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<h2>HL Paper 1</h2><div class="question">
<p>The first term in an arithmetic sequence is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> and the fifth term is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>625</mn></math>.</p>
<p>Find the common difference of the sequence, expressing your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mi>p</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the integral <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_1^t {\frac{{ - 1}}{{x + {x^2}}}{\text{ }}} dx">
<munderover>
<mo>∫<!-- ∫ --></mo>
<mn>1</mn>
<mi>t</mi>
</munderover>
<mrow>
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
</mrow>
<mi>d</mi>
<mi>x</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t > 1">
<mi>t</mi>
<mo>></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Very briefly, explain why the value of this integral must be negative.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{ - 1}}{{x + {x^2}}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> in partial fractions.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use parts (a) and (b) to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t < {\text{ln}}\,2">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
<mo><</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Three planes have equations:</p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y + z = 5"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>5</mn> </math></span></p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 3y - z = 4"> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mn>4</mn> </math></span> , where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{, }}b \in \mathbb{R}"> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x - 5y + az = b"> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> <mo>+</mo> <mi>a</mi> <mi>z</mi> <mo>=</mo> <mi>b</mi> </math></span></p>
<p>Find the set of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> such that the three planes have no points of intersection.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><mi mathvariant="normal">i</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>. The roots of this equation are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>3</mn></msub></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Im</mi><mfenced><msub><mi>ω</mi><mn>2</mn></msub></mfenced><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Im</mi><mfenced><msub><mi>ω</mi><mn>3</mn></msub></mfenced><mo><</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>3</mn></msub></math> are represented by the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> respectively on an Argand diagram.</p>
</div>
<div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>=</mo><mtext>i</mtext><msup><mi>z</mi><mn>3</mn></msup><mo>,</mo><mo> </mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></msup></math> is a root of this equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>3</mn></msub></math>, expressing these in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mi>θ</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plot the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> on an Argand diagram.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AC</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using de Moivre’s theorem, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mstyle></mrow></msup></mrow></mfrac></math> is a root of this equation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mi>α</mi></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = 2\left( {{\text{cos}}\frac{\pi }{3} + {\text{i}}\,{\text{sin}}\frac{\pi }{3}} \right)">
<mi>w</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>These four points form the vertices of a quadrilateral, <em>Q</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup> in modulus-argument form.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch on an Argand diagram the points represented by <em>w</em><sup>0</sup> , <em>w</em><sup>1</sup> , <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the area of the quadrilateral <em>Q</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{21\sqrt 3 }}{2}"> <mfrac> <mrow> <mn>21</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 2\left( {{\text{cos}}\frac{\pi }{n} + {\text{i}}\,{\text{sin}}\frac{\pi }{n}} \right),\,\,n \in {\mathbb{Z}^ + }"> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>. The points represented on an Argand diagram by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^0},\,\,{z^1},\,\,{z^2},\, \ldots \,,\,\,{z^n}"> <mrow> <msup> <mi>z</mi> <mn>0</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mn>1</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> </math></span> form the vertices of a polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_n}"> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></span>.</p>
<p>Show that the area of the polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_n}"> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></span> can be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {{b^n} - 1} \right){\text{sin}}\frac{\pi }{n}"> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>b</mi> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,\,\,b\, \in \mathbb{R}"> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mo>…</mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>p</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Consider the case where the series is geometric.</p>
</div>
<div class="specification">
<p>Now consider the case where the series is arithmetic with common difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, show that the series is convergent.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of the series is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="question">
<p>Determine the roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(z + 2{\text{i}})^3} = 216{\text{i}}">
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>216</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}">
<mi>z</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>, giving the answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = a\sqrt 3 + b{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,{\text{ }}b \in \mathbb{Z}">
<mi>a</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2x - 4}}{{{x^2} - 1}}{\text{, }} - 1 < x < 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>, </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>For the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span>, then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>show that there are no <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph, showing clearly any asymptotic behaviour.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}"> <mfrac> <mn>3</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The area enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 4"> <mi>y</mi> <mo>=</mo> <mn>4</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,v"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>v</mi> </math></span>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A farmer has six sheep pens, arranged in a grid with three rows and two columns as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Five sheep called Amber, Brownie, Curly, Daisy and Eden are to be placed in the pens. Each pen is large enough to hold all of the sheep. Amber and Brownie are known to fight.</p>
<p>Find the number of ways of placing the sheep in the pens in each of the following cases:</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Each pen is large enough to contain five sheep. Amber and Brownie must not be placed in the same pen.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Each pen may only contain one sheep. Amber and Brownie must not be placed in pens which share a boundary.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two distinct lines, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, intersect at a point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span>. In addition to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span>, four distinct points are marked out on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and three distinct points on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>. A mathematician decides to join some of these eight points to form polygons.</p>
