N</mi><mfenced><mi>β</mi></mfenced><mo>=</mo><mi>p</mi></math> <em><strong>R1</strong></em></p>
<p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mtext>i</mtext></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mi>p</mi></math> which is a contradiction <em><strong>R1</strong></em></p>
<p>hence a prime number, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, that is not of the form <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 a Gaussian prime <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">j.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">j.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will explore some of the properties of special functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">g</mi></math> and their relationship with the trigonometric functions, sine and cosine.</strong></p>
<p><br>Functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> are defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>,</mo><mo> </mo><mi>u</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math>, find expressions, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>u</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>u</mi></math>, for</p>
</div>
<div class="specification">
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi></math> are known as circular functions as the general point (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>,</mo><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></math>) defines points on the unit circle with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>.</p>
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> are known as hyperbolic functions, as the general point ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>θ</mi><mo>)</mo><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math> ) defines points on a curve known as a hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>. This hyperbola has two asymptotes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> satisfies the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>u</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find, and simplify, an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>, stating the coordinates of any axis intercepts and the equation of each asymptote.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math> can be rotated to coincide with the curve defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mi>k</mi><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></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><mrow><mn>2</mn><mi>t</mi></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>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo> </mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p><em><strong><br></strong></em><strong>Note: </strong>Accept combinations of METHODS 1 & 2 that meet at equivalent expressions.</p>
<p><em><strong><br></strong></em><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> into the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> <em><strong>(M1)</strong></em></p>
<p>obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>-i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi><mo>+</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong></em> can be awarded for the use of sine and cosine being odd and even respectively.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi><mo>-</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p>substituting and attempt to simplify <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><em><strong>[2 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"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p>substituting expressions found in part (c) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi></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"><mi>f</mi><mfenced><mrow><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></msup></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>u</mi></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept equivalent final answers that have been simplified removing all imaginary parts eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>−</mo><mn>1</mn></math>etc</p>
<p><em><strong><br></strong></em><em><strong>[2 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><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>4</mn><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for a value of 1 obtained from either LHS or RHS of given expression.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn></math> (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>) <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Award full marks for showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>z</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>z</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>∀</mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.<br><br><em><strong><br></strong></em><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct curves in the upper quadrants, <em><strong>A1</strong></em> for correct curves in the lower quadrants, <em><strong>A1</strong></em> for correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercepts of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> (condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>), <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>−</mo><mi>x</mi></math>.</p>
<p><br><em><strong><br></strong></em><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to rotate by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>°</mo></math> in either direction <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Evidence of an attempt to relate to a sketch of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mi>k</mi></math> would be sufficient for this <em><strong>(M1)</strong></em>.</p>
<p><br>attempting to rotate a particular point, eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> rotates to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>,</mo><mo> </mo><mo>±</mo><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></mfrac></mrow></mfenced></math> (or similar) <em><strong>(A1)</strong></em></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to investigate and prove a geometric property involving the roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>=</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> for integers <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>2</mn></math>.</strong></p>
<p><br>The roots of the equation <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>=</mo><mn>1</mn></math></strong> where <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math></strong> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><msup><mtext>e</mtext><mfrac><mrow><mn>2</mn><mi>πi</mi></mrow><mi>n</mi></mfrac></msup></math>. Each root can be represented by a point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>2</mn></msub><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math>, respectively, on an Argand diagram.</p>
<p>For example, the roots of the equation <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></strong> where <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math></strong> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>. On an Argand diagram, the root <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> can be represented by a point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub></math> and the root <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> can be represented by a point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>1</mn></msub></math>.</p>
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math>.</p>
<p>The roots of the equation <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn></math></strong> where <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math></strong> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>2</mn></msup></math>. On the following Argand diagram, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>2</mn></msub></math> lie on a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Line segments <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>]</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>]</mo></math> are added to the Argand diagram in part (a) and are shown on the following Argand diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math>is the length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>]</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub></math> is the length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>]</mo></math>.</p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>.</p>
<p>The roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>4</mn></msup><mo>=</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>3</mn></msup></math>.</p>
</div>
<div class="specification">
<p>On the following Argand diagram, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>3</mn></msub></math> lie on a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>]</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>]</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>]</mo></math> are line segments.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>For the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math>, the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>5</mn></msup><mo>=</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>3</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>4</mn></msup></math>.</p>
<p>It can be shown that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>4</mn></msub><mo>=</mo><mn>5</mn></math>.</p>
<p>Now consider the general case for integer values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>2</mn></math>.</p>
<p>The roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>=</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>. On an Argand diagram, these roots can be represented by the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>2</mn></msub><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math> respectively where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>]</mo><mo>,</mo><mo> </mo><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>]</mo><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>]</mo></math> are line segments. The roots lie on a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mn>1</mn><mo>-</mo><mi>ω</mi><mo>|</mo></math>.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>-</mo><mn>1</mn><mo>=</mo><mo>(</mo><mi>z</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo> </mo><mo>…</mo><mo> </mo><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo> </mo></math>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</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"><mo>(</mo><mi>ω</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></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, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>=</mo><mn>3</mn></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>By factorizing <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>4</mn></msup><mo>-</mo><mn>1</mn></math>, or otherwise, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>3</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mn>4</mn></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>Suggest a value for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><mo> </mo><mo>…</mo><mo> </mo><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo> </mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo> </mo><mo>…</mo><mo> </mo><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn></math> as a product of linear factors over the set <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">ℂ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, using the part (g)(i) and part (f) results, or otherwise, prove your suggested result to part (e).</p>
