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<h2>HL Paper 2</h2><div class="specification">
<p>Consider the identity <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>+</mo><mn>7</mn><mi>x</mi></mrow><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, expand <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>+</mo><mn>7</mn><mi>x</mi></mrow><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></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>Give a reason why the series expansion found in part (b) is not valid for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mtext>d</mtext><mi>n</mi></msup><mrow><mtext>d</mtext><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>=</mo><msup><mi>p</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mi>n</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo> </mo><mo>∈</mo><mo> </mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>p</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℚ</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>At a gathering of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> teachers, seven are male and five are female. A group of five of these teachers go out for a meal together. Determine the possible number of groups in each of the following situations:</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>There are more males than females in the group.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Two of the teachers, Gary and Gerwyn, refuse to go out for a meal together.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>2</mn><mfenced><mrow><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>5</mn></mfrac><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>5</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mn>8</mn><mfenced><mrow><mi>cos</mi><mfrac><mrow><mn>2</mn><mi>k</mi><mi mathvariant="normal">π</mi></mrow><mn>5</mn></mfrac><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mfrac><mrow><mn>2</mn><mi>k</mi><mi mathvariant="normal">π</mi></mrow><mn>5</mn></mfrac></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="specification">
<p>Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi><mo> </mo><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the modulus of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the argument of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> found in part (i), find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} > 2x + 1">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use mathematical induction to prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^{n + 1}} > {n^2}">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z}">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 3">
<mi>n</mi>
<mo>⩾</mo>
<mn>3</mn>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the set of six-digit positive integers that can be formed from the digits <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>7</mn><mo>,</mo><mo> </mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math>.</p>
<p>Find the total number of six-digit positive integers that can be formed such that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the digits are distinct.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the digits are distinct and are in increasing order.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>Find the coordinates where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> crosses the</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><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">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the vertical asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The oblique asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>30</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>30</mn></math>, clearly indicating the points of intersection with each axis and any asymptotes.</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>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac></math> in partial fractions.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>3</mn></munderover><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac><mo>d</mo><mi>x</mi></math>, expressing your answer as a single logarithm.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _{10}}\left( {\frac{1}{{2\sqrt 2 }}\left( {p + 2q} \right)} \right) = \frac{1}{2}\left( {{{\log }_{10}}p + {{\log }_{10}}q} \right),{\text{ }}p > 0,{\text{ }}q > 0">
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>p</mi>
<mo>+</mo>
<mn>2</mn>
<mi>q</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mrow>
<mi>log</mi>
</mrow>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mi>p</mi>
<mo>+</mo>
<mrow>
<msub>
<mrow>