</div>
<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> has vector equation <em><strong>r</strong></em><sub>1</sub> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 1 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 1 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ<!-- λ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda \in \mathbb{R}">
<mi>λ<!-- λ --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> has vector equation <em><strong>r</strong></em><sub>2</sub> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 1} \\ 0 \\ 2 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}} 5 \\ 6 \\ 2 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>μ<!-- μ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu \in \mathbb{R}">
<mi>μ<!-- μ --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> has coordinates (4, 6, 4).</p>
</div>
<div class="specification">
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> has coordinates (3, 4, 3) and lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> has coordinates (−1, 0, 2) and lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how many sets of four points can be selected which can form the vertices of a quadrilateral.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how many sets of three points can be selected which can form the vertices of a triangle.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> is the point of intersection of the two lines.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> corresponding to the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PA}}} ">
<mover>
<mrow>
<mtext>PA</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PB}}} ">
<mover>
<mrow>
<mtext>PB</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}">
<mrow>
<mtext>C</mtext>
</mrow>
</math></span> be the point on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> with coordinates (1, 0, 1) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{D}}">
<mrow>
<mtext>D</mtext>
</mrow>
</math></span> be the point on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> with parameter <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = - 2">
<mi>μ</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span>.</p>
<p>Find the area of the quadrilateral <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{CDBA}}">
<mrow>
<mtext>CDBA</mtext>
</mrow>
</math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Use the method of mathematical induction to prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^n} + 15n - 1">
<mrow>
<msup>
<mn>4</mn>
<mi>n</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> is divisible by 9 for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>In the following Argand diagram, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>2</mtext></msub></math> are the vertices of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>1</mtext></msub><msub><mtext>OZ</mtext><mtext>2</mtext></msub></math> described anticlockwise.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub></math> represents the complex number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>α</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></math>. The point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>2</mn></msub></math> represents the complex number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>θ</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></math>.</p>
<p>Angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>θ</mi></math> are measured anticlockwise from the positive direction of the real axis such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>,</mo><mo> </mo><mi>θ</mi><mo><</mo><mn>2</mn><mi>π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math>.</p>
</div>
<div class="specification">
<p>In parts (c), (d) and (e), consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mtext>1</mtext></msub><msub><mtext>OZ</mtext><mtext>2</mtext></msub></math> is an equilateral triangle.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math> be the distinct roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>z</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>r</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><mtext>i</mtext><mfenced><mrow><mi>α</mi><mo>-</mo><mi>θ</mi></mrow></mfenced></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup></math> is the complex conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msup><msub><mi>z</mi><mn>2</mn></msub><mo>∗</mo></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub><msub><mtext>OZ</mtext><mn>2</mn></msub></math> is a right-angled triangle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>z</mi><mn>2</mn></msub><mn>2</mn></msup><mo>=</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the result from part (c)(ii) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>b</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>z</mi><mo>+</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>α</mi><mo>-</mo><mi>θ</mi><mo><</mo><mi>π</mi></math>, deduce that only one equilateral triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Z</mtext><mn>1</mn></msub><msub><mtext>OZ</mtext><mn>2</mn></msub></math> can be formed from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the roots of this equation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following system of equations where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in \mathbb{R}">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + 4y - z = 10">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2y + az = 5">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mi>a</mi>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5x + 12y = 2a">
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>a</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> for which the system of equations does not have a unique solution.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the solution of the system of equations when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 2">
<mi>a</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A team of four is to be chosen from a group of four boys and four girls.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of different possible teams that could be chosen.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of different possible teams that could be chosen, given that the team must include at least one girl and at least one boy.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Find the solution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}x - {\log _2}5 = 2 + {\log _2}3">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>5</mn>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>3</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <em>S</em> be the sum of the roots found in part (a).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^{24}} = 1"> <mrow> <msup> <mi>z</mi> <mrow> <mn>24</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>1</mn> </math></span> which satisfy the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < {\text{arg}}\left( z \right) < \frac{\pi }{2}"> <mn>0</mn> <mo><</mo> <mrow> <mtext>arg</mtext> </mrow> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, expressing your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{e^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mi>e</mi> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta \in {\mathbb{R}^ + }"> <mi>θ</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Re <em>S</em> = Im <em>S</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>, find the value of cos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sqrt a + \sqrt b }}{c}"> <mfrac> <mrow> <msqrt> <mi>a</mi> </msqrt> <mo>+</mo> <msqrt> <mi>b</mi> </msqrt> </mrow> <mi>c</mi> </mfrac> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> are integers to be determined.