<div class="marks">[4]</div>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ω</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>)</mo></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>-</mo><mi>ω</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><msup><mi>ω</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts polynomial division on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>ω</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><mi>ω</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ω</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> In part (a), award marks as appropriate where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> has been converted into Cartesian, modulus-argument (polar) or Euler form.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> is a root of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn></math>)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>≠</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> In part (a), award marks as appropriate where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> has been converted into Cartesian, modulus-argument (polar) or Euler form.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts to find either <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub></math> <em><strong> (M1)</strong></em></p>
<p>accept any valid method</p>
<p><em>e.g.</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mo> </mo><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo> </mo><mi>cos</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mo> </mo><mfrac><mn>1</mn><mrow><mi>sin</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></mrow><mrow><mi>sin</mi><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac></mstyle></mrow></mfrac></math>from either <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>ΔOP</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>ΔOP</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub></math></p>
<p><em>e.g.</em> use of Pythagoras’ theorem</p>
<p><em>e.g.</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><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></mrow></mfenced><mo>,</mo><mo> </mo><mo> </mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><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></mrow></mfenced></mrow></mfenced></math> by calculating the distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> points</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mi mathvariant="normal">=</mi><msqrt><mn>3</mn></msqrt></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mi mathvariant="normal">=</mi><msqrt><mn>3</mn></msqrt></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award a maximum of <em><strong>M1A1A0</strong></em> for any decimal approximation seen in the calculation of either <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub></math> or both.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>=</mo><mn>3</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>=</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi>ω</mi></mrow></mfenced><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>=</mo><mfenced open="|" close="|"><mrow><msup><mi>ω</mi><mn>3</mn></msup><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>-</mo><mi>ω</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mfenced><mrow><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>2</mn></mrow></mfenced></math> and since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>=</mo><mn>3</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>4</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mfenced><mrow><mi>z</mi><mo>−</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>3</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> is a root hence) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>4</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>≠</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Condone the use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> throughout.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>considers the sum of roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>4</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong> (M1)</strong></em></p>
<p>the sum of roots is zero (there is no <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>3</mn></msup></math> term) <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>substitutes for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> <em><strong> (M1)</strong></em></p>
<p><em>e.g.</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHS</mtext><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mrow><mn>3</mn><mtext>π</mtext></mrow><mn>2</mn></mfrac></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mi>πi</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfrac><mtext>π</mtext><mn>2</mn></mfrac></mrow></msup><mo>+</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mtext>i</mtext><mo>-</mo><mn>1</mn><mo>+</mo><mtext>i</mtext><mo>+</mo><mn>1</mn></math> <em><strong> A1</strong></em></p>
<p><br><strong>Note:</strong> This can be demonstrated geometrically or by using vectors. Accept Cartesian or modulus-argument (polar) form.<br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></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><mi>ω</mi><mn>3</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mfrac><mrow><msup><mi>ω</mi><mn>4</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><mi>ω</mi><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><mn>0</mn><mrow><mi>ω</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>≠</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mi mathvariant="normal">2</mi></msub><mo>=</mo><mn>2</mn></math> <em><strong> A1</strong></em></p>
<p>attempts to find either <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub></math> <em><strong> (M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> For example <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mi mathvariant="normal">=</mi><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">i</mi></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mi mathvariant="normal">=</mi><mfenced open="|" close="|"><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">i</mi></mrow></mfenced></math>.<br> Various geometric and trigonometric approaches can be used by candidates.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mi mathvariant="normal">=</mi><msqrt><mn>2</mn></msqrt><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mi mathvariant="normal">=</mi><msqrt><mn>2</mn></msqrt></math> <em><strong> A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award a maximum of <em><strong>A1M1A1A0</strong></em> if labels such as <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math> are not clearly shown.<br> Award full marks if the lengths are shown on a clearly labelled diagram.<br> Award a maximum of <em><strong>A1M1A1A0</strong></em> for any decimal approximation seen in the calculation of either <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub></math> or both.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mn>4</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi>ω</mi></mrow></mfenced><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>3</mn></msup></mrow></mfenced></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mfenced open="|" close="|"><mrow><mo>-</mo><msup><mi>ω</mi><mn>6</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>5</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>4</mn></msup><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>-</mo><mi>ω</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="|" close="|"><mrow><mo>-</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><msup><mi>ω</mi><mn>5</mn></msup><mo>+</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>ω</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>6</mn></msup><mo>=</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>4</mn></msup><mo>=</mo><mn>1</mn></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>ω</mi><mn>5</mn></msup><mo>-</mo><mi>ω</mi><mo>+</mo><mn>4</mn></mrow></mfenced></math> and since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>5</mn></msup><mo>=</mo><mi>ω</mi></math> <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mn>4</mn></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"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mi mathvariant="normal">2</mi></msub><mo>=</mo><mn>2</mn></math> <em><strong> A1</strong></em></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi>ω</mi></mrow></mfenced><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>3</mn></msup></mrow></mfenced></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mfenced open="|" close="|"><mrow><msup><mi>ω</mi><mn>4</mn></msup><mo>-</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>-</mo><mi>ω</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="|" close="|"><mrow><mn>2</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mi>ω</mi></mrow></mfenced><mo>-</mo><mi>ω</mi></mrow></mfenced></math> since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>4</mn></msup><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>3</mn></msup><mo>=</mo><mo>-</mo><mi>ω</mi></math> <em><strong>R1</strong></em> </p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mn>4</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 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"><mfenced><mrow><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><mo> </mo><mo>…</mo><mo> </mo><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mfenced><mo>=</mo><mi>n</mi></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>=</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>3</mn></msup></mrow></mfenced></math> <em><strong> A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> <em><strong> A1</strong></em><em><strong>A1</strong></em></p>
<p> <br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi>ω</mi></mrow></mfenced></math> from symmetry.</p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>-</mo><mn>1</mn><mo>=</mo><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo> </mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo> </mo><mo>…</mo><mo> </mo><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math></p>
<p>considers the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo> </mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo> </mo><mo>…</mo><mo> </mo><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong> (M1)</strong></em></p>
<p>the roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mo>…</mo><mo> </mo><mo>,</mo><mo> </mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong> (A1)</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>ω</mi></mrow></mfenced><mfenced><mrow><mi>z</mi><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mo>…</mo><mfenced><mrow><mi>z</mi><mo>-</mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>1</mn></math>into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>ω</mi></mrow></mfenced><mfenced><mrow><mi>z</mi><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mo>…</mo><mfenced><mrow><mi>z</mi><mo>-</mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>≡</mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo> </mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo> </mo><mo>…</mo><mo> </mo><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</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>ω</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mo>…</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>=</mo><mi>n</mi></math> <em><strong> (A1)</strong></em></p>