<mi>log</mi>
</mrow>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mi>q</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>p</mi>
<mo>></mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>q</mi>
<mo>></mo>
<mn>0</mn>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = a + b{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{b}} \in {\mathbb{R}^ + }">
<mrow>
<mtext>b</mtext>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span> and let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arg}}\,z = \theta ">
<mrow>
<mtext>arg</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>=</mo>
<mi>θ<!-- θ --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show the points represented by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z - 2a">
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</math></span> on the following Argand diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression in terms of <em>θ</em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arg}}\left( {z - 2a} \right)">
<mrow>
<mtext>arg</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression in terms of <em>θ</em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arg}}\left( {\frac{z}{{z - 2a}}} \right)">
<mrow>
<mtext>arg</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise find the value of <em>θ</em> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Re}}\left( {\frac{z}{{z - 2a}}} \right) = 0">
<mrow>
<mtext>Re</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^2} = 4{{\text{e}}^{\frac{\pi }{2}{\text{i}}}}">
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>, giving your answers in the form</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{{\text{e}}^{{\text{i}}\theta }}">
<mi>r</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</msup>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta \in \mathbb{R}">
<mi>θ</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r > 0">
<mi>r</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{i}}b">
<mi>a</mi>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>b</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{R}">
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>2</mn><mi>x</mi></math>. It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method, with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math>, to find an approximate value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi></math> to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mi>v</mi><mo>-</mo><mn>2</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 solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math>, suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove the identity <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has two real roots, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math>.</p>
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>n</mi><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> and which has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>α</mi><mn>3</mn></msup></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>β</mi><mn>3</mn></msup></mfrac></math>.<br>Without solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider a geometric sequence with a first term of 4 and a fourth term of −2.916.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio of this sequence.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum to infinity of this sequence.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> moves in a straight line such that after time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, its velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>−</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mo> </mo><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> has displacement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>; at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>At successive times when the acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>0</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo> </mo></math>, the velocities of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> form a geometric sequence. The acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is zero at times <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub><mo><</mo><msub><mi>t</mi><mn>3</mn></msub></math> and the respective velocities are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>3</mn></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the times when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> comes to instantaneous rest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum displacement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, in metres, from its initial position.