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, show that <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left( {1 + \sqrt 2 } \right)\left( {1 + \sqrt 3 } \right)\left( {1 + {\text{i}}} \right)"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove by mathematical induction that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>n</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo>+</mo><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, determine the Maclaurin series of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced open="[" close="]"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}"> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi "> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>≡</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>k</mi> <mi>π</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the principle of mathematical induction to prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x + \ldots + \sin (2n - 1)x = \frac{{1 - \cos 2nx}}{{2\sin x}},{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}x \ne k\pi "> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>k</mi> <mi>π</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x = \cos x"> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> in the interval <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \pi "> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>π</mi> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Three girls and four boys are seated randomly on a straight bench. Find the probability that the girls sit together and the boys sit together.</p>
</div>
<br><hr><br><div class="question">
<p>Prove by mathematical induction that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \ldots + \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} n \\ 3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z},n \geqslant 3">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>,</mo>
<mi>n</mi>
<mo>⩾</mo>
<mn>3</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^4} = - 4">
<mrow>
<msup>
<mi>z</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>4</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}">
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation, giving the solutions in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{i}}b"> <mi>a</mi> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mi>b</mi> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{, }}b \in \mathbb{R}"> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>4</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mn>2</mn>
<mi>n</mi>
</msup>
</mrow>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> </math></span> is an odd or even function, justifying your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using mathematical induction, prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mfrac> <mrow> <mi>m</mi> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{Z}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find an expression for the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 1"> <mi>n</mi> <mo>></mo> <mn>1</mn> </math></span>, the equation of the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_n}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y - \pi = 0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Consider integers <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> is exactly divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>. Prove by contradiction that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> cannot both be odd.</p>
</div>
<br><hr><br><div class="specification">
<p>Let the roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^3} = - 3 + \sqrt 3 {\text{i}}">
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>+</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>On an Argand diagram, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span> are represented by the points U, V and W respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + \sqrt 3 {\text{i}}"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> <mrow> <mtext>i</mtext> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{{\text{e}}^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r > 0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \pi < \theta \leqslant \pi "> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo>⩽</mo> <mi>π</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> expressing your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{{\text{e}}^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r > 0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \pi < \theta \leqslant \pi "> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo>⩽</mo> <mi>π</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle UVW.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the sum of the roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span>, show that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{18}} + {\text{cos}}\frac{{7\pi }}{{18}} + {\text{cos}}\frac{{17\pi }}{{18}} = 0"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>17</mn> <mi>π</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>9</mn></msubsup><mfenced><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></mfenced><mo>d</mo><mi>x</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the three planes</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle><mo>:</mo><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>4</mn></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle><mo>:</mo><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>5</mn></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle><mo>:</mo><mo>-</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>32</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the three planes do not intersect.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> lies on both <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>, the line of intersection of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Prove by contradiction that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has no integer roots.</p>
</div>
<br><hr><br><div class="question">
<p>Consider the quartic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>4</mn></msup><mo>+</mo><mn>4</mn><msup><mi>z</mi><mn>3</mn></msup><mo>+</mo><mn>8</mn><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mn>80</mn><mi>z</mi><mo>+</mo><mn>400</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p>Two of the roots of this equation are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>+</mo><mi>a</mi><mtext>i</mtext></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></msqrt></mfrac><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><mi>x</mi></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>≠</mo><mn>0</mn></math>.</p>
<p>The binomial expansion of this expression, in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, as far as the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>b</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the restriction which must be placed on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for this expansion to be valid.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 1 - \cos 2\theta - {\text{i}}\sin 2\theta ,{\text{ }}z \in \mathbb{C},{\text{ }}0 \leqslant \theta \leqslant \pi ">
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>θ<!-- θ --></mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ ">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msup>
<mn>0</mn>
<mo>∘</mo>
</msup>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<msup>
<mn>180</mn>
<mo>∘</mo>