<p>takes modulus of both sides <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>ω</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mo>…</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced open="|" close="|"><mi>n</mi></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi>ω</mi></mrow></mfenced><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mo>…</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>=</mo><mi>n</mi></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><mo>…</mo><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>n</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>M1A1FTM1A0</strong></em> from part (e).</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mi>ω</mi></mrow></mfenced><mo>,</mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> are the roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>v</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>v</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo> </mo><mo>…</mo><mo>+</mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>v</mi></mrow></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong> M1</strong></em></p>
<p>coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>v</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> and the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>A1</strong></em></p>
<p>product of the roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mi>n</mi></mrow><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mfrac><mo>=</mo><mi>n</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi>ω</mi></mrow></mfenced><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfenced><mo>…</mo><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>=</mo><mi>n</mi></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><mo>…</mo><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>n</mi></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore some properties of polygonal numbers and to determine and prove interesting results involving these numbers.</strong></p>
<p><br>A polygonal number is an integer which can be represented as a series of dots arranged in the shape of a regular polygon. Triangular numbers, square numbers and pentagonal numbers are examples of polygonal numbers.</p>
<p>For example, a triangular number is a number that can be arranged in the shape of an equilateral triangle. The first five triangular numbers are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math>.</p>
<p>The following table illustrates the first five triangular, square and pentagonal numbers respectively. In each case the first polygonal number is one represented by a single dot.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>For an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>-sided regular polygon, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>3</mn></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th polygonal number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></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>.</p>
<p style="text-align: left;">Hence, for square numbers, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>4</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>4</mn><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mn>4</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup></math>.</p>
</div>
<div class="specification">
<p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th pentagonal number can be represented by the arithmetic series</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>7</mn><mo>+</mo><mo>…</mo><mo>+</mo><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For triangular numbers, verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></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>The number <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>351</mn></math> is a triangular number. Determine which one it is.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≡</mo><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State, in words, what the identity given in part (b)(i) shows for two consecutive triangular numbers.</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>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>, sketch a diagram clearly showing your answer to part (b)(ii).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.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"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> is the square of an odd number 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>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></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 class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using a suitable table of values or otherwise, determine the smallest positive integer, greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, that is both a triangular number and a pentagonal number.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A polygonal number, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced></math>, can be represented by the series</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>3</mn></math>.</p>
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></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>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">f.</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>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>3</mn><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mo>-</mo><mi>n</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0A1</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow><mn>2</mn></mfrac></math> only is seen.</p>
<p>Do not award any marks for numerical verification.</p>
<p> </p>
<p>so for triangular numbers, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>uses a table of values to find a positive integer that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>351</mn></math> <em><strong>(M1)</strong></em></p>
<p>for example, a list showing at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> consecutive terms <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>…</mo><mn>325</mn><mo>,</mo><mo> </mo><mn>351</mn><mo>,</mo><mo> </mo><mn>378</mn><mo>…</mo></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for use of a GDC’s numerical solve or graph feature.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>26</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>26</mn></math>th triangular number) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>−</mo><mn>27</mn><mo>,</mo><mn>26</mn></math>. Award <em><strong>A0</strong></em> if additional solutions besides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>26</mn></math> are given.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mn>351</mn><mo> </mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>-</mo><mn>702</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mrow><mo>-</mo><mn>702</mn></mrow></mfenced></msqrt></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>-</mo><mn>26</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>27</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>26</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>26</mn></math>th triangular number) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>−</mo><mn>27</mn><mo>,</mo><mn>26</mn></math>. Award <em><strong>A0</strong></em> if additional solutions besides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>26</mn></math> are given.</p>
<p> </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>attempts to form an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≡</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>≡</mo><mfrac><mrow><mn>2</mn><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≡</mo><mfenced><mrow><mfrac><msup><mi>n</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>n</mi><mn>2</mn></mfrac></mrow></mfenced><mo>+</mo><mfenced><mrow><mfrac><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><mfenced><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mfenced><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>≡</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></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"><mo>≡</mo><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the sum of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>th triangular numbers</p>
<p>is the <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>th square number <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept equivalent single diagrams, such as the one above, where the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>th and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>th triangular numbers and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>th square number are clearly shown.<br>Award <em><strong>A1</strong> </em>for a diagram that show <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mn>4</mn></mfenced></math> (a triangle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> dots) and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mn>5</mn></mfenced></math> (a triangle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> dots) and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>4</mn></msub><mfenced><mn>5</mn></mfenced></math> (a square with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo></math>dots).</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.iii.</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"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>8</mn><mfenced><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><mn>4</mn><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to expand their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> is odd <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>8</mn><mfenced><mrow><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>1</mn><mfenced><mrow><mo>=</mo><mn>8</mn><mfenced><mrow><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced><mspace linebreak="newline"></mspace></math> <em><strong>A1</strong></em></p>
<p>attempts to expand their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>4</mn><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo> </mo><mfenced><mrow><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></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> is odd <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Method 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>8</mn><mfenced><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><mrow><mi>A</mi><mi>n</mi><mo>+</mo><mi>B</mi></mrow></mfenced><mn>2</mn></msup></mrow></mfenced></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>) <em><strong>A1</strong></em></p>
<p>attempts to expand their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><em><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><msup><mi>A</mi><mn>2</mn></msup><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>A</mi><mi>B</mi><mi>n</mi><mo>+</mo><msup><mi>B</mi><mn>2</mn></msup></mrow></mfenced></math></em></p>
<p>now equates coefficients and obtains <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> is odd <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p>substitutes their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math> and their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>d</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></math> <em><strong>(A1)</strong></em></p>
