File "HL-paper1.html"
Path: /IB QUESTIONBANKS/5 Fifth Edition - PAPER/HTML/Math AA/Topic 2/HL-paper1html
File size: 259.42 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-02ef852527079acf252dc4c9b2922c93db8fde2b6bff7c3c7f657634ae024ff1.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-9717ccaf4d6f9e8b66ebc0e8784b3061d3f70414d8c920e3eeab2c58fdb8b7c9.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../../../../../../index.html">Home</a>
</li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li class="dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Help
<b class="caret"></b>
</a>
<ul class="dropdown-menu">
<li><a href="https://questionbank.ibo.org/video_tour?locale=en">Video tour</a></li>
<li><a href="https://questionbank.ibo.org/instructions?locale=en">Detailed instructions</a></li>
<li><a target="_blank" href="https://ibanswers.ibo.org/">IB Answers</a></li>
</ul>
</li>
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
<div class="pull-right screen_only"><a class="btn btn-small btn-info" href="https://questionbank.ibo.org/updates?locale=en">Updates to Questionbank</a></div>
<p class="muted language_chooser">
User interface language:
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=en">English</a>
|
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=es">Español</a>
</p>
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="https://mirror.ibdocs.top/qb.png" alt="Ib qb 46 logo">
</div>
</div>
</div>
<h2>HL Paper 1</h2><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 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>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></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><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>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> has a vertical asymptote and a horizontal asymptote.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the vertical asymptote.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the horizontal asymptote.</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>On the set of axes below, 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>.</p>
<p>On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.</p>
<p><img src="data:image/png;base64,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"></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, solve the inequality <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo><</mo><mn>2</mn></math>.</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>Solve the inequality <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mfrac><mrow><mn>2</mn><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>-</mo><mn>1</mn></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mn>1</mn></mrow></mfrac><mo><</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</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>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="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>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. </p>
</div>
<div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>8</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> may be obtained by transforming the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> using a sequence of three transformations.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> does not exist, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>a</mi><mi>c</mi><mo>></mo><mn>0</mn></math>.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> exists.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>(</mo><mi>x</mi><mo>−</mo><mn>2</mn><msup><mo>)</mo><mn>3</mn></msup><mo>+</mo><mi>q</mi></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State each of the transformations in the order in which they are applied.</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>Sketch the graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math> on the same set of axes, indicating the points where each graph crosses the coordinate axes.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</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>The cubic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>k</mi><mo>=</mo><mn>0</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>0</mn></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>β</mi></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mi>β</mi><mo>=</mo><mo>-</mo><mfrac><msup><mi>k</mi><mn>2</mn></msup><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</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{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</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>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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> has a local maximum at A. Find the coordinates of A.