</msup>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the modulus and argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>. Express each answer in its simplest form.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cube roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in modulus-argument form.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the binomial theorem to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>4</mn></msup></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mi mathvariant="normal">i</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> are expressed in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use de Moivre’s theorem and the result from part (a) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cot</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cot</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cot</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the identity from part (b) to show that the quadratic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce a quadratic equation with integer coefficients, having roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Use the principle of mathematical induction to prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + \, \ldots \, + n{\left( {\frac{1}{2}} \right)^{n - 1}} = 4 - \frac{{n + 2}}{{{2^{n - 1}}}}">
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<mo>…</mo>
<mspace width="thinmathspace"></mspace>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i, </mtext>
</mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = \frac{{{z_1}}}{{{z_2}}}">
<mi>w</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form write down the modulus of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>;</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form write down the argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the smallest positive integer value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^n}">
<mrow>
<msup>
<mi>w</mi>
<mi>n</mi>
</msup>
</mrow>
</math></span> is a real number.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = x\,{{\text{e}}^{2x}}"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}"> <mi>x</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>. The <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{n^{{\text{th}}}}"> <mrow> <msup> <mi>n</mi> <mrow> <mrow> <mtext>th</mtext> </mrow> </mrow> </msup> </mrow> </math></span> derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> is denoted by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( n \right)}}\left( x \right)"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<p> </p>
<p>Prove, by mathematical induction, that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( n \right)}}\left( x \right) = \left( {{2^n}x + n{2^{n - 1}}} \right){{\text{e}}^{2x}}"> <mrow> <msup> <mi>f</mi> <mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }"> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{4x - 5}}{{{x^2} - 3x + 2}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 1{\text{,}}\,x \ne 2">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>1</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> in partial fractions.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use part (a) to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is always decreasing.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use part (a) to find the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_{ - 1}^0 {f(x)dx} ">
<munderover>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>0</mn>
</munderover>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>d</mi>
<mi>x</mi>
</mrow>
</math></span>, giving the answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,q">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Q}">
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>b</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mn>2</mn><mi>b</mi><mtext>i</mtext></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arg</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.<br>Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_14.39.35.png" alt="N17/5/MATHL/HP1/ENG/TZ0/10"></p>
<p>Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.<br>Chloe wins if <strong>no</strong> matches occur; otherwise Selena wins.</p>
</div>
<div class="specification">
<p>Chloe and Selena repeat their game so that they play a total of 50 times.<br>Suppose the discrete random variable <em>X </em>represents the number of times Chloe wins.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability that Chloe wins the game is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{8}">
<mfrac>
<mn>3</mn>
<mn>8</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the mean of <em>X</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the variance of <em>X</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi><mo> </mo><mo>\</mo><mo> </mo><mfenced open="{" close="}"><mi>k</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>≠</mo><mn>5</mn></math>. </p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the vertical asymptote on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the horizontal asymptote on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an algebraic method to determine whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is a self-inverse function.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region bounded by the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>7</mn></math> is rotated through <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. Find the volume of the solid generated, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>)</mo><mo> </mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\omega ">
<mi>ω<!-- ω --></mi>
</math></span> be one of the non-real solutions of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^3} = 1">
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the complex numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 1 - 3{\text{i}}">
<mi>p</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = x + (2x + 1){\text{i}}">
<mi>q</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of</p>
<p>(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \omega + {\omega ^2}">
<mn>1</mn>
<mo>+</mo>
<mi>ω</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>;</p>
<p>(ii) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \omega {\text{*}} + {(\omega {\text{*}})^2}">
<mn>1</mn>
<mo>+</mo>
<mi>ω</mi>
<mrow>
<mtext>*</mtext>
</mrow>
<mo>+</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>ω</mi>
<mrow>
<mtext>*</mtext>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\omega - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13">
<mo stretchy="false">(</mo>
<mi>ω</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>ω</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>13</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> that satisfy the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| p \right| = \left| q \right|">
<mrow>
<mo>|</mo>
<mi>p</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mi>q</mi>
<mo>|</mo>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2}">
<mi>Re</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>8</mn>
<mo><</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>Im</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mi>q</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>An arithmetic sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}{\text{, }}{u_2}{\text{, }}{u_3} \ldots ">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>…<!-- … --></mo>
</math></span> has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 1">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> and common difference <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \ne 0">
<mi>d</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2}{\text{, }}{u_3}">
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6}">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