<p>substitutes their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math> and their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mi>n</mi></msub></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfenced><mrow><mn>3</mn><mfenced><mn>1</mn></mfenced><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>3</mn><mfenced><mn>2</mn></mfenced><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>3</mn><mfenced><mn>3</mn></mfenced><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mo>…</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfenced><mrow><mn>3</mn><mfenced><mn>1</mn></mfenced><mo>+</mo><mn>3</mn><mfenced><mn>2</mn></mfenced><mo>+</mo><mn>3</mn><mfenced><mn>3</mn></mfenced><mo>+</mo><mo>…</mo><mo>+</mo><mn>3</mn><mi>n</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>n</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>3</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo>+</mo><mo>…</mo><mo>+</mo><mi>n</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>n</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> into their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>3</mn><mfenced><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo>-</mo><mn>2</mn><mi>n</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>3</mn><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to find the arithmetic mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms <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><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>multiplies the above expression by the number of terms <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>forms a table of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced></math> values that includes some values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p>forms a table of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>m</mi></mfenced></math> values that includes some values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>></mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> if at least one <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced></math> value is correct. Award <strong><em>(M1)</em></strong> if at least one <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>m</mi></mfenced></math> value is correct. Accept as above for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow></mfenced></math> values and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mi>m</mi></mrow></mfenced></math> values.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> for triangular numbers <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>12</mn></math> for pentagonal numbers <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> is seen in or out of a table. Award <em><strong>(A1)</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>12</mn></math> is seen in or out of a table. Condone the use of the same parameter for triangular numbers and pentagonal numbers, for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> for triangular numbers and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>12</mn></math> for pentagonal numbers.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> (is a triangular number and a pentagonal number) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award all five marks for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> seen anywhere with or without working shown.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p>attempts to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>m</mi></mfenced></math> as a quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> (or equivalent)</p>
<p>attempts to solve their quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><mn>12</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>m</mi><mo>+</mo><mn>1</mn></msqrt></mrow><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mrow><mi>m</mi><mo>-</mo><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup></mrow></mfenced></msqrt></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p style="text-align:left;"><strong>OR</strong></p>
<p>attempts to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>m</mi></mfenced></math> as a quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mi>m</mi><mo>-</mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> (or equivalent)</p>
<p>attempts to solve their quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>±</mo><msqrt><mn>12</mn><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>12</mn><mi>n</mi><mo>+</mo><mn>1</mn></msqrt></mrow><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>12</mn><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow></mfenced></msqrt></mrow><mn>6</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> for triangular numbers <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>12</mn></math> for pentagonal numbers <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> (is a triangular number and a pentagonal number) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>m</mi><mfenced><mrow><mn>3</mn><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>m</mi><mo>+</mo><mi>k</mi><mo> </mo><mfenced><mrow><mi>n</mi><mo>></mo><mi>m</mi></mrow></mfenced></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mi>m</mi><mo>=</mo><mfenced><mrow><mi>m</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mfenced><mrow><mi>m</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>m</mi><mo>-</mo><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempts to find the discriminant of their quadratic</p>
<p>and recognises that this must be a perfect square <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>8</mn><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>N</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>8</mn><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>4</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math></p>
<p>determines that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>8</mn></math> leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mi>m</mi><mo>-</mo><mn>72</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>m</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>,</mo><mn>12</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>12</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> (is a triangular number and a pentagonal number) <em><strong>A1</strong></em></p>
<p> </p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>m</mi><mfenced><mrow><mn>3</mn><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>n</mi><mo>-</mo><mi>k</mi><mo> </mo><mfenced><mrow><mi>m</mi><mo><</mo><mi>n</mi></mrow></mfenced></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>=</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mfenced><mrow><mn>3</mn><mfenced><mrow><mi>n</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>n</mi><mo>+</mo><mfenced><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempts to find the discriminant of their quadratic</p>
<p>and recognises that this must be a perfect square <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>8</mn><mfenced><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>N</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>8</mn><mfenced><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>4</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math></p>
<p>determines that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>8</mn></math> leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>50</mn><mi>n</mi><mo>+</mo><mn>200</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>n</mi><mo>=</mo><mn>5</mn><mo>,</mo><mn>20</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> (is a triangular number and a pentagonal number) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> Award a maximum of <em><strong>R1M0M0A1M1A1A1R0</strong></em> for a ‘correct’ proof using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math>.</p>
<p> </p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>:</mo><mo> </mo><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><msup><mn>1</mn><mn>2</mn></msup></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mfenced><mn>1</mn></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></math></p>
<p>so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math> <em><strong>R1</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><msup><mn>1</mn><mn>2</mn></msup></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mfenced><mn>1</mn></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></math>.<br>Do not accept one-sided considerations such as '<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>1</mn></math> and so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math>'.<br>Subsequent marks after this <em><strong>R1</strong> </em>are independent of this mark can be awarded.</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>, <em>ie.</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>k</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi></mrow><mn>2</mn></mfrac></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> ”. The assumption of truth must be clear.<br>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"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>:</mo></math></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> can be represented by the sum</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></munderover><mfenced><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mfenced><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>k</mi><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> and so</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>k</mi><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><msub><mi>P</mi><mi>r</mi></msub><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>k</mi></mfenced><mo>+</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>k</mi><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi><mo>+</mo><mn>2</mn><mo>+</mo><mn>2</mn><mi>k</mi><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>hence true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math> 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 <em><strong>R1</strong></em></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></p>
<p> </p>
<p><strong>Note:</strong> Only award the final <em><strong>R1</strong> </em>if the first five marks have been awarded. Award marks as appropriate for solutions that expand both the LHS and (given) RHS of the equation.</p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) (i) was generally well done. Unfortunately, some candidates adopted numerical verification. Part (a) (ii) was generally well done with the majority of successful candidates using their GDC judiciously and disregarding <em>n </em>= −27 as a possible solution. A few candidates interpreted the question as needing to deal with P<sub>3</sub>(351).</p>
<p>Although part (b) (i) was generally well done, a significant number of candidates laboured unnecessarily to show the required result. Many candidates set their LHS to equal the RHS throughout the solution. Part (b) (ii) was generally not well done with many candidates unable to articulate clearly in words and symbols what the given identity shows for the sum of two consecutive triangular numbers. In part (b) (iii), most candidates were unable to produce a clear diagram illustrating the identity stated in part (b) (i). </p>
<p>Part (c) was reasonably well done. Most candidates were able to show algebraically that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo><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>. A good number of candidates were then able to express <math xmlns="http://www.w3.org/1998/Math/MathML"><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> as <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></math> and conclude that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math> is odd. Rather than making the connection that <math xmlns="http://www.w3.org/1998/Math/MathML"><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> is a perfect square, many candidates attempted instead to analyse the parity of either <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>n</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><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>. As with part (b) (i), many candidates set their LHS to equal the RHS throughout the solution. A number of candidates unfortunately adopted numerical verification.</p>
<p>Part (d) was not answered as well as anticipated with many candidates not understanding what was<br>required. Instead of using the given arithmetic series to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>(</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></mfrac></math>, a large number of<br>candidates used <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mo>(</mo><mn>5</mn><mo>-</mo><mn>2</mn><mo>)</mo><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mo>(</mo><mn>5</mn><mo>-</mo><mn>4</mn><mo>)</mo><mi>n</mi></mrow><mn>2</mn></mfrac></math> . Unfortunately, a number of candidates adopted numerical verification.</p>