</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 there is exactly one point of inflexion, B, on 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 class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The coordinates of B can be expressed in the form B<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{2^a},\,b \times {2^{ - 3a}}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>a</mi> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mn>3</mn> <mi>a</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> where <em>a</em>, <em>b</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \in \mathbb{Q}"> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> </math></span>. Find the value of <em>a</em> and the value of <em>b</em>.</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>Sketch 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> showing clearly the position of the points A and B.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</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">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {x^2} - {a^2},{\text{ }}x \in \mathbb{R}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is a positive constant.</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(x) = x\sqrt {f(x)} ">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<msqrt>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| x \right| > a">
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>></mo>
<mi>a</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </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>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{f(x)}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span>;</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left| {\frac{1}{{f(x)}}} \right|"> <mi>y</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</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="\int {f(x)\cos x{\text{d}}x} "> <mo>∫</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </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>By finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> is an increasing function.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{1 - 3x}}{{x - 2}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>x</mi> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span>, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.</p>
<p><img src="images/Schermafbeelding_2018-02-07_om_17.42.06.png" alt="N17/5/MATHL/HP1/ENG/TZ0/06.a"></p>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfrac></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>3</mn></math>.</p>
</div>
<div class="specification">
<p>Write down the equation of</p>
</div>
<div class="specification">
<p>Find the coordinates where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> crosses</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the vertical asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the horizontal asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</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>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> on the axes below.</p>
<p><img src="data:image/png;base64,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"></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>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>-</mo><mi>x</mi></mrow></mfrac></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>3</mn></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"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced></math>, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</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{{{x^2} - 10x + 5}}{{x + 1}}{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \ne - 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>10</mn>
<mi>x</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the co-ordinates of all stationary points.</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>Write down the equation of the vertical asymptote.</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>With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{ln}}\left| x \right|">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> \ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ 0 \right\}">