</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>Find the total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> seconds of its motion.</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, at these times, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math>.</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>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Find the term independent of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac><msup><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mfrac><mi>x</mi><mn>2</mn></mfrac></mrow></mfenced><mn>9</mn></msup></math>.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express the binomial coefficient <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> as a polynomial in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the least value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right) > {10^6}">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>></mo>
<mrow>
<msup>
<mn>10</mn>
<mn>6</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mi>r</mi><mi>x</mi><mo>)</mo></math> . The graph has a local maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mn>5</mn></mrow></mfenced></math> and a local minimum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the area of the shaded region.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Mary, three female friends, and her brother, Peter, attend the theatre. In the theatre there is a row of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> empty seats. For the first half of the show, they decide to sit next to each other in this row.</p>
</div>
<div class="specification">
<p>For the second half of the show, they return to the same row of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> empty seats. The four girls decide to sit at least one seat apart from Peter. The four girls do not have to sit next to each other.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of ways these five people can be seated in this row.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of ways these five people can now be seated in this row.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down and simplify the first three terms, in ascending powers of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, in the Extended Binomial expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - x} \right)^{\frac{1}{3}}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{9}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
</math></span> find a rational approximation to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt[3]{9}">
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^5} - 3{x^4} + m{x^3} + n{x^2} + px + q = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>m</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mi>x</mi>
<mo>+</mo>
<mi>q</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{R}">
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>The equation has three distinct real roots which can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,b">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,c">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
</math></span>.</p>
<p>The equation also has two imaginary roots, one of which is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d{\text{i}}">
<mi>d</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}">
<mi>d</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The values <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> are consecutive terms in a geometric sequence.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="abc = 8"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that one of the real roots is equal to 1.</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 8{d^2}"> <mi>q</mi> <mo>=</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span>, find the other two real roots.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} = {\text{si}}{{\text{n}}^3}\,y">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant y \leqslant \pi ">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>y</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