<p>In part (e), the overwhelming majority of candidates who successfully determined that 210 is the smallest positive integer greater than 1 that is both triangular and pentagonal used a table of values. Unfortunately, a large proportion of these candidates seemingly spent quite a few minutes listing the first 20 triangular numbers and the first 12 pentagonal numbers. And it can be surmised that a number of these candidates constructed their table of values either without the use of a GDC or with the arithmetic functionality of a GDC rather than with a GDC's table of values facility. Candidates should be aware that a relevant excerpt from a table of values is sufficient evidence of correct working. A number of candidates started constructing a table of values but stopped before identifying 210. Disappointingly, a significant number of candidates attempted to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mi>P</mi><mn>5</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<p>Part (f) proved beyond the reach of most with only a small number of candidates successfully proving the given result. A significant number of candidates were unable to show that the result is true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math>. A number of candidates established the validity of the base case for the RHS only while a number of other candidates attempted to prove the base case for <em>r</em> = 3. A large number of candidates did not state the inductive step correctly with the assumption of truth not clear. A number of candidates then either attempted to work backwards from the given result or misinterpreted the question and attempted to prove the result stated in the question stem rather than the result stated in the question. Some candidates who were awarded the first answer mark when considering the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> case were unable to complete the square or equivalent simplification correctly. Disappointingly, a significant number listed the steps involved in an induction proof without engaging in the actual proof.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore cubic polynomials of the form</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> <strong>and corresponding cubic equations with one real root and two complex roots of the form </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>z</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><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><mo>)</mo><mo>=</mo><mn>0</mn></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p> </p>
</div>
<div class="specification">
<p>In parts (a), (b) and (c), let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn></math>.</p>
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>z</mi><mo>+</mo><mn>17</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><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><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i</mtext></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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="specification">
<p>On the Cartesian plane, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mo>-</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> represent the real and imaginary parts of the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><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><mo>=</mo><mn>0</mn></math>.</p>
<p><br>The following diagram shows a particular curve of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced></math> and the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mn>80</mn></mrow></mfenced></math>. The curve and the tangent both intersect the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math> are also shown.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≠</mo><mi>r</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mo>(</mo><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> are as defined in part (d)(ii). The curve has a point of inflexion at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="specification">
<p>Consider the special case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mtext>i</mtext></math> are roots of the equation, write down the third root.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that the mean of the two complex roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></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>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and the tangent to the curve at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, clearly showing where the tangent crosses the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, prove that the tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow></mfenced></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce from part (d)(i) that the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><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><mo>=</mo><mn>0</mn></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this diagram to determine the roots of the corresponding equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><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><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced></math>.</p>
<p>You are <strong>not</strong> required to demonstrate a change in concavity.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence describe numerically the horizontal position of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> relative to the horizontal positions of the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</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></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>, state in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, the coordinates of points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mtext>i</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>mean<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>4</mn><mo>+</mo><mtext>i</mtext><mo>+</mo><mn>4</mn><mo>-</mo><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts product rule differentiation <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for attempting to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn><mi>x</mi><mo>-</mo><mn>17</mn></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mn>4</mn></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> is correct, award <em><strong>A1</strong></em> for solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn></math> and obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>1</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>=</mo><mn>4</mn><mo>+</mo><mi>c</mi><mo>⇒</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>states the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and verifies that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> lies on the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is the tangent to the curve at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>(M0)A0A1A1</strong></em> to a candidate who does not attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to solve a correct cubic equation <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>4</mn></math></p>
<p><strong><br>OR</strong></p>
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mn>1</mn></math> and forms <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn><mo>=</mo><mn>1</mn><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to solve a correct quadratic equation <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>4</mn></math></p>
<p><strong><br>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> is a double root <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is the tangent to the curve at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Candidates using this method are not required to verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math>.</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 style="padding-left:60px;"><img src="data:image/png;base64,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"></p>
<p>a positive cubic with an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math>, and a local maximum and local minimum in the first quadrant both positioned to the left of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> As the local minimum and point A are very close to each other, condone graphs that seem to show these points coinciding.<br>For the point of tangency, accept labels such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>,</mo><mo> </mo><mfenced><mrow><mn>4</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math> or the point labelled from both axes. Coordinates are not required.</p>
<p> </p>
<p>a correct sketch of the tangent passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and crossing the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the same point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math> as the curve <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if both graphs cross the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at distinctly different points.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(M1)A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi></mrow></mfenced><mi>x</mi><mo>-</mo><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mi>r</mi></math></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</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>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>a</mi><mi>x</mi><mo>-</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>a</mi><mi>r</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>attempts to substitute their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR </strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mo>-</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mi>a</mi></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>A1</strong></em><br><strong><br>THEN</strong></p>
<p>so the tangent intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</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>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>attempts to substitute their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mi>x</mi><mo>+</mo><mi>c</mi></math> and attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>-</mo><mi>r</mi><mo>=</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>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>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>the line through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> parallel to the tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has equation<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>≠</mo><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>since there is a double root <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow></mfenced></math>, this parallel line through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> is the required tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>⇒</mo><mi>b</mi><mo>=</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em><br><br><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>±</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i=</mtext></mrow></mfenced><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>R1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>hence the complex roots can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>4</mn></math> (seen anywhere) <em><strong>A1</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p>attempts to find the gradient of the tangent in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and equates to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math> <em><strong>(M1)</strong></em><br><br><br><strong>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>80</mn></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo>=</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>80</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>16</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></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"><mfrac><mn>80</mn><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>16</mn><mo>⇒</mo><mi>a</mi><mo>=</mo><mn>3</mn></math></p>