<mrow>
<mo>{</mo>
<mn>0</mn>
<mo>}</mo>
</mrow>
</math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {\text{ln}}\left| {x + k} \right|">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>k</mi>
</mrow>
<mo>|</mo>
</mrow>
</math></span>, <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> \ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ { - k} \right\}">
<mrow>
<mo>{</mo>
<mrow>
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
<mo>}</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k > 2">
<mi>k</mi>
<mo>></mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> intersect at the point P .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe the transformation by which <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 transformed to <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>.</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 range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span>.</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>Sketch the graphs 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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> on the same axes, clearly stating the points of intersection with any axes.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The region <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is bounded by the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>6</mn></msqrt></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> be the area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> divides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> into two regions of equal area.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> be the gradient of a tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>6</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>32</mn></mfrac><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Sketch the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{x}{2} + 1"> <mi>y</mi> <mo>=</mo> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mn>1</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left| {x - 2} \right|"> <mi>y</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>|</mo> </mrow> </math></span> on the following axes.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></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><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></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>3</mn></math>.</p>
</div>
<div class="specification">
<p>The inverse of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfrac><mi>x</mi><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></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>a</mi></math>.</p>
<p>Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mi>q</mi><mn>2</mn></mfrac><msqrt><mi>r</mi></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</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) = {{\text{e}}^{2x}} - 6{{\text{e}}^x} + 5{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \leqslant a">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>a</mi>
</math></span>. 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> is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> has an inverse function.</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>For this value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}\left( x \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, stating its domain.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let <em>f</em>(<em>x</em>) = <em>x</em><sup>4</sup> + <em>px</em><sup>3</sup> + <em>qx</em> + 5 where <em>p</em>, <em>q</em> are constants.</p>
<p>The remainder when <em>f</em>(<em>x</em>) is divided by (<em>x</em> + 1) is 7, and the remainder when <em>f</em>(<em>x</em>) is divided by (<em>x</em> − 2) is 1. Find the value of <em>p</em> and the value of <em>q</em>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the polynomial <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q(x) = 3{x^3} - 11{x^2} + kx + 8">
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>11</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>k</mi>
<mi>x</mi>
<mo>+</mo>
<mn>8</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q(x)"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> has a factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x - 4)"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, factorize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q(x)">