</math></span> are the first three terms of a geometric sequence</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_N} = - 15">
<mrow>
<msub>
<mi>u</mi>
<mi>N</mi>
</msub>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>15</mn>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^N {{u_r}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>r</mi>
</msub>
</mrow>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^x} + {2^{x + 2}} = 3">
<mrow>
<msup>
<mn>4</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Solve the simultaneous equations</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}6x = 1 + 2\,{\text{lo}}{{\text{g}}_2}y">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mn>6</mn>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mi>y</mi>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + {\text{lo}}{{\text{g}}_6}x = {\text{lo}}{{\text{g}}_6}\left( {15y - 25} \right)">
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>6</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>15</mn>
<mi>y</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}(x + 3) + {\log _2}(x - 3) = 4">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>The 1st, 4th and 8th terms of an arithmetic sequence, with common difference <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \ne 0">
<mi>d</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>, are the first three terms of a geometric sequence, with common ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>. Given that the 1st term of both sequences is 9 find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>;</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>In the following Argand diagram the point A represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 + 4{\text{i}}"> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mrow> <mtext>i</mtext> </mrow> </math></span> and the point B represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + 0{\text{i}}"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>0</mn> <mrow> <mtext>i</mtext> </mrow> </math></span>. The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_06.11.20.png" alt="M17/5/MATHL/HP1/ENG/TZ2/05"></p>
</div>
<br><hr><br><div class="question">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r,\,x \in {\mathbb{R}^ + }">
<mi>r</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{{1 - {x^2}}}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 < x < 1">
<mo>−</mo>
<mn>1</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>1</mn>
</math></span>. Use partial fractions to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f\left( x \right)} {\text{ }}dx">
<mo>∫</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mi>d</mi>
<mi>x</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>A</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>B</mi><mo>≡</mo><mi>sin</mi><mfenced><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow></mfenced><mo>-</mo><mi>sin</mi><mfenced><mrow><mi>A</mi><mo>-</mo><mi>B</mi></mrow></mfenced></math>. (Do <strong>not</strong> prove this identity.)</p>
<p>Using mathematical induction and the above identity, prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mi mathvariant="normal">Σ</mi><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>cos</mi><mfenced><mrow><mn>2</mn><mi>r</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle><mstyle displaystyle="true"><mi>θ</mi></mstyle></mrow><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mo> </mo></mstyle><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<br><hr><br><div class="question">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>z</mi></mrow><mrow><mn>3</mn><mo>-</mo><mi>z</mi><mo>*</mo></mrow></mfrac><mo>=</mo><mtext>i</mtext></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mtext>i</mtext><mi>y</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, with asymptotes at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe a sequence of transformations that transforms the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan </mtext><mi>x</mi></math> to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>p</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi><mo>≡</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo><</mo><mn>1</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan </mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mtext>arctan </mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi mathvariant="normal">+</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using mathematical induction and the result from part (b), prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>n</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Use mathematical induction to prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^n {r\left( {r{\text{!}}} \right)} = \left( {n + 1} \right){\text{!}} - 1">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>r</mi>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^4} + a{z^3} + b{z^2} + cz + d = 0"> <mrow> <msup> <mi>z</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mi>a</mi> <mrow> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mi>b</mi> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>c</mi> <mi>z</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}"> <mi>d</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}"> <mi>z</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> </math></span>.</p>
<p>Two of the roots of the equation are log<sub>2</sub>6 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\sqrt 3 "> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </math></span> and the sum of all the roots is 3 + log<sub>2</sub>3.</p>
<p>Show that 6<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d"> <mi>d</mi> </math></span> + 12 = 0.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider two events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> defined in the same sample space.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = \frac{4}{9},{\text{ P}}(B|A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B|A') = \frac{1}{6}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span>;</p>
<p>(ii) hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{ax + b}}{{cx + d}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mrow>
<mi>c</mi>
<mi>x</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
</mfrac>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R},\,\,x \ne - \frac{d}{c}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>d</mi>
<mi>c</mi>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = \frac{{2x - 3}}{{x - 2}},\,\,x \in \mathbb{R},\,\,x \ne 2">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>2</mn>
</math></span></p>
</div>
<div class="question">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right)"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A + \frac{B}{{x - 2}}"> <mi>A</mi> <mo>+</mo> <mfrac> <mi>B</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span> where A, B are constants.</p>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>3</mn></msub><mo> </mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>3</mn></mrow></mfrac><mo>+</mo><msub><mi>log</mi><mn>3</mn></msub><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<br><hr><br><div class="question">
<p>Consider the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></mfenced><mi>n</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>. Determine all possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> for which the expansion has a non-zero constant term.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mi>n</mi></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>n</mi></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>2</mn></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mo>,</mo><mo> </mo><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>×</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
<p>It is given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> term in the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br>