<p>roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> (seen anywhere) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>±</mo><mn>4</mn><mtext>i</mtext></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>±</mo><mn>4</mn><mtext>i</mtext></math>. Do not accept coordinates.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Accept “<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><mo>−</mo><mn>4</mn></math>”.<br>Do not award <em><strong>A1FT</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mo>−</mo><mn>4</mn><mo>)</mo></math>. </p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</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>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>a</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>r</mi><mo>-</mo><mn>4</mn><mi>a</mi></mrow></mfenced></math></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> and correctly solves for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p>for example, obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>r</mi><mo>+</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if the answer does not lead to the <em><strong>A</strong><strong>G</strong></em>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math> of the horizontal distance (way) from point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> to point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept equivalent numerical statements or a clearly labelled diagram displaying the numerical relationship.<br>Award <em><strong>A0</strong></em> for non-numerical statements such as “<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is between <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, closer to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>”.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><img 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"></p>
<p>a positive cubic with no stationary points and a non-stationary point of inflexion at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Graphs may appear approximately linear. Award this <em><strong>A1</strong> </em>if a change of concavity either side of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> is apparent.<br>Coordinates are not required and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept need not be indicated.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) (i) was generally well done with a significant majority of candidates using the conjugate root theorem to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mtext>i</mtext></math> as the third root. A number of candidates, however, wasted considerable time attempting an algebraic method to determine the third root. Part (a) (ii) was reasonably well done. A few candidates however attempted to calculate the product of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mtext>i</mtext></math>.</p>
<p>Part (b) was reasonably well done by a significant number of candidates. Most were able to find a correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> and a good number of those candidates were able to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mn>4</mn><mo>)</mo><mo>=</mo><mn>1</mn></math>. Candidates that did not determine the equation of the tangent had to state that the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is also 1 and verify that the point (4,3) lies on the line. A few candidates only met one of those requirements. Weaker candidates tended to only verify that the point (4,3) lies on the curve and the tangent line without attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<p>Part (c) was not answered as well as anticipated. A number of sketches were inaccurate and carelessly drawn with many showing both graphs crossing the <em>x-</em>axis at distinctly different points.</p>
<p>Part (d) (i) was reasonably well done by a good number of candidates. Most successful responses involved use of the product rule. A few candidates obtained full marks by firstly expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, then differentiating to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math>and finally simplifying to obtain the desired result. A number of candidates made elementary mistakes when differentiating. In general, the better candidates offered reasonable attempts at showing the general result in part (d) (ii). A good number gained partial credit by determining that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>(</mo><mi>a</mi><mo>-</mo><mi>r</mi><mo>)</mo></math>. Only the very best candidates obtained full marks by concluding that as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<p>In general, only the best candidates were able to use the result <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> to deduce that the complex roots of the equation can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo></msqrt></math>. Although given the complex roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i</mtext></math>, a significant number of candidates attempted, with mixed success, to use the quadratic formula to solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><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><mo>=</mo><mn>0</mn></math>.</p>
<p>In part (f) (i), only a small number of candidates were able to determine all the roots of the equation. Disappointingly, a large number did not state <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> as a root. Some candidates determined that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>4</mn></math> but were unable to use the diagram to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>3</mn></math>. Of the candidates who determined all the roots in part (f) (i), very few gave the correct coordinates for C<sub>2</sub> . The most frequent error was to give the <em>y-</em>coordinate as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mn>4</mn><mtext>i</mtext></math>.</p>
<p>Of the candidates who attempted part (g) (i), most were able to find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> and a reasonable number of these were then able to convincingly show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi><mo>)</mo></math>. It was very rare to see a correct response to part (g) (ii). A few candidates stated that P is between R and A with some stating that P was closer to A. A small number restated <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi><mo>)</mo></math> in words.</p>
<p>Of the candidates who attempted part (h) (i), most were able to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced></math>. However, most graphs were poorly drawn with many showing a change in concavity at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> rather than at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>. In part (h) (ii), only a very small number of candidates determined that A and P coincide at (<em>r</em>,0).</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>This question investigates some applications of differential equations to modeling population growth.</p>
<p>One model for population growth is to assume that the rate of change of the population is proportional to the population, i.e. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = kP">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>P</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
<mi>P</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is the time (in years) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> is the population</p>
</div>
<div class="specification">
<p>The initial population is 1000.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.003">
<mi>k</mi>
<mo>=</mo>
<mn>0.003</mn>
</math></span>, use your answer from part (a) to find</p>
</div>
<div class="specification">
<p>Consider now the situation when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is not a constant, but a function of time.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.003 + 0.002t">
<mi>k</mi>
<mo>=</mo>
<mn>0.003</mn>
<mo>+</mo>
<mn>0.002</mn>
<mi>t</mi>
</math></span>, find</p>
</div>
<div class="specification">
<p>Another model for population growth assumes</p>
<ul>
<li>there is a maximum value for the population, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>.</li>
<li>that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is not a constant, but is proportional to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - \frac{P}{L}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mfrac>
<mi>P</mi>
<mi>L</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</li>
</ul>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the general solution of this differential equation is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = A{{\text{e}}^{kt}}"> <mi>P</mi> <mo>=</mo> <mi>A</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mi>k</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span>, 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>
<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 population after 10 years</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the number of years it will take for the population to triple.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{t \to \infty } P"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mi>P</mi> </math></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the solution of the differential equation, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = f\left( t \right)"> <mi>P</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the number of years it will take for the population to triple.</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{R}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </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>Solve the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span>, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = g\left( t \right)"> <mi>P</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the initial population is 1000, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L = 10000"> <mi>L</mi> <mo>=</mo> <mn>10000</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 0.003"> <mi>m</mi> <mo>=</mo> <mn>0.003</mn> </math></span>, find the number of years it will take for the population to triple.