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22s3KL+1h4RvaHdn/9KKwsd6iz1aLg/GrNMjACurq4GHyAxZiSFN1EAKINubUe+jGhv8FfqDf+TMttr2tq8RJ3OEm0M91PHw5qYd1IfYPa4NMbE0Eo9Zqw53kQBIFc9uf8vvO0t7favUWdhjQZ78XtABvL8+fO0vLwcnqlqynHwAMp8n2E+3L+kQfd9WugOiCulvHEM66S/rEcJ8b0MU0Kf4IG93+3C/S99f9wNEdo22unT54tfUn/HRyMM/2DwMCX+gUGf4LvSGrwHPHfuHAXZ9zf6N/U//xNtjD0THv2ExNgIIx4bfYJHdg4aQBn1Hl71c49ebN6iW4MdGn/yOzJMzABKgHaIc/UcWUjnVdAAcjWGPWBY2xvaHWxQd/NXGm/3HL+LmAFENfTI1kEDyPCF9SyRD99o9190a+Onw4YYNk/MAHJVlKexR2hawNMSSvUznGoMw/ctLeW2cr5P3cHLo59FBwBKaFo49hszj9qbYD1gmNXP4naL3QOyFzx58mRgNZji9it6ZrAAhlf9LGqSd+d5ABCtoYFWQcOrfpaDj8/2ACBaQwMFkKuf3JkbYud7URQ9ACitoZ5nTQuyCsqxn0F2vhelz4kH5OdAroZevHixhDJxnRocgDLvS3id7+UKjgcPyAB6jw0NDsAwhh6Vgy3vbC8AyhAlvl+PW3AA8rCjMIcelSteXgBkL8hDlLyOlA8KQG504cYXDw/tngCUkfIxN6pN+vkNCkDpfvCw5JUnAD1HxQQFoHQ/TPo1iem4JwC5GuqlZpMto0EB6KH7QQzkDUBeV97jVBXBACgzn3HB9LB5A1Cmr7c9sZZ+yQsGQO7342qKl7lEvAF4//79xL6x9+9mEQ4GwNhHP2QN4w1Ar8+BwQDIfX88+7WXzSOA/Bz42WefeTFxcp9BACj9f54mdPUIoDwHeuoPDAJAT/1/8vPvEUB5DvTQzyt2DgJAif/09MvoEUAZnuSpphMEgPzs5yH+U34Vee8RQG6I4eWs19fX01JE/ToIAGOffiKvhHkFkGdLO3v2bJ4kUR4zD6C3DngpZV4B9NYQYx5AWXzF04M5Q+gVQG8NMeYB9NgA4xlAbw0x5gHkuV9CWHpaqo5ae68ekBtieClrjnzysJkH0FsEjBQ6zwByRMzXX38tUkS9Nw8gB2B76heS0uYZQBkhL1rEvDcNoNcWUC5wngGUmdI8DE0yDaDXFlDvAHpqCTUNoHgBD7+E2WqW3Dt7A26Y8PQnUxV6GBtoGkDpgsgWTg/vPQPIPzZenv1NA+i1C8J7FdRTV4RpADkI29Mg3LRn9+4BuSvCw2S9pgHkPsDYF2FJQ5d+DQA/pT//+c9pSaJ8bRpAL88BeSXLO4ASlJ2nTUzHAKBRawLAm0lDjFHzqGXLLICeO+HZugDwHYCxz4JgFkDPnfAA8Al56YwHgGqVCd2EvHtAAKhbnkqnBg/4Y/IM5DEShvsBAWBpZHS/AAABILeCczmIeTNbBZUqmMc4UC5wcv/ePSDPCRvzZhZAz3GgAPBd8LmHfmAAaPTn1bsH9BKQDQABoNmhTvCALRZOVEF9N8LAA7YIH18aAAJAeMAWIQSAABAAtgigLEmNbgh/U1KgI75F8OTS6Ij37QERCSMktLQHgACQq6CIhAGArSjgvR8QHrCVYnd0UXhAeEB4wCMeGn/FAzHZAOwJPG7ePSCmpDBQ6j00Q0+SGQBiSopJZaOx4wCwQ15HQ8ADNobZ5At5XZqMFfHuAXle0K+++mpy4YjkE7PB2KwvJub16wEZwBs3bkSC2eTbMA0gr5J67ty5ybmP+BPvHtDLKrmmAfQcD+odQC/P/6YBlHjQ7e3tiH1d/q15BvDBgwdJFxSWJ8svG40d9dwZ7xlAL1EwDJJpDyid8VwV9bZ5BlC6IGKfFds8gEkGOx16+PChN/5cd0Osrq7S2bNnXdjctAdkC3jtivDsAb2sDZg4GOs/M167IjwDyC2gXmo95j2gFERvI+Plvr2Foj169MhVEL55AL22hHoFsNfrJQDGPhBXap7mAfTaEuoVQGkBlQIa+948gGyATz75xN1a8V4B5AYYD2vDyw9LEADeuXMngVAy7WHvFUAvMaBShoMAUAqjp4YYuWdPjTDeGmAYwiAAlIaY2Jeqkl9F3nsE8O7du0kDjKfY3yAATH4pHPUNeQWQI2C4D9DTFszdckQMj5D3snn0gOfPn6fr1697MXFyn8EAKGMDPQTosmW8Afj48ePE+3kLvA8GQG/Pgd4A9NYBL24+GAA5w55iBL0B6PH5LynTQmIIe0/Pgd4A/OCDD9w9/wUHoDwHeugP9ASgx/4/cXhBVUG5f4iroR7mCvEEIAcbsF099f8FCSBnmqcp5NC02DdPAF6+fJkuXrwYu0lz7y8oD8h34GWAricA2fuxXT1uwQHI4WhssNjD0rwAKN0Psdtz0o9LcADK+MDYpyzwAuDVq1eTH1QvARZZEIMDkG/Aw/AkLwBy98P6+nq2XLp5HySAHmbM9gCgpxmwJ/2iBAkg9wPyc2DM1VAPAEr0i9fqJ0MZJICccR4ZwVNVxLp5AJCrn99++22sJix0X8ECKAU01s5bub9YR8Sj+vmOz2ABjL0aGjuAqH4GDiBnP+bW0NgB5MmXPLd+Sv00WA/INyCtoTF24sYMoHS+e4jpFdAm7YMGUDrlYwxjihlA753vaRiDBpBvRGJDY2vKjhVAnnri5MmTdO/evXQ5dPs6eAAlNjS26kysAMrUgzE+NlT5FQkeQL5pHqLEo+Vj2mIFkKee9zr0KK98RgEgR8RwZExMI+VjBFBGvsccwZQH2bRjUQAofYIxTWkXI4DS9xfTD+U0uIp8FgWAfKOxhabFCCD3/XmbeHcWhNEAGE6f4Ij2t36i/maXljrXqL/7NtdGsQEojS+xNZblGq/EwWgA5G6IIOaL2RtQd6GTPLN2HAHIjS9nzpwpUTR9nBoNgGyucBpjXtPW5iXyAiAaXyb/mEQFoDTG2G9le0u7/WtuAETjixMA+Ta5P5Crora3yQByd8qkvydPnlBofxz5wo0vN27csG2SlnIXlQdkDaUxxvbD/mQApRzE0ggjjS+IfBHLju+jA5Bvj0fK215L0A+APOp9eXl5vNTh3aECUQIoa0jwkmY2Nx8A3r9/P6lOszfHlq9AlADKMCW7U9j7AJCnnD99+jTFNlIlH6VqR6MEkKXgYUp240OlG+ISbW69zrVc6M+A0vUQ41jNXINVPBgtgNIlYa4AvB3SxuJ4S+fixpCy8TChAyiDbhH3OZ3MaAHk25YuiRCrQCEDKINuMefLdPj406gBlHXlQxwlETKAN2/eTKr/dhvBZoPR1BlRA8gicneE/Y754+YOFUDpeMeoh+M2zTsSPYBhdMwfN02oAErHu+1AiON6t3UkegBZWO6YD20a+1AB5I73jz/+uK3yHNx1XQAYohcMEUB4v/L8uwCQZeHnwJC8YIgAsvfDhEvlIHQDoP3wtHHDhQagzHYdYovzuPLNvnMDoIyYD2X6wtAA5BHvZ8+eRdhZSX7dAMi6hOQFQwJQgq7h/UrSF3tHfFaOkLxgSADC+2VLWvH3rjwgyxKKFwwFQHi/4rDlnekOwFC8YCgAsvfDkKM8tIodcwcgyxKCFwwBQHi/YpBNO8slgCF4wRAAhPebhlaxz1wCyNJY94LWAYT3KwbYrLPcAmjdC1oHEN5vFlrFPncLIMtj2QtaBhDerxhcRc5yDaBlL2gZQPT7FUGr2DmuAWSJrHpBqwDC+xUDq+hZ7gG06gWtAsgjHhDzWRSv2ee5B5AlEi9oaRS3RQBlvB/rhU1HAQBIlETwWxsvaBFAeD8d6NKpAMADNax5QWsAivezVEtIF+RQXwPAlOUszR1jDUD2flhkJVVYlF4CwJSQluaOsQSgzPMJ75cqLEovAWBGSPaC/DzIraNtblYAlHk+MdNZPaUBAGZ0FS/YdkufFQDh/TIFRPktAMwRVGbTbtMLWgBQ1njALNc5hUTpEADMEdLCmhIWAFxdXcUaDznlQ/MQAJygJs+e1ub6gm0DKOv73bhxY4JCOKyhAACcoKJ4wbbWF2wbQFnfDyscTSggSocB4BQheYnrtrxgmwA+ePAgue979+5NUQcfaSgAAKeoKKvstjGZb5sA8nCjtn54ppgjyo8A4AyzylrzTVfF2gJQhhu1VfWeYY7oPgaAM0zKXpA75pv2gm0ByN7vzJkzrQcizDBLNB8DwAKmlEDt58+fFzhb55Q2AJSA67aDEHQUDCMVAFjATjJot8nlzdoAEMONChQG5VMAYEFBBYimApLlerdv36YnT57U/ifer6n7Kyh79KcBwBImluFKTYSoNQkgAq5LFALlUwFgCUGbDNRuEkAEXJcoBMqnAsCSgjYVqN0UgAi4LlkAlE8HgCUFlRC1hw8flvxmudObAhAhZ+Xson02AKygaBOB2k0AiIDrCsZX/goArCDo9vZ2EqpVZ7RIEwCK9+P7wdaOAgCwou4SolZX4a0bQIScVTS88tcAYEVB6w7UrhtABFxXNLzy1wDgHIJyQwyPGqgjULtOAMX7IeRsDuMrfRUAziGkhKjVEahdJ4Dnz5/H+g5z2F3zqwBwTjUlUFs7hKsuACXkDN5vTsMrfR0AzimkeEHtQO26AETA9ZwGV/46AFQQVGDR9IKSpmYwtng/zXwqyOc6CQCoZH4J1FZKjrQB5JAz9n6Y4VrLQjrpAEAdHUk7UFsbQARcKxlaORkAqCgoe0GtdSU0AZThRleuXFG8WySloQAA1FDxIA0J1NZoYdQEULxfHf2VivK5TAoAKpud+wQ1vKAWgDLcCDNcKxtaKTkAqCSkJKPlBbUAxPoOYhmbewBYg13EC3K8aNVNA0AMN6qqfnPfA4A1aC1ecJ7hShoAynAjPPvVYGSlJAGgkpDZZOYdtDsvgOL91tfXs1nDe0MKAMCajDGvF5wXQPF+81SDa5IGyaYUAIApMbRfyqDdKhDMA6B4P6xupG1R/fQAoL6mhynKoN0qz4LzAAjvd2gC8y8AYM0mquoFqwII71ezQZWTB4DKgmaTq+oFqwII75e1gO33ALAB+1TxglUAhPdrwJjKlwCAyoLmJVelRbQKgPB+eerbPgYAG7KPRMcUXdilLIAc88kTRKHfryGDKl0GACoJOSsZ8YJFR0qUBRAjHmZZwObnALBBu5TxgmUAxIiHBo2ofCkAqCzotOTKeMEyAML7TVPd9mcAsGH78PJmRWZQKwMgz/Vy/fr1hu8El9NQAABqqFgiDZk7ZtbMZEUBxExnJcQ3eCoAbMEo7AHZE07bigLI3m95eXlaUvjMsAIAsAXjyGza/Ew4aSsCoKzxwOdiC1MBANiC3WQ27Tt37ky8ehEAL1++TKdPn6aifYsTL4YPWlMAALYk/azwtFkASthZ3UtltySPm8sCwJZMLUHakwCaBaBMtlRlrGFLt4zL5igAAHNEaeqQdMznXW8WgCdOnCBMNZinXFjHAGCL9nr+/HkSv5nXJTENQOl64O9jC1sBANiy/bhLgj1hdpsGIC8vffHixexX8D5ABQBgy0aTLonss9wkAKXxpWhQd8u3h8vPUAAAzhCo7o8nNcZMAhCNL3VbpNn0AWCzeudejaug2fjQSQAi7jNXwmAPAkADppP40HRkTB6AvV5vYqONgdtAFiooAAAriKb9FY5k4dHs6ekL8wDkKScQ+aKtfrvpAcB29T+8OsPHy5rJlgcg9/1hsl1RKI49ADRiR+kTlL69LIBS/ZTPjWQb2ZhTAQA4p4CaX2cPKNXQLIBS/dS8HtJqXwEA2L4NDnPAoyOkGpoFENXPQ5miegEADZlTWkO3t7cpDaCM+8sLWTOUfWSlggIAsIJodX1FWkM5yiUNoHS+Y9xfXcq3ly4AbE/73CvzVBX8lwbw/PnzmHQpV63wDwJAYzaU2NCnT58mfYNra2vJHrGfxgyllB0AqCSkVjIyd+hf/vKXBLyVlZVkn46S0boW0mlfAQDYvg2O5fqPOHQAAAO5SURBVICjYi5dupSA9+GHHyb7YyfhQBQKAECDZuTgbIZQ/jDprkEjKWUJACoJqZkMzxMj8PEez3+a6tpKCwDaskeSG+kPFAgRfmbQSEpZAoBKQmomwx3xAh/vs6PlNa+FtNpVAAC2q//Eq6cBnHgSPgheAQBo1ITSCpodKW80u8hWRQUAYEXh6v4aR8OwF5y1iEvd+UD69SoAAOvVt3LqMnW9DE+qnBC+aFoBAGjUPADQqGGUs1UYQK4OYWtOgc3NzaQK+sMPPzR3UcdXmla++cew3EiU36i/spjYb+HzPu2MiEa7z6i3coo6nSXaGO4fKl2YqmkZPEwNL9QUkNEQvMdWvwLTyjd/xgOly9liRPvDHi11LtHm8CfqdR/Ri/3RsRsBgMcksXEAADZrh1kA8uf8x63SxQPjDzzhwpfU33mTe0MAMFeW9g8CwGZtUBRAAZHjdWcHSLymrc1L1Fnp0+6E2ykFoFwc+6NA6Tq1+N3vfpf86tZ5DaRdzZYMIEcsTd8OAFxYo8He8eonf7cUgNMvhk+1FWA4sDWjwDSt0z9SXAUtGps72nlKa93/oZWF96k7eJl7I4UtPC2DuSnj4NwKQPO5JSycwDSt+TNuhCk1KmX/V9rsfk/D/f+jQfdDurD5gt7uPKVbf39BaV8IAAubqPkTpxWK5nMT9xWnac3dELOf9w70Gb2gzQsL1Flao/7WHhGNaG+wRgudU7TSe0a7afrKVEHjlt/m3U0rFDZzjFyVVaCwByybMM6fXwEAOL+G1lMAgIYtBAANG0cpa9UA3N+iQf9v1F06RSv935SygmSyCgDArCLa70e0v/UT9Te7tNS5Rv3dt9oXmJleBQBf0qD7/kH/1CIAnClx9RMAYHXtCn1zb0DdBekHDAbAd7c22tqkCx0AWMjQFU8CgBWFK/W1g87ycDzgwd3t9mkFAJYyddmTAWBZxaqc/5Z2+9eoAwCriBf3dwBgE/YFgE2oHOQ1AGATZrMAYFKdlIfRCftsRDeqoLWXDgBYu8REZAHAKvcJAKuoVuo7ALCUXBVPBoAVhYv/awCwCRsHCuC7boiFJMo7E1/ahGourgEAmzCzdENcos2t101ccOwaFTri92m4sTQ+UHRxg4bNBxGM3UiMbwBgzVZ9O6SNxfE2j8WNITVZlCsAWLMoSP5QAQB4KEW0LwCgYdMCQMPGUcoaAFQSso5kAGAdqtpKEwDasgdy40wBAOjM4LhdWwoAQFv2QG6cKQAAnRkct2tLAQBoyx7IjTMF/h9MpJfyhzWAOwAAAABJRU5ErkJggg=="></p>
</div>
<div class="specification">
<p>The shaded region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is the area bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \pi ">
<mi>y</mi>
<mo>=</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using implicit differentiation, find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the tangent to the curve at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span> is now rotated about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis, through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> radians, to form a solid.</p>
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{si}}{{\text{n}}^3}\,y}"> <mrow> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {\text{co}}{{\text{s}}^2}\,y} \right){\text{sin}}\,y"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </math></span>, show that the volume of the solid formed is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Phil takes out a bank loan of $150 000 to buy a house, at an annual interest rate of 3.5%. The interest is calculated at the end of each year and added to the amount outstanding.</p>
</div>
<div class="specification">
<p>To pay off the loan, Phil makes annual deposits of $<em>P </em>at the end of every year in a savings account, paying an annual interest rate of 2% . He makes his first deposit at the end of the first year after taking out the loan.</p>
</div>
<div class="specification">
<p>David visits a different bank and makes a single deposit of $<em>Q </em>, the annual interest rate being 2.8%.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.</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 the total value of Phil’s savings after 20 years is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mn>1.02</mn> </mrow> <mrow> <mn>20</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>P</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1.02</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that Phil’s aim is to own the house after 20 years, find the value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P"> <mi>P</mi> </math></span> to the nearest dollar.</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>David wishes to withdraw $5000 at the end of each year for a period of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> years. Show that an expression for the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q"> <mi>Q</mi> </math></span> is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q"> <mi>Q</mi> </math></span> that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Eight boys and two girls sit on a bench. Determine the number of possible arrangements, given that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the girls do not sit together.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the girls do not sit on either end.</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>the girls do not sit on either end and do not sit together.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>The complex numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> satisfy the equations</p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{w}{z} = 2{\text{i}}">