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{P}} {\text{d}}P = \int {k{\text{d}}t} "> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mi>k</mi> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,P = kt + c"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>=</mo> <mi>k</mi> <mi>t</mi> <mo>+</mo> <mi>c</mi> </math></span> <em><strong> A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = {e^{kt + c}}"> <mi>P</mi> <mo>=</mo> <mrow> <msup> <mi>e</mi> <mrow> <mi>k</mi> <mi>t</mi> <mo>+</mo> <mi>c</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="P = A{e^{kt}}"> <mi>P</mi> <mo>=</mo> <mi>A</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>k</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = {e^c}"> <mi>A</mi> <mo>=</mo> <mrow> <msup> <mi>e</mi> <mi>c</mi> </msup> </mrow> </math></span> <em><strong> AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0{\text{,}}\,\,P = 1000"> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>=</mo> <mn>1000</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow A = 1000"> <mo stretchy="false">⇒</mo> <mi>A</mi> <mo>=</mo> <mn>1000</mn> </math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( {10} \right) = 1000{e^{0.003\left( {10} \right)}} = 1030"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mn>10</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1000</mn> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mrow> <mo>(</mo> <mrow> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mo>=</mo> <mn>1030</mn> </math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 = 1000{e^{0.003t}}"> <mn>3000</mn> <mo>=</mo> <mn>1000</mn> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mi>t</mi> </mrow> </msup> </mrow> </math></span> <em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = \frac{{{\text{ln}}\,3}}{{0.003}} = 366"> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> <mrow> <mn>0.003</mn> </mrow> </mfrac> <mo>=</mo> <mn>366</mn> </math></span> years <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{t \to \infty } P = \infty "> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mi>P</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{P}} {\text{d}}P = \int {\left( {0.003 + 0.002t} \right){\text{d}}t} "> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.003</mn> <mo>+</mo> <mn>0.002</mn> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,P = 0.003t + 0.001{t^2} + c"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>=</mo> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>c</mi> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = {e^{0.003t + 0.001{t^2} + c}}"> <mi>P</mi> <mo>=</mo> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>c</mi> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0{\text{,}}\,\,P = 1000"> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>=</mo> <mn>1000</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {e^c} = 1000"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>e</mi> <mi>c</mi> </msup> </mrow> <mo>=</mo> <mn>1000</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = 1000{e^{0.003t + 0.001{t^2}}}"> <mi>P</mi> <mo>=</mo> <mn>1000</mn> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mrow> </msup> </mrow> </math></span></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 = 1000{e^{0.003t + 0.001{t^2}}}"> <mn>3000</mn> <mo>=</mo> <mn>1000</mn> <mrow> <msup> <mi>e</mi> <mrow> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mrow> </msup> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,3 = 0.003t + 0.001{t^2}"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>=</mo> <mn>0.003</mn> <mi>t</mi> <mo>+</mo> <mn>0.001</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>Use of quadratic formula or GDC graph or GDC polysmlt <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 31.7"> <mi>t</mi> <mo>=</mo> <mn>31.7</mn> </math></span> years <em><strong>A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = m\left( {1 - \frac{P}{L}} \right)"> <mi>k</mi> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mi>P</mi> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> , where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m"> <mi>m</mi> </math></span> is the constant of proportionality <em><strong>A1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = m\left( {1 - \frac{P}{L}} \right)P"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mi>P</mi> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>P</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{P\left( {L - P} \right)}}} {\text{d}}P = \int {\frac{m}{L}{\text{d}}t} "> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{P\left( {L - P} \right)}} = \frac{A}{P} + \frac{B}{{L - P}}"> <mfrac> <mn>1</mn> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>A</mi> <mi>P</mi> </mfrac> <mo>+</mo> <mfrac> <mi>B</mi> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \equiv A\left( {L - P} \right) + BP"> <mn>1</mn> <mo>≡</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mi>P</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \frac{1}{L}{\text{,}}\,\,B = \frac{1}{L}"> <mi>A</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>L</mi> </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}{L}\int {\left( {\frac{1}{P} + \frac{1}{{L - P}}} \right){\text{d}}P} = \int {\frac{m}{L}{\text{d}}t} "> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> <mo>∫</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{L}\left( {{\text{ln}}\,P - {\text{ln}}\left( {L - P} \right)} \right) = \frac{m}{L}t + c"> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>P</mi> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>t</mi> <mo>+</mo> <mi>c</mi> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {\frac{P}{{L - P}}} \right) = mt + d"> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>P</mi> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>m</mi> <mi>t</mi> <mo>+</mo> <mi>d</mi> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = cL"> <mi>d</mi> <mo>=</mo> <mi>c</mi> <mi>L</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{P}{{L - P}} = C{e^{mt}}"> <mfrac> <mi>P</mi> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> </mfrac> <mo>=</mo> <mi>C</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = {e^d}"> <mi>C</mi> <mo>=</mo> <mrow> <msup> <mi>e</mi> <mi>d</mi> </msup> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( {1 + C{e^{mt}}} \right) = CL{e^{mt}}"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>C</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>C</mi> <mi>L</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = \frac{{CL{e^{mt}}}}{{\left( {1 + C{e^{mt}}} \right)}}{\text{ }}\left( { = \frac{L}{{\left( {D{e^{ - mt}} + 1} \right)}}{\text{,}}\,\,{\text{where}}\;D = \frac{1}{C}} \right)"> <mi>P</mi> <mo>=</mo> <mfrac> <mrow> <mi>C</mi> <mi>L</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>C</mi> <mrow> <msup> <mi>e</mi> <mrow> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mi>L</mi> <mrow> <mrow> <mo>(</mo> <mrow> <mi>D</mi> <mrow> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>m</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>where</mtext> </mrow> <mspace width="thickmathspace"></mspace> <mi>D</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>C</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[10 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1000 = \frac{{10000}}{{D + 1}}"> <mn>1000</mn> <mo>=</mo> <mfrac> <mrow> <mn>10000</mn> </mrow> <mrow> <mi>D</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = 9"> <mi>D</mi> <mo>=</mo> <mn>9</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 = \frac{{10000}}{{9{e^{ - 0.003t}} + 1}}"> <mn>3000</mn> <mo>=</mo> <mfrac> <mrow> <mn>10000</mn> </mrow> <mrow> <mn>9</mn> <mrow> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mn>0.003</mn> <mi>t</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 450"> <mi>t</mi> <mo>=</mo> <mn>450</mn> </math></span> years <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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.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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>In parts (b) and (c), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {abc \ldots } \right)_n}">
<mrow>
<msub>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mi>b</mi>
<mi>c</mi>
<mo>…<!-- … --></mo>
</mrow>
<mo>)</mo>
</mrow>
<mi>n</mi>
</msub>
</mrow>
</math></span> denotes the number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{abc \ldots }">
<mrow>
<mi>a</mi>
<mi>b</mi>
<mi>c</mi>
<mo>…<!-- … --></mo>
</mrow>
</math></span> written in base <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</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>. For example, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {359} \right)_n} = 3{n^2} + 5n + 9">
<mrow>
<msub>
<mrow>
<mo>(</mo>
<mrow>
<mn>359</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mi>n</mi>
<mo>+</mo>
<mn>9</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State Fermat’s little theorem.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the remainder when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{15^{1207}}"> <mrow> <msup> <mn>15</mn> <mrow> <mn>1207</mn> </mrow> </msup> </mrow> </math></span> is divided by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="13"> <mn>13</mn> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Convert <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {7A2} \right)_{16}}"> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>7</mn> <mi>A</mi> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mn>16</mn> </mrow> </msub> </mrow> </math></span> to base <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5"> <mn>5</mn> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( A \right)_{16}} = {\left( {10} \right)_{10}}"> <mrow> <msub> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mrow> <mn>16</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>10</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mn>10</mn> </mrow> </msub> </mrow> </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>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1251} \right)_n} + {\left( {30} \right)_n} = {\left( {504} \right)_n} + {\left( {504} \right)_n}"> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>1251</mn> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>30</mn> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msub> </mrow> <mo>=</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>504</mn> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>504</mn> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msub> </mrow> </math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<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="{a^p} \equiv a\left( {{\text{mod}}\,p} \right)"> <mrow> <msup> <mi>a</mi> <mi>p</mi> </msup> </mrow> <mo>≡</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>mod</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> is prime <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^{p - 1}} \equiv 1\left( {{\text{mod}}\,p} \right)"> <mrow> <msup> <mi>a</mi> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>≡</mo> <mn>1</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>mod</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> is prime and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> does not divide <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> (or equivalent statement) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{15^{1207}} \equiv {2^{1207}}\left( {{\text{mod}}\,13} \right)"> <mrow> <msup> <mn>15</mn> <mrow> <mn>1207</mn> </mrow> </msup> </mrow> <mo>≡</mo> <mrow> <msup> <mn>2</mn> <mrow> <mn>1207</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>mod</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>13</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^{12}} \equiv 1\left( {{\text{mod}}\,13} \right)"> <mrow> <msup> <mn>2</mn> <mrow> <mn>12</mn> </mrow> </msup> </mrow> <mo>≡</mo> <mn>1</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>mod</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>13</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^{1207}} = {\left( {{2^{12}}} \right)^{100}}{2^7}"> <mrow> <msup> <mn>2</mn> <mrow> <mn>1207</mn> </mrow> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mn>12</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mn>100</mn> </mrow> </msup> </mrow> <mrow> <msup> <mn>2</mn> <mn>7</mn> </msup> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^{1207}}\left( { \equiv {2^7}} \right) \equiv 11\left( {{\text{mod}}\,13} \right)"> <mrow> <msup> <mn>2</mn> <mrow> <mn>1207</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo>≡</mo> <mrow> <msup> <mn>2</mn> <mn>7</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≡</mo> <mn>11</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>mod</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>13</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1</strong></em></p>
<p>the remainder is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="11"> <mn>11</mn> </math></span></p>
<p><strong>Note:</strong> Award as above for using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15"> <mn>15</mn> </math></span> instead of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2"> <mn>2</mn> </math></span>.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {7A2} \right)_{16}} = 7 \times {16^2} + 10 \times 16 + 2"> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>7</mn> <mi>A</mi> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mn>16</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>7</mn> <mo>×</mo> <mrow> <msup> <mn>16</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>10</mn> <mo>×</mo> <mn>16</mn> <mo>+</mo> <mn>2</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1954"> <mo>=</mo> <mn>1954</mn> </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="5\left| \!{\underline {\, {1954} \,}} \right. "> <mn>5</mn> <mrow> <mo>|</mo> <mspace width="negativethinmathspace"></mspace> <mrow> <munder> <mrow> <mspace width="thinmathspace"></mspace> <mrow> <mn>1954</mn> </mrow> <mspace width="thinmathspace"></mspace> </mrow> <mo>_</mo> </munder> </mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> </mrow> </math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="390\,r\,4"> <mn>390</mn> <mspace width="thinmathspace"></mspace> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mn>4</mn> </math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="78\,r\,0"> <mn>78</mn> <mspace width="thinmathspace"></mspace> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mn>0</mn> </math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15\,r\,3"> <mn>15</mn> <mspace width="thinmathspace"></mspace> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mn>3</mn> </math></span> <em><strong>M1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,r\,0"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mn>0</mn> </math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0\,r\,3"> <mn>0</mn> <mspace width="thinmathspace"></mspace> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mn>3</mn> </math></span></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1954 = 3 \times {5^4} + 0 \times {5^3} + 3 \times {5^2} + 0 \times {5^1} + 4"> <mn>1954</mn> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mrow> <msup> <mn>5</mn> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mn>0</mn> <mo>×</mo> <mrow> <msup> <mn>5</mn> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mo>×</mo> <mrow> <msup> <mn>5</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>0</mn> <mo>×</mo> <mrow> <msup> <mn>5</mn> <mn>1</mn> </msup> </mrow> <mo>+</mo> <mn>4</mn> </math></span> <em><strong>M1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {7A2} \right)_{16}} = {\left( {30304} \right)_5}"> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>7</mn> <mi>A</mi> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mn>16</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>30304</mn> </mrow> <mo>)</mo> </mrow> <mn>5</mn> </msub> </mrow> </math></span> <em><strong>A1</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>the equation can be written as</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{n^3} + 2{n^2} + 5n + 1 + 3n = 2\left( {5{n^2} + 4} \right)"> <mrow> <msup> <mi>n</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>4</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=" \Rightarrow {n^3} - 8{n^2} + 8n - 7 = 0"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>n</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>8</mn> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mi>n</mi> <mo>−</mo> <mn>7</mn> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(</strong><strong>M1)</strong></em></p>
<p><strong>Note:</strong> The <em><strong>(M1)</strong></em> is for an attempt to solve the original equation.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 7"> <mi>n</mi> <mo>=</mo> <mn>7</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the remainder when <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>14</mn><mn>2022</mn></msup></math> is divided by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Fermat’s little theorem to find the remainder when <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>14</mn><mn>2022</mn></msup></math> is divided by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</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>Prove that a number in base <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math> is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> if, and only if, the sum of its digits is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>.</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>The base <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math> number <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mi>y</mi><mn>93</mn><mi>y</mi><mn>25</mn></math> is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>. Find the possible values of the digit <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>.</p>
<div class="marks">[4]</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>the remainder is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>14</mn><mn>16</mn></msup><mo>≡</mo><mn>1</mn><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>17</mn></mrow></mfenced></math> (from Fermat’s little theorem) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>14</mn><mn>2022</mn></msup><mo>=</mo><msup><mn>14</mn><mrow><mn>16</mn><mo>×</mo><mn>126</mn><mo>+</mo><mn>6</mn></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for a <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext></math> exponent consistent with the correct use of Fermat’s little theorem.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>14</mn><mn>2022</mn></msup><mo>≡</mo><msup><mn>14</mn><mn>6</mn></msup><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>17</mn></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>≡</mo><mn>15</mn><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>17</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>the remainder is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><msub><mi>a</mi><mi>n</mi></msub><msup><mn>13</mn><mi>n</mi></msup><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><msup><mn>13</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mn>13</mn><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> The above <em><strong>M1</strong></em> is independent of the <em><strong>A</strong></em> marks below.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo>≡</mo><mn>1</mn><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>6</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>13</mn><mi>x</mi></msup><mo>≡</mo><mn>1</mn><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>6</mn></mrow></mfenced></math> (for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math>) <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>≡</mo><msub><mi>a</mi><mi>n</mi></msub><msup><mn>1</mn><mi>n</mi></msup><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><msup><mn>1</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mn>1</mn><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>6</mn></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mi>N</mi><mo>≡</mo><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>6</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>≡</mo><mn>0</mn><mo> </mo><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>6</mn></mrow></mfenced></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub><mo>≡</mo><mn>0</mn><mo> </mo><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>6</mn></mrow></mfenced></math> <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><menclose notation="left"><mi>N</mi></menclose></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><menclose notation="left"><mfenced><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mfenced></menclose></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><msub><mi>a</mi><mi>n</mi></msub><msup><mn>13</mn><mi>n</mi></msup><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><msup><mn>13</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mn>13</mn><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></math> <em><strong>(</strong><strong>M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfenced><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>13</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msub><mi>a</mi><mi>n</mi></msub><mfenced><mrow><msup><mn>13</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mn>13</mn><mn>0</mn></msup></mrow></mfenced><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mfenced><mrow><msup><mn>13</mn><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mn>13</mn><mn>0</mn></msup></mrow></mfenced><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msup><mn>13</mn><mn>0</mn></msup></mrow></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempting to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfenced><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>13</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mi>M</mi></math>.</p>
<p><br>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><menclose notation="left"><mfenced><mrow><mn>13</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mi>M</mi></menclose></math> <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><menclose notation="left"><mi>N</mi></menclose></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><menclose notation="left"><mfenced><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mfenced></menclose></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>the sum of the digits is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mo>+</mo><mn>20</mn></math> <em><strong>(A</strong><strong>1)</strong></em></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mo>+</mo><mn>20</mn><mo>=</mo><mn>6</mn><mi>k</mi></math> (or equivalent) to attempt to find a valid value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>(M</strong><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>,</mo><mo> </mo><mn>8</mn><mo>,</mo><mo> </mo><mn>11</mn><mfenced><mi>B</mi></mfenced></math> <em><strong>A</strong><strong>1</strong></em><em><strong>A</strong><strong>1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>,</mo><mo> </mo><mn>8</mn></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>11</mn><mo>(</mo><mi>B</mi><mo>)</mo></math>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><mrow><mn>1</mn><mi>y</mi><mn>93</mn><mi>y</mi><mn>25</mn></mrow></mfenced><mn>13</mn></msub><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>13</mn><mn>6</mn></msup><mo>+</mo><mi>y</mi><mo>×</mo><msup><mn>13</mn><mn>5</mn></msup><mo>+</mo><mn>9</mn><mo>×</mo><msup><mn>13</mn><mn>4</mn></msup><mo>+</mo><mn>3</mn><mo>×</mo><msup><mn>13</mn><mn>3</mn></msup><mo>+</mo><mi>y</mi><mo>×</mo><msup><mn>13</mn><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>×</mo><msup><mn>13</mn><mn>1</mn></msup><mo>+</mo><mn>5</mn><mo>×</mo><msup><mn>13</mn><mn>0</mn></msup></math> <em><strong>(A</strong><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>371462</mn><mi>y</mi><mo>+</mo><mn>5090480</mn></math></p>
<p>attempts to find a valid value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> such that</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>371462</mn><mi>y</mi><mo>+</mo><mn>5090480</mn><mo>≡</mo><mn>0</mn><mfenced><mrow><mtext>mod</mtext><mo> </mo><mn>6</mn></mrow></mfenced></math> <em><strong>(</strong><strong>M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>,</mo><mo> </mo><mn>8</mn><mo>,</mo><mo> </mo><mn>11</mn><mfenced><mi>B</mi></mfenced></math> <em><strong>A</strong><strong>1</strong></em><em><strong>A</strong><strong>1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>,</mo><mo> </mo><mn>8</mn></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>11</mn><mo>(</mo><mi>B</mi><mo>)</mo></math>.</p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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>
<br><hr><br>