<mfrac>
<mi>w</mi>
<mi>z</mi>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{z}}^ * } - 3w = 5 + 5{\text{i}}">
<mrow>
<msup>
<mrow>
<mtext>z</mtext>
</mrow>
<mo>∗</mo>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>w</mi>
<mo>=</mo>
<mn>5</mn>
<mo>+</mo>
<mn>5</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>.</p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + b{\text{i}}">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{b}} \in \mathbb{Z}">
<mrow>
<mtext>b</mtext>
</mrow>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Boxes of mixed fruit are on sale at a local supermarket.</p>
<p>Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.</p>
<p>Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.</p>
<p>Box C contains 5 bananas and 4 kiwifruit and costs $3.00.</p>
<p>Find the cost of each type of fruit.</p>
</div>
<br><hr><br><div class="specification">
<p>A random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> has probability density function</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3a}&{\text{,}}&{0 \leqslant x < 2}&{} \\ {a\left( {x - 5} \right)\left( {1 - x} \right)}&{\text{,}}&{2 \leqslant x \leqslant b}&{a{\text{, }}b \in {\mathbb{R}^ + }{\text{, }}3 < b \leqslant 5.} \\ 0&{\text{,}}&{{\text{otherwise}}}&{} \end{array}} \right.">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>a</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mn>3</mn>
<mo><</mo>
<mi>b</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>5.</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span></p>
<p> </p>
</div>
<div class="specification">
<p>Consider the case where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 5">
<mi>b</mi>
<mo>=</mo>
<mn>5</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Find the value of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, the probability that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span> lies between 1 and 3.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>. State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right)"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the median of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>Eight runners compete in a race where there are no tied finishes. Andrea and Jack are two of the eight competitors in this race.</p>
<p>Find the total number of possible ways in which the eight runners can finish if Jack finishes</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>in the position immediately after Andrea.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>in any position after Andrea.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Prove by contradiction that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>5</mn></math> is an irrational number.</p>
</div>
<br><hr><br><div class="question">
<p>A geometric sequence has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_4} = - 70"> <mrow> <msub> <mi>u</mi> <mn>4</mn> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mn>70</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_7} = 8.75"> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> </mrow> <mo>=</mo> <mn>8.75</mn> </math></span>. Find the second term of the sequence.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{{2 + 7{\text{i}}}}{{6 + 2{\text{i}}}}">
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>7</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mrow>
<mn>6</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{i}}b">
<mi>a</mi>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>b</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,\,b \in \mathbb{Q}">
<mi>a</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact value of the modulus of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span>, giving your answer to 4 decimal places.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>It is known that the number of fish in a given lake will decrease by 7% each year unless some new fish are added. At the end of each year, 250 new fish are added to the lake.</p>
<p>At the start of 2018, there are 2500 fish in the lake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there will be approximately 2645 fish in the lake at the start of 2020.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the approximate number of fish in the lake at the start of 2042.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span> in the expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{1}{x} + 5x} \right)^8}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> <mo>+</mo> <mn>5</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mn>8</mn> </msup> </mrow> </math></span> is equal to the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^4}"> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </math></span> in the expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a + 5x} \right)^7},{\text{ }}a \in \mathbb{R}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mn>5</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mn>7</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>a</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the first three terms of the binomial expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><msup><mo>)</mo><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and the result from part (a), show that the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi></math> up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>x</mi></math> and the result from part (b), find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>x</mi><mtext> arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>A biased coin is weighted such that the probability, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, of obtaining a tail is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>6</mn></math>. The coin is tossed repeatedly and independently until a tail is obtained.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> be the event “obtaining the first tail on an even numbered toss”.</p>
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced></math>.</p>
</div>
<br><hr><br><div class="question">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>≠</mo><mn>1</mn></math>.</p>
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><mi>z</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow></mfrac></mfenced><mo>=</mo><mn>0</mn></math>.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3} = 8{\text{i}}">
<mrow>
<msup>
<mi>w</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w \in \mathbb{C}">
<mi>w</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>. Give your answers in Cartesian form.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>One of the roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_1}">
<mrow>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> satisfies the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Re}}\left( {{w_1}} \right) = 0">
<mrow>
<mtext>Re</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_1} = \frac{z}{{z - {\text{i}}}}">
<mrow>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</mfrac>
</math></span>, express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + b{\text{i}}">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{Q}">
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a derivative given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>o</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>k</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is a positive constant.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, the population of a colony of ants, which has an initial value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn></math>.</p>
<p>The rate of change of the population can be modelled by the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow><mrow><mn>5</mn><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in days, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is the upper bound for the population.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn></math> the population of the colony has doubled in size from its initial value.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></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 solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math>.</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>, giving your answer correct to four significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when the rate of change of the population is at its maximum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = a{z^3} - 37{z^2} + 66z - 10"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>37</mn> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>66</mn> <mi>z</mi> <mo>−</mo> <mn>10</mn> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}"> <mi>z</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in \mathbb{Z}"> <mi>a</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<p>One of the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = 0"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 + {\text{i}}"> <mn>3</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </math></span>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>The population, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, of a particular species of marsupial on a small remote island can be modelled by the logistic differential equation</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in years and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mo> </mo><mi>N</mi></math> are positive constants.</p>
<p>The constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> represents the maximum population of this species of marsupial that the island can sustain indefinitely.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> be the initial population of marsupials.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of the population model, interpret the meaning of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></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>Hence show that the population of marsupials will increase at its maximum rate when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math>. Justify your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></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>By solving the logistic differential equation, show that its solution can be expressed in the form</p>
<p style="padding-left:150px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years, the population of marsupials is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></math>. It is known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> for this population model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question">
<p>In a trial examination session a candidate at a school has to take 18 examination papers including the physics paper, the chemistry paper and the biology paper. No two of these three papers may be taken consecutively. There is no restriction on the order in which the other examination papers may be taken.</p>
<p>Find the number of different orders in which these 18 examination papers may be taken.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the polynomial <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) \equiv {z^4} - 6{z^3} - 2{z^2} + 58z - 51,\,\,z \in \mathbb{C}">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>≡<!-- ≡ --></mo>
<mrow>
<msup>
<mi>z</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>58</mn>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mn>51</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {x^4} - 6{x^3} - 2{x^2} + 58x - 51"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>58</mn> <mi>x</mi> <mo>−</mo> <mn>51</mn> </math></span>, stating clearly the coordinates of any maximum and minimum points and intersections with axes.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, state the condition on <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> such that all roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = k"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> </math></span> are real.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Use mathematical induction to prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - a} \right)^n} > 1 - na">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>n</mi>
</msup>
</mrow>
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mi>n</mi>
<mi>a</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {n\,{\text{:}}\,n \in {\mathbb{Z}^ + },\,n \geqslant 2} \right\}">
<mrow>
<mo>{</mo>
<mrow>
<mi>n</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</mrow>
<mo>}</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < a < 1">
<mn>0</mn>
<mo><</mo>
<mi>a</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>The 3rd term of an arithmetic sequence is 1407 and the 10th term is 1183.</p>
</div>
<div class="question">
<p>Calculate the number of positive terms in the sequence.</p>
</div>
<br><hr><br><div class="question">
<p>Consider the expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {2 + x} \right)^n}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 3"> <mi>n</mi> <mo>⩾</mo> <mn>3</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z}"> <mi>n</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<p>The coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}"> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </math></span> is four times the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span>. 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>
<br><hr><br>