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> as a product of linear factors.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></math>.</p>
<p>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> is shown below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWUAAAD5CAYAAAD/ViQ3AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAA9KSURBVHhe7d1/bJXVGcDxB2PMcKhbA4hQbixZ+dVsVdwQWpbZFYeBQQrdxg87+TFIZIWONkugxWZ/QAskDEiRuKgDZI2iG3IDQmiow6gtIBOBWCarWimFQmYaYSiGGFif0/dC1QKltPc+73u/n3/ee99LiEu6bw/nnnPebpebCQDAhNu8KwDAAKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogyDTkh49o8kFEqU0JA/SPjUxZbb596QxUMmyuqDn7W8BwKIKMOg/pL1/CF5f9Ms+e7n78qB/5xrud1joDz66zukpv5/Le+BACLKMOo2uXvwMEmXc3LmswverQTpN2CojErp1fIeCCCiDLt6hiQloUn+Vfdf+ar57aVTu6Ts45/JpOTvtHwOBBBRhk98Jof+USuZT6bJ3d4dIIiIMuy6vZck/ThBmmo+kbqDr0g4NFkm9L3D+xAIJqIM+xq2yZ/D98mTE0L8wCLw+BmHYd+XUMp9zdcfyJTfj5G+/LQiDvBjDsO+lHOfJkp+6Tx5pA/TFogPPA4KRl2U029slFfklzLvkb6MHhA3iDIMuSTnDz4tv3q8XqaU3Cu158ZK4YwU6eF9CsQDBiAw5dK5T+X458ek9rYJBBlxiZEyABjCSBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZZgWDodl7ty5cuGCd9A9EHBEGWY1NTXJ6tWrZMeO7VJRUeHdBYKNKMOslStXytix49zrvLx5cvLkSfcaCDKiDJOqq6ulquptmTNnjnufmztfli5d6l4DQUaUYY7OHxcWLpLi4mJJSEhw9/Ly8uTo0RqprNzt3gNBRZRhTllZmaSnj5LRox/17oh07969+f5aWbJkiZtrBoKKKMMUnTdet26t5ObmeneuSk1NdbF+8803vTtA8HBKHMzRkXBk2kKFQolSX9/gvfv250CQMFKGOTcKLkFGkBFlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMIQoA4AhRBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMIQoA4AhRBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGdLvczHsNmBQKJUp9fYP3zt9qa2ulpqZG3nnnHfnwww9l375q75OrkpIGSHr6KBk6dKhkZGRIv379vE8QD4gyzPNzlC9cuCDvvfeevPXWW7Ju3Vrvrsi4cePl/vvvl0GDBnl3rmpsbJQjR47Ijh3b3fsRI9JkwYIFkpaW5t4j2IgyzPNjlHVEXFlZKcuWlbj3GlYd9WpYBw4cKN27d3f3rycS9DVr1rgRtf4dJSUlkpyc7P0JBBFRhnl+inJ1dbW89tprUl6+yb0vLFzsQpyamured5T+vZE4l5U9LVlZWd4nCBqiDPP8EOXDhw+7UaxGU+eE8/MLZMyYMe0aEbeXjpzLysrcNIjGfu7cud4nCBKiDPMsR1mnKTZs2OBGxhrjZcuWd/nc7zPPPOOmRQhzMLEkDuiApqYmF8fMzAypqnrbTSns2lURlS/jNMQaZA1zeXm5dxdBwUgZ5lkbKVdW7pYlS5ZIXd3HLo4zZszo1GmK9oqMmLdv33HLc9awgyjDPCtRPnnypCxdutQtVdMlbQUFBTFdCaFzzNOnT5czZ067UXosfjGg8zF9AbRDOByWkSMflqNHa9xUhY5SY700TSOsXy7qiF2/AEQwEGXgOnTuuKioSPLy5rnR8datYVPL0fQXQ2npcrciQ790hP8RZeAadJnbxIlZbmVFZHSckJDgfWpHdna2W/mhq0Dgf0QZaIOuahg/fpzce28f2bt3v+nNGjqNoeui9ZcHo2X/I8pAKzpdoUvOiooWSW7ufHnhhRd8cSCQblRhtBwMRBnw6ChTpyt0dcX69Rtk4cKFvlnRoP+dU6ZMdaNl/cUC/yLKQDNde6wbQZROV4we/ah77SeTJ0921507d7or/IkoI+7pF3izZs10qyt0va9fzy/WLyH1f8O2bdu8O/Ajooy4pZsvdP44co6ExtnvGzCysye5Q5F0owv8iSgjLmm0dEdeZP44KAf7DBv2kLvu2bPHXeE/RBlxR4M8bdpUtztv8+ZXfDl/fC1MYfgfUUZc0cPiNcjqxRdfCuQjlnR5nE5hsArDn4gy4oYGecqU37gNIRrkoD6QNCUlxV0/+OADd4W/EGXEBV3ypkHWf9r7ZUNIR0UOStJt4vAfoozAa73kbdWqVb5fYdEeOTlP8GWfTxFlBJoGWZe86ZbpeAmyGj58uJtX1mV/8BeijMCKBFnXIPtpy3RnSEpKcteGBn88BRxXEWUEUusgx+PDRfv37++uNTU17gr/IMoInHgPsoqc+9zY2Oiu8A+ijEAhyFfpl30nTpzw3sEviDICgyB/3T333CNVVW977+AXRBmBQJC/bdCgQe6hqvAXogzfI8ht6927t7tyYpy/EGX4GkG+tl69ernrF1984a7wB6IM3wqHwwT5Ou688053Jcr+QpThS3q4UF7ePLd1miC3LXK+R11dnbuii5wOy+xQooRCw+XJcL1ckoty+sBfZPaQ5nsPrJKDX3l/rp2IMnwnctpb5CwLIKb6ZMnz9e9LOL+f7Fy/Ww4d2Chra34qa/7dIPWHCmTY7d6fayeiDF/RJ063DnI8bZ2GZd+TYVOnyy8O/Uke/2tvyX0iRXp4n9wsogzf0FUEs2f/TpKSBhDkdtINJMeOHfPeoS36ZbH+sr9lvYfIqAf6SvpjD0nfWygrUYYv6FM0Wj8xhCC339mzZ71XaMtdd90lmZkZUl5e3gmn6p2VqgMfyTnvXUcQZfjC4sWL3UaIID8xBLGRk5Mje/ful6qqKnnssTEdfDjARTm1bavU/nCkyKv/lIPnLnn3bx5Rhi/oU6f1IacEGV1Bf650GiM/v0Dy8ubLihUrbuIZh5fk/LG/y7r6TClcOEMmyUfyyZnzcipcJn+r/dL7M+3X7XIz73WbQqFE7xUAxJdDh45cOXGvLZdqN0pW5jK5+NsVsrZwgiT3aJI3FmfJE68Okj9uKpF5P+lz0yPfG0YZt05/sZWWLnf/TEL7RXbrqfp6DmtH19M55S1btkhR0aKY/X+W6Yso6dGjowtk4lPr3XpANOhc8vTp093c8uuv74nZIIoow5zIbj1dzsVuPXQ1HR3rHPL48eNk2rRp7l9okSeCxwJRhimtd+sVFxd7d4GuoysudNmgzh9nZWV5d2OHKMOMb26fZi0yokGXWZaWll73C71oIspRwK6qGyPIiBVryyyJcpSwq+raCDJwFVFGTBFk4OuIMmKGIAPfRpSjYPjw4VJevsl7B0WQgbYRZURdZeVuggxcA1GOAp4qfJUuzJ81ayZBBq6BKEcBTxVuETnLQrdOE2SgbUQ5ChITW07aq6mpcdd41DrIunWaIANtI8pRoAHSRxg1NjZ6d+KHniugEW4dZADXRpSjJD19lBw5csR7Fx80yAUFBe6A+rKypwky0A5EOUp0WZzG6dafAeYP+qWmHoMYeWKIhYNeAD8gylGSlJTkrg0NwT+sXYOsDzk9c+a0bN++Q9LS0rxPANwIUY6SgQMHuuv+/fvdNah0U8jIkQ+713r6VmpqqnsNoH2IcpTol326NlefahBU+oj2yKaQrVvDPOQU6ACiHEXp6elujrX9T8n1h8iTG/S5ZnpMqa5BtnI2LeA3RDmKMjIy3PXgwXfdNQj0F4yusFi3bq1b8qaHhbMGGeg4ohxF+s/5ESPSZMuWV707/lZbWysTJ2ZdWWHBkjfg1hHlKJswYUIgpjD0adOZmS0j/71797PCAugkRDnKxo4d6647d+50V7+JzB9Hnja9a1cFX+gBnYgoR5l+AaYxe+65Z323kUSnK3RDiM4fl5YuZ/4Y6AJEOQZmzpwpdXUfS0VFhXfHvsh0RWRDSE5OjvcJgM5ElGMgOTnZjZZXr15lfrSsc99FRUVXpit0/TEbQoCuQ5RjJDJaLisr8+7Yo08I0dUV+igrPVBIpytYfwx0LaIcIzpa1nlZnZ89fPiwd9eGyOhYnxAydGiKW13BgUJAdHS73Mx7jSjTqQv94kznaXUVg4UvzXTuWKdVWkbxT5uIcSiUKPX1wT/ICVCMlGNII1xSUuICqLviYjm/rCsrdPOHzh0zOgZihyjHmE5j6G443VCyceNG72706FSFrjvWlRVHj9bI+vUb3KObWHsMxAbTF0boCWuRA32Ki4u7fCpDY/zyyy+7xzQpnd/Ozs42ue6Y6QvEE6JsiK52iDx+/6mnnuqS0eo3Y5ybO1/mzJljelUFUUY8IcrG6CHxhYWL3Ov8/IJOm9fVOePKysqvxXjSpElu+sQ6oox4QpQN0tHsypUr3fpgPVVuwYIF8uCDD9701IL+PXpM6LPPPif79lW7e3q85uTJk3213pgoI54QZcN01LxmzRoX1KSkATJlylS3m27w4MFtRlWfjXf8+HG37nnPnj1XQqzTIdnZk9wTtS3OGd8IUUY8Ico+oJHVQG/e/JJbPtce+oXhz3+eIcOGPeT7XXhEGfGEKPuMzg0fP/6JnD//uXfnKn1ids+ePQO3nI0oI54QZZhHlBFP2DwCAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMIQoA4AhRBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMIQoA4AhRBkADCHKAGAIUQYAQ4gyABhClAHAEKIMAIYQZQAwhCgDgCFEGQAMIcoAYAhRBgBDiDIAGEKUAcAQogwAhhBlADCEKAOAIUQZAAwhygBgCFEGAEOIMgAYQpQBwBCiDACGEGUAMKTb5WbeawBAjDFSBgBDiDIAGEKUAcAQogwAhhBlADBD5P9XEnnjzfwFdgAAAABJRU5ErkJggg=="></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></math> is an odd function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The range of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>b</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[6]</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="f\left( x \right) = \frac{{2x + 6}}{{{x^2} + 6x + 10}}{\text{,}}\,\,x \in \mathbb{R}.">
<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>6</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>.</mo>
</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="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> has no vertical asymptotes.</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 equation of the horizontal asymptote. </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {f\left( x \right)} \,dx">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</math></span>, giving the answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,q{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>p</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mi>p</mi></math> has two real, distinct roots.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the possible values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</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>Consider the case when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>4</mn></math>. The roots of the equation can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>±</mo><msqrt><mn>13</mn></msqrt></mrow><mn>6</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</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="question">
<p>The following diagram shows 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>. The graph has a horizontal asymptote at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span>. The graph crosses the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>On the following set of axes, sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\left[ {f\left( x \right)} \right]^2} + 1">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>[</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span>, clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAxYAAAJCCAYAAABd1lHyAAAgAElEQVR4Ae3dD5jVdZ0v8M8BrlKL4JrddUZNN7mgba4V/tknfXYRAvTpyppsZm4uBrXXQZQrT8ribre6uZjkg2kKmomi2W5sztX+g0Xce6NWYtS29iazWtnKjFs+Lk6zpaTn3OcMDAw4HA5zzvzm9+c1zwMz5/fn+/1+Xp8fZ3ifPzOlSqVSCR8ECBAgQIAAAQIECBBoQGBUA+c6lQABAgQIECBAgAABAn0CgoULgQABAgQIECBAgACBhgXG9I9QKpX6v/SZAAECBAgQIECAAAECgwrs750Uu4NF9YBquNjfgYOOaiMBAgQIECBAgAABAoUQOFBW8FKoQlwGiiRAgAABAgQIECAwvAKCxfD6Gp0AAQIECBAgQIBAIQQEi0K0WZEECBAgQIAAAQIEhldAsBheX6MTIECAAAECBAgQKISAYFGINiuSAAECBAgQIECAwPAKCBbD62t0AgQIECBAgAABAoUQECwK0WZFEiBAgAABAgQIEBheAcFieH2NToAAAQIECBAgQKAQAoJFIdqsSAIECBAgQIAAAQLDKyBYDK+v0QkQIECAAAECBAgUQkCwKESbFUmAAAECBAgQIEBgeAUEi+H1NToBAgQIECBAgACBQggIFoVosyIJECBAgAABAgQIDK+AYDG8vkYnQIAAAQIECBAgUAgBwaIQbVYkAQIECBAgQIAAgeEVECyG19foBAgQIECAAAECBAohIFgUos2KJECAAAECBAgQIDC8AoLF8PoanQABAgQIECBAgEAhBASLQrRZkQQIECBAgAABAgSGV0CwGF5foxMgQIAAAQIECBAohIBgUYg2K5IAAQIECBAgQIDA8AoIFsPra3QCBAgQIECAAAEChRAQLArRZkUSIJBbge72mFsqRan65+yboqO3vLPUng2xpPXkmN/+dOzaklsChREgQIBAOgQEi3T0wSoIECAwNIGWC2JN5Tex9a53R2z8Xvyga8fOccZNjlmXHhfPbP/N0MZ1FgECBAgQOEgBweIgwRxOgACB9AmMjTeecnqcEL+M53/18s7ljXpdHDPxD2LGKUeFO/r0dcyKCBAgkEcB32/y2FU1ESBQOIFRhx0RJ8S/xg+f/ve+2svbvhYrNp8R733r4YWzUDABAgQIjIyAYDEy7mYlQIBAUwVGHXZ4HLV7xO3x2NrOmPnh/xpHu5ffreILAgQIEBheAd9yhtfX6AQIEEhG4HcOj6NanopNP/l5/OuGz8YDx703zj/6kGTmNgsBAgQIEIjw0ltXAQECBHIh8NoJ8frXRvz6sdXxqY2nxBXnH+cOPheNVQQBAgSyI+AZi+z0ykoJECCwf4FRvxNHnNASMeaP44PXTI8W9+77t7KHAAECBIZFwLeeYWE1KAECBEZA4JjL43OfeG+cOM5d+wjom5IAAQKFF/Ddp/CXAAACBLIvsD06bv5CxNVXxLQW76vIfj9VQIAAgWwKjMnmsq2aAAECRRfYHh03vi/O674g7j7mZ/H//nhxXHXi+KKjqJ8AAQIERlDAMxYjiG9qAgQIDF3g5Xjhl89G96Nb49k/botFU/y+iqFbOpMAAQIEmiFQqlQqlf6BSqVSDLjZv9lnAgQIECBAgAABAgQKLnCgrOAZi4JfIMonQIAAAQIECBAg0AwBwaIZisYgQIAAAQIECBAgUHABwaLgF4DyCRDIl8Ctt94a1T8+CBAgQIBA0gJ+KlTS4uYjQIDAMAlUA8UVV1yxe/SFCxfu/toXBAgQIEBguAW8eXu4hY1PgACBBATuvffemDt37l4zffrTnw7hYi8SNwgQIECgAQFv3m4Az6kECBDIgsCmTZv6QsVpp522e7nVr6vPXlT3+SBAgAABAkkIeI9FEsrmIECAwDAJVIPDWWedFdUg0d7evnuW6tfVbdV9wsVuFl8QIECAwDAKeCnUMOIamgABAsMp0B8qqnM89thj8Za3vCWqT1NXP6q/k6izszMmT57cd/s73/lOnHnmmX1f+4sAAQIECAxF4EAvhRIshqLqHAIECIywwPPPPx+ve93r+lYxMDQMDBbVnQPDx9atW2PSpEkjvHLTEyBAgEBWBQ4ULPxUqKx21roJECi0wBFHHBHVN2e/9a1vrflMRPVZimrwqD6jceyxxxbaTPEECBAgMLwCnrEYXl+jEyBAIFGBfZ+xSHRykxEgQIBArgUO9IyFN2/nuv2KI0CAAAECBAgQIJCMgGCRjLNZCBAgQIAAAQIECORaQLDIdXsVR4AAAQIECBAgQCAZAcEiGWezECBAgAABAgQIEMi1gGCR6/YqjgABAgQIECBAgEAyAoJFMs5mIUCAAAECBAgQIJBrAcEi1+1VHAECBAgQIECAAIFkBASLZJzNQoAAAQIECBAgQCDXAoJFrturOAIECBAgQIAAAQLJCAgWyTibhQABAgQIECBAgECuBQSLXLdXcQQIECBAgAABAgSSERAsknE2CwECBAgQIECAAIFcCwgWuW6v4ggQIECAAAECBAgkIyBYJONsFgIECDRRYEd0d3wtvnjj3GgtlaJ1yYboaeLohiJAgAABAkMRECyGouYcAgQIjJRAuTs23/TBmHLeF+Jnx78/Nv7qlei6YVqMH6n1mJcAAQIECOwSGEOCAAECBLIisD06Vnwwzv/hufFgxwfj9JZDsrJw6yRAgACBAggIFgVoshIJEMiDQDl6O+6OD604Pm79vlCRh46qgQABAnkT8FKovHVUPQQI5FOg3Blrr70t4pJTovx3H+x7b0WpdW7c+HBn9OazYlURIECAQMYEBIuMNcxyCRAopkD5ye/GF9afEKe/6ZQ4c/Ga6Kq8ED++bkysmLk4bu/YXkwUVRMgQIBAqgQEi1S1w2IIECAwmEA5ep95Mn7YMiVmvevUaOm75x4fJ156dVw389G4+tr26CwPdp5tBAgQIEAgOQHvsUjOupAzrVq1qpB1K5pAcwV+G7/47rro/vXr40t33R5bx5Z2Df/b+MV/bon48tqYPHr+XlOWSjuPWbly5V7b3SBAYGQF2traRnYBZicwjAKCxTDiGnqnQN7uRKthSU3ZvLqz27ty9Ex+Nu743JPxh++cF/Mmjd3VgBej89DvxEd/cU7E+nWDNiVv12p/kdntZX8F9X1WZ31OWTmq2k8fBPIs4KVQee6u2ggQyInAqBh/6jvikpYfx6Yf/VvsedVTbzyzdVvMfM/b45VKJSqVyu56q18PvL17hy8IECBAgMAwCQgWwwRrWAIECDRVYPxZceWtfxJfX/g/454nqr9ne0d0b/5CrHn8z2LZhZPCnXlTtQ1GgAABAkMQ8L1oCGhOIUCAQPICh8TRFyyLjWveHP972oQolY6Pix8YH1fec3lMGeeuPPl+mJEAAQIE9hXwHot9RdwmQIBAagXGx6QZV8WarqtiTWrXaGEECBAgUFQBD3MVtfPqJkCAAAECBAgQINBEAcGiiZiGIkCAAAECBAgQIFBUAcGiqJ1XNwECBAgQIECAAIEmCggWTcQ0FAECBAgQIECAAIGiCggWRe28ugkQIECAAAECBAg0UUCwaCKmoQgQIECAAAECBAgUVUCwKGrn1U2AAAECBAgQIECgiQKCRRMxDUWAAAECBAgQIECgqAKCRVE7r24CBAgQIECAAAECTRQQLJqIaSgCBAgQIECAAAECRRUQLIraeXUTIECAAAECBAgQaKKAYNFETEMRIECAAAECBAgQKKqAYFHUzqubAAECBAgQIECAQBMFBIsmYhqKAAECBAgQIECAQFEFBIuidl7dBAgQIECAAAECBJooIFg0EdNQBAgQIECAAAECBIoqIFgUtfPqJkCAAAECBAgQINBEAcGiiZiGIkCAAAECBAgQIFBUAcGiqJ1XNwECBAgQIECAAIEmCggWTcQ0FAECBAgQIECAAIGiCggWRe28ugkQIECAAAECBAg0UUCwaCKmoQgQIECAAAECBAgUVUCwKGrn1U2AAAECBAgQIECgiQKCRRMxDUWAAAECBAgQIECgqAKCRVE7r24CBAgQIECAAAECTRQQLJqIaSgCBAgQIECAAAECRRUQLIraeXUTIECAAAECBAgQaKKAYNFETEMRIECAAAECBAgQKKqAYFHUzqubAAECBAgQIECAQBMFBIsmYhqKAAECBAgQIECAQFEFBIuidl7dBAgQIECAAAECBJooIFg0EdNQBAgQIECAAAECBIoqIFgUtfPqJkCAAAECBAgQINBEAcGiiZiGIkCAAAECBAgQIFBUAcGiqJ1XNwECBAgQIECAAIEmCggWTcQ0FAECBAgQIECAAIGiCggWRe28ugkQIECAAAECBAg0UUCwaCKmoQgQIECAAAECBAgUVUCwKGrn1U2AAAECBAgQIECgiQKCRRMxDUWAAAECBAgQIECgqAKCRVE7r24CBAgQIECAAAECTRQQLJqIaSgCBAgQIECAAAECRRUQLIraeXUTIECAAAECBAgQaKKAYNFETEMRIECAAAECBAgQKKqAYFHUzqubAAECBAgQIECAQBMFBIsmYhqKAAECBAgQIECAQFEFBIuidl7dBAhkU6BnQyxpLUWp1P/nwljd+WI2a7FqAgQIEMiVgGCRq3YqhgCBfAvsiG3fbI/7ugdUOfOcOGvi2AEbfEmAAAECBEZGYMzITGtWAgQIEDhogd7H4+/uOCI+98IrMW28x4UO2s8JBAgQIDCsAoLFsPIanAABAs0SKEfP5gdjxfp7YvLfHhvbZ/1JzJw2KcY1a3jjECBAgACBBgU85NUgoNMJECCQiEC5M754wz3RHd2xcflfxpzpU+O8Je3R2VtOZHqTECBAgACBAwkIFgcSsp8AAQJpEBh1Ysxb1xWVV7piy4OfiWumRmxcvjD+2+1bojdi95u5+5fa/+bu/ts+EyBAgACB4RYoVSqVSv8k1W9EA272b/aZwJAFVq1aNeRznUiAQA2ByvbY+qX74uZ/fEMs+sjsuHnx5YMevHLlykG320iAwMgItLW1jczEZiXQBIEDZQXvsWgCsiFqC+TtTrQaltRUu+dp3Zu73r3vjDj0xGsiTrkwKpUFfezVO/3qR94fJMpdL/fzj0ad+4HJ6OZqP30QyLOAYJHn7qqNAIF8C4xrjcknT444xlu4891o1REgQCAbAt5jkY0+WSUBAgReLdDbFU9OmR9/NsnvsXg1ji0ECBAgkLSAYJG0uPkIECAwBIFyd0c89FBHdO/6IVDl7k1x098+Gu+44qwYP4TxnEKAAAECBJotIFg0W9R4BAgQGBaB5+ORT50XraNLUWqdGys2/Tbe+eHFMa3lkGGZzaAECBAgQOBgBbzH4mDFHE+AAIEREBjVMiOWfbsrlo3A3KYkQIAAAQL1CHjGoh4lxxAgQIAAAQIECBAgUFNAsKjJYycBAgQIECBAgAABAvUICBb1KDmGAAECBAgQIECAAIGaAoJFTR47CRAgQIAAAQIECBCoR0CwqEfJMQQIECBAgAABAgQI1BQQLGry2EmAAAECBAgQIECAQD0CgkU9So4hQIAAAQIECBAgQKCmgGBRk8dOAgQIECBAgAABAgTqERAs6lFyDAECBAgQIECAAAECNQUEi5o8dhIgQIAAAQIECBAgUI+AYFGPkmMIECBAgAABAgQIEKgpIFjU5LGTAAECBAgQIECAAIF6BASLepQcQ4AAAQIECBAgQIBATQHBoiaPnQQIECBAgAABAgQI1CMgWNSj5BgCBAgQIECAAAECBGoKCBY1eewkQIAAAQIECBAgQKAeAcGiHiXHECBAgAABAgQIECBQU0CwqMljJwECBAgQIECAAAEC9QgIFvUoOYYAAQIECBAgQIAAgZoCgkVNHjsJECBAgAABAgQIEKhHQLCoR8kxBAgQIECAAAECBAjUFBAsavLYSYAAAQIECBAgQIBAPQKCRT1KjiFAgAABAgQIECBAoKaAYFGTx04CBAgQIECAAAECBOoRECzqUXIMAQIECBAgQIAAAQI1BQSLmjx2EiBAgAABAgQIECBQj4BgUY+SYwgQIECAAAECBAgQqCkgWNTksZMAAQIECBAgQIAAgXoEBIt6lBxDgAABAgQIECBAgEBNAcGiJo+dBAgQIECAAAECBAjUIyBY1KPkGAIECBAgQIAAAQIEagoIFjV57CRAgAABAgQIECBAoB4BwaIeJccQIECAAAECBAgQIFBTQLCoyWMnAQIECBAgQIAAAQL1CAgW9Sg5hgABAgQIECBAgACBmgKCRU0eOwkQIECAAAECBAgQqEdAsKhHyTEECBAgQIAAAQIECNQUECxq8thJgAABAgQIECBAgEA9AoJFPUqOIUCAAAECBAgQIECgpoBgUZPHTgIECBAgQIAAAQIE6hEQLOpRcgwBAgQIECBAgAABAjUFBIuaPHYSIECAAAECBAgQIFCPgGBRj5JjCBAgQIAAAQIECBCoKSBY1OSxkwABAgQIECBAgACBegQEi3qUHEOAAAECBAgQIECAQE0BwaImj50ECBBIqUD56Wiff2rMWv1ElFO6RMsiQIAAgWIJCBbF6rdqCRDIhcCO2PbgJ2Ph6q5cVKMIAgQIEMiHgGCRjz6qggCBwgiUo7fj9rh206Exo6UwRSuUAAECBDIgIFhkoEmWSIAAgd0CvVvi9lsirlx8Thy1e6MvCBAgQIDAyAsIFiPfAysgQIBAnQLbo+P2z0dc+Rcx5bDRdZ7jMAIECBAgkIyAYJGMs1kIECDQoED1JVD3xi1xcVw25fBXjVUqlaL6p/9j39v9230mQIAAAQLDJVCqVCqV/sGr34gG3Ozf7DOBIQusWrVqyOc6kQCBAQIv/Sw2fKMn3jb7D+Pwan54cWs8+LF74l/P+++x8O2/F5cvWDDg4D1frly5cs8NXxEgMOICbW1tI74GCyAwVIEDZYUxQx3YeQTqFcjbnWg1LKmp3u6n67js9m57dNx0e/zeXR+PC44+ZCdqz4b42SfXRpx9UVw278RYsOs/K/3PWuT9QaLs9vLg/k2o8+C80n50tZ8+CORZwEuh8txdtREgkA+Bnkdj7SeXxpxjDu17uVPfy5wmTI/l3d2xfv5JMbp0YazufDEftaqCAAECBDIrIFhktnUWToBAYQTGT4sbuip9L1WtPhPR9+eFb8U1LS0x864fxyuVtTFv0tjCcCiUAAECBNIpIFiksy9WRYAAAQIECBAgQCBTAoJFptplsQQIECBAgAABAgTSKeDN2+nsi1URIECgtkDfy6O6ah9jLwECBAgQSFDAMxYJYpuKAAECBAgQIECAQF4FBIu8dlZdBAgQIECAAAECBBIUECwSxDYVAQIECBAgQIAAgbwKCBZ57ay6CBAgQIAAAQIECCQoIFgkiG0qAgQIECBAgAABAnkVECzy2ll1ESBAgAABAgQIEEhQQLBIENtUBAgQIECAAAECBPIqIFjktbPqIkCAAAECBAgQIJCggGCRILapCBAgQIAAAQIECORVQLDIa2fVRYAAAQIECBAgQCBBAcEiQWxTESBAgAABAgQIEMirgGCR186qiwABAgQIECBAgECCAoJFgtimIkCAAAECBAgQIJBXAcEir51VFwECBAgQIECAAIEEBQSLBLFNRYAAAQIECBAgQCCvAoJFXjurLgIECBAgQIAAAQIJCggWCWKbigABAgQIECBAgEBeBQSLvHZWXQQIECBAgAABAgQSFBAsEsQ2FQECBAgQIECAAIG8CggWee2suggQIECAAAECBAgkKCBYJIhtKgIECBAgQIAAAQJ5FRAs8tpZdREgQIAAAQIECBBIUECwSBDbVAQIECBAgAABAgTyKiBY5LWz6iJAgAABAgQIECCQoIBgkSC2qQgQIECAAAECBAjkVUCwyGtn1UWAAAECBAgQIEAgQQHBIkFsUxEgQIAAAQIECBDIq4BgkdfOqosAAQIECBAgQIBAggKCRYLYpiJAgAABAgQIECCQVwHBIq+dVRcBAgQIECBAgACBBAUEiwSxTUWAAAECBAgQIEAgrwKCRV47qy4CBAgQIECAAAECCQoIFglim4oAAQIECBAgQIBAXgUEi7x2Vl0ECBAgQIAAAQIEEhQQLBLENhUBAgQIECBAgACBvAoIFnntrLoIECBAgAABAgQIJCggWCSIbSoCBAgQIECAAAECeRUQLPLaWXURIECAAAECBAgQSFBAsEgQ21QECBAgQIAAAQIE8iogWOS1s+oiQIAAAQIECBAgkKCAYJEgtqkIECBAgAABAgQI5FVAsMhrZ9VFgAABAgQIECBAIEEBwSJBbFMRIECAAAECBAgQyKuAYJHXzqqLAAECBAgQIECAQIICgkWC2KYiQIAAAQIECBAgkFcBwSKvnVUXAQIECBAgQIAAgQQFBIsEsU1FgACBhgTK3bH5prnRWipFqTQrlrQ/Eb0NDehkAgQIECDQPAHBonmWRiJAgMAwCmyPxx7cFHHRndFVeSm6Hpkdzy58X3x8w3PDOKehCRAgQIBA/QKCRf1WjiRAgMCICZSf+ud44YzZcXrLIRFxSLSc/p6Ye0nEfev+KXpGbFUmJkCAAAECewQEiz0WviJAgEBqBUadcGZMPboaKnZ9lJ+Lnz1+bCy+8G0xvn+bzwQIECBAYAQFSpVKpdI/f6lUigE3+zf7TGDIAqtWrRryuU4kQGAwgUq89IvO6Pjuo/HLk86N2ZMPj1JELFiwYLCDY+XKlYNut5EAgZERaGtrG5mJzUqgCQIHygpjmjCHIQjUFMjbnWg1LKmpZstTuzP7vXsuNiw5J6Yv74iIlpj6lsnxjkvOj0njRu03WOTtWu2/uLLfy/5Kan9WZ22frO2t9tMHgTwLeClUnrurNgIEciZwZEy7YUtUXumKLfdcErF8Tkxd9GBsK0ffs80Dn3Gufj3wds4glEOAAAECKRQQLFLYFEsiQIBATYFRLTFl7sfijrveHd1f3xJbe8s1D7eTAAECBAgkISBYJKFsDgIECDRdYGxMPOucmNn0cQ1IgAABAgSGJiBYDM3NWQQIEBhhgXL0PvNkPHXuqTF5nLvyEW6G6QkQIEAgInw3chkQIEAg9QI7Ylv7wmg9e0ms6eiO6gufyt3fik/c8Hxce/U74mj35KnvoAUSIECgCAK+HRWhy2okQCDjAmNiwptOixlbl8elp7bG6NLJ8f77fxVz7rkl5p3ot1hkvLmWT4AAgdwI+HGzuWmlQggQyK/AqBh34txY0zU31uS3SJURIECAQMYFPGOR8QZaPgECBAgQIECAAIE0CAgWaeiCNRAgQIAAAQIECBDIuIBgkfEGWj4BAgQIECBAgACBNAgIFmnogjUQIECAAAECBAgQyLiAYJHxBlo+AQIECBAgQIAAgTQICBZp6II1ECBAgAABAgQIEMi4gGCR8QZaPgECBAgQIECAAIE0CAgWaeiCNRAgQIAAAQIECBDIuIBgkfEGWj4BAgQIECBAgACBNAgIFmnogjUQIECAAAECBAgQyLiAYJHxBlo+AQIECBAgQIAAgTQICBZp6II1ECBAgAABAgQIEMi4gGCR8QZaPgECBAgQIECAAIE0CAgWaeiCNRAgQIAAAQIECBDIuIBgkfEGWj4BAgQIECBAgACBNAgIFmnogjUQIECAAAECBAgQyLiAYJHxBlo+AQIECBAgQIAAgTQICBZp6II1ECBAgAABAgQIEMi4gGCR8QZaPgECBAgQIECAAIE0CAgWaeiCNRAgQIAAAQIECBDIuIBgkfEGWj4BAgQIECBAgACBNAgIFmnogjUQIECAAAECBAgQyLiAYJHxBlo+AQIECBAgQIAAgTQICBZp6II1ECBAgAABAgQIEMi4gGCR8QZaPgECBAgQIECAAIE0CAgWaeiCNRAgQIAAAQIECBDIuIBgkfEGWj4BAgQIECBAgACBNAgIFmnogjUQIECAAAECBAgQyLiAYJHxBlo+AQIECBAgQIAAgTQICBZp6II1ECBAgAABAgQIEMi4gGCR8QZaPgECBAgQIECAAIE0CAgWaeiCNRAgQIAAAQIECBDIuIBgkfEGWj4BAgQIECBAgACBNAgIFmnogjUQIECAAAECBAgQyLiAYJHxBlo+AQIECBAgQIAAgTQICBZp6II1ECBAgAABAgQIEMi4gGCR8QZaPgECBAgQIECAAIE0CAgWaeiCNRAgQIAAAQIECBDIuIBgkfEGWj4BAgQIECBAgACBNAgIFmnogjUQIECAAAECBAgQyLiAYJHxBlo+AQIECBAgQIAAgTQICBZp6II1ECBAgAABAgQIEMi4gGCR8QZaPgECBAgQIECAAIE0CAgWaeiCNRAgQIAAAQIECBDIuIBgkfEGWj4BAkURKEdv57q4ce7JUSqVolSaFUvWbI7uclHqVycBAgQIpF1AsEh7h6yPAAECEVHe9o2486sRs2/7QVQqL0XXI7Pj2aXnx8UrNkcvIQIECBAgkAIBwSIFTbAEAgQI1BZ4MX76L0fEhYtmxaRx1bvtQ6Ll9Pnx19edFRtXPBibezxtUdvPXgIECBBIQmBMEpOYgwABAgQaERgbJ0z9o30GOCSOOn5itOyz1U0CBAgQIDBSAp6xGCl58xIgQKAJAq8999SY3PcsRhMGMwQBAgQIEGhAoFSpVCr951ffEDjgZv9mnwkMWWDVqlVDPteJBAjUEuiNrQ8+ED9/24Ux4w2viQULFgx68MqVKwfdbiMBAiMj0NbWNjITm5VAEwQOlBW8FKoJyIaoLZC3O9FqWFJT7Z6ndW9+eleO3o5b4/L/uCXu/dDpMS5iv8Eib9dq/7WVn172VzT4Z3UO7pLVrdV++iCQZwEvhcpzd9VGgEA+BXq3xJ1rfzeWXnZqX6ioFll9tnngM8773s4nhKoIECBAIE0CgkWaumEtBAgQOJBA+el46M6fxrkf/vM40XsrDqRlPwECBAgkKCBYJIhtKgIECDQkUN4WG27+ehx20bv2hIreH8Wa1Zuip6GBnRR+sswAACAASURBVEyAAAECBBoXECwaNzQCAQIEhl+g94loXzovpi9ui+mth+767dulKB02Mz4fr9v9kqjhX4gZCBAgQIDA4ALevD24i60ECBBIj0D56Whf9O6Ys/pHg6zprHjPWceHR4kGobGJAAECBBIVECwS5TYZAQIEhiAw6ri44K4fRuWuIZzrFAIECBAgkJCAB7kSgjYNAQIECBAgQIAAgTwLCBZ57q7aCBAgQIAAAQIECCQkIFgkBG0aAgQIECBAgAABAnkWECzy3F21ESBAgAABAgQIEEhIQLBICNo0BAgQIECAAAECBPIsIFjkubtqI0CAAAECBAgQIJCQgGCRELRpCBAgQIAAAQIECORZQLDIc3fVRoAAAQIECBAgQCAhAcEiIWjTECBAgAABAgQIEMizgGCR5+6qjQABAgQIECBAgEBCAoJFQtCmIUCAAAECBAgQIJBnAcEiz91VGwECBAgQIECAAIGEBASLhKBNQ4AAAQIECBAgQCDPAoJFnrurNgIECBAgQIAAAQIJCQgWCUGbhgABAgQIECBAgECeBQSLPHdXbQQIECBAgAABAgQSEhAsEoI2DQECBAgQIECAAIE8CwgWee6u2ggQIECAAAECBAgkJCBYJARtGgIECBAgQIAAAQJ5FhAs8txdtREgQIAAAQIECBBISECwSAjaNAQIECBAgAABAgTyLCBY5Lm7aiNAgAABAgQIECCQkIBgkRC0aQgQIECAAAECBAjkWUCwyHN31UaAAAECBAgQIEAgIQHBIiFo0xAgQIAAAQIECBDIs4Bgkefuqo0AAQIECBAgQIBAQgKCRULQpiFAgAABAgQIECCQZwHBIs/dVRsBAgQIECBAgACBhAQEi4SgTUOAAAECBAgQIEAgzwKCRZ67qzYCBAgQIECAAAECCQkIFglBm4YAAQIECBAgQIBAngUEizx3V20ECBAgQIAAAQIEEhIQLBKCNg0BAgQIECBAgACBPAsIFnnurtoIECBAgAABAgQIJCQgWCQEbRoCBAgQIECAAAECeRYQLPLcXbURIECAAAECBAgQSEhAsEgI2jQECBAgQIAAAQIE8iwgWOS5u2ojQIAAAQIECBAgkJCAYJEQtGkIECBAgAABAgQI5FlAsMhzd9VGgAABAgQIECBAICEBwSIhaNMQIECAAAECBAgQyLOAYJHn7qqNAAECBAgQIECAQEICgkVC0KYhQIAAAQIECBAgkGcBwSLP3VUbAQIECBAgQIAAgYQEBIuEoE1DgAABAgQIECBAIM8CgkWeu6s2AgRyJlCO3s6N0f7FG2PuxGtjQ085Z/UphwABAgSyLCBYZLl71k6AQKEEyp33xEUfuzseuPLquPfXhSpdsQQIECCQAQHBIgNNskQCBAhUBUZNmhdfuf+O+Mh17wZCgAABAgRSJyBYpK4lFkSAAAECBAgQIEAgewKCRfZ6ZsUECBAgQIAAAQIEUidQqlQqlf5VlUqlGHCzf7PPBIYssGrVqiGf60QCBAYT+G384rtr4qNffn0s+sjsmDy21HfQggULBjs4Vq5cOeh2GwkQGBmBtra2kZnYrASaIHCgrDCmCXMYgkBNgbzdiVbDkppqtjy1O/PRuxej89DvxEe/NTFmz78spo3f+cTz/oJF3q7V/osrH73sr2b/n9W5f5ss7qn20weBPAsIFnnurtoIECiMQP+zzdVHk6of/bcLA6BQAgQIEBhxAe+xGPEWWAABAgQIECBAgACB7AsIFtnvoQoIECBAgAABAgQIjLiAYDHiLbAAAgQIECBAgAABAtkXECyy30MVECBQFIGeDbGk9TUxef4/RHRfH9MnHBOzVj8R5aLUr04CBAgQSLWAN2+nuj0WR4AAgQEC46fFDV2VuGHAJl8SIECAAIG0CHjGIi2dsA4CBAgQIECAAAECGRYQLDLcPEsnQIAAAQIECBAgkBYBwSItnbAOAgQIECBAgAABAhkWECwy3DxLJ0CAAAECBAgQIJAWAcEiLZ2wDgIECBAgQIAAAQIZFhAsMtw8SydAgAABAgQIECCQFgHBIi2dsA4CBAgQIECAAAECGRYQLDLcPEsnQIAAAQIECBAgkBYBwSItnbAOAgQIECBAgAABAhkWECwy3DxLJ0CAAAECBAgQIJAWAcEiLZ2wDgIECBAgQIAAAQIZFhAsMtw8SydAgAABAgQIECCQFgHBIi2dsA4CBAgQIECAAAECGRYQLDLcPEsnQIAAAQIECBAgkBYBwSItnbAOAgQIECBAgAABAhkWECwy3DxLJ0CAAAECBAgQIJAWAcEiLZ2wDgIECBAgQIAAAQIZFhAsMtw8SydAgAABAgQIECCQFgHBIi2dsA4CBAgQIECAAAECGRYQLDLcPEsnQIAAAQIECBAgkBYBwSItnbAOAgQIECBAgAABAhkWECwy3DxLJ0CAAAECBAgQIJAWAcEiLZ2wDgIECBAgQIAAAQIZFhAsMtw8SydAgAABAgQIECCQFgHBIi2dsA4CBAgQIECAAAECGRYQLDLcPEsnQIAAAQIECBAgkBYBwSItnbAOAgQIECBAgAABAhkWECwy3DxLJ0CAAAECBAgQIJAWAcEiLZ2wDgIECBAgQIAAAQIZFhAsMtw8SydAgAABAgQIECCQFgHBIi2dsA4CBAgQIECAAAECGRYQLDLcPEsnQIAAAQIECBAgkBYBwSItnbAOAgQIECBAgAABAhkWECwy3DxLJ0CAAAECBAgQIJAWAcEiLZ2wDgIECBAgQIAAAQIZFhAsMtw8SydAgAABAgQIECCQFgHBIi2dsA4CBAgQIECAAAECGRYQLDLcPEsnQIAAAQIECBAgkBYBwSItnbAOAgQIECBAgAABAhkWECwy3DxLJ0CAAAECBAgQIJAWAcEiLZ2wDgIECBAgQIAAAQIZFhAsMtw8SydAgAABAgQIECCQFgHBIi2dsA4CBAgQIECAAAECGRYQLDLcPEsnQIAAAQIECBAgkBYBwSItnbAOAgQIECBAgECdAps2bYp77723zqMdRiAZgTHJTGMWAgQIECBAgACBZgj85je/iauuuiq+//3vR09PTyxcuLAZwxqDQMMCnrFomNAABAgQIECAAIHkBF7zmtdEe3t7nHbaaXHFFVfErbfeWnvy7vaYWypFqfrn7Juio7cc5e5NcdPck6NUOjtu7Oitfb69BOoUECzqhHIYAQIECBAgQCAtAsccc0xfuPjTP/3TvnCxbNmyqD6TMehHywWxpvJSPPPA5dGy8f5Yu/7LcfOnfhLn3vaDqFS+HR+aMm7Q02wkcLACpUqlUuk/qZpkB9zs3+wzgSEJVK+n6keerik1DelSSMVJeezdYLDqHEwlu9uK0M8i1Fi9AoerzmqYuPLKK+Ozn/1sfOADH4hbbrklqs9oDPrRsyGWnDg9lsfl8cD3V8QFRx8y6GE2EtifQPU6rvX/Ou+x2J+c7QQIECBAgACBlAtUQ0Q1TFQ/quGi+rHfcDH+D2PWJVNi+eNvize3CBV9WP5qqoBg0VROgw0m0P8ozWD7srpNTVnt3J5HDbNbQX0rz+M1Oljl6hxMJZvb9LI5fauGi1NOOWXwN3SX/yO2P/dSxPpvxHeevDgmTRrbnEmNQmCXgPdYuBQIECBAgAABAjkSmD9//iDV7IhtD66Mrx75JzE1fhJbn/GG7UGQbGpQwDMWDQI6/cACtV6Ld+Cz03VE/yNqakpXX+pZTR57N1jd6hxMJbvbitDPItRYvQKHq85632NR3vaV+Pj6M+J/3vqmWPd4e/zNusfiyglPxW1PnxvXXXBceKQ5u/cTaVq56yhN3cjZWqr/+V65cmWuqlJTdtuZx95ltxuNr7x635KngL8/kSLUWZR/m8NR5/PPP3/AN26/3HFjTCy1xvRbIhbfeH4cPeaoOGXG26J7+Y1xy9NT41qhYn///GwfgoBnLIaA5hQCBAgQIECAwEgKPPPMM3HBBRf0/ZK8T3/604O/pyIixkz5UDxZ+dCApR4eUz70ldhr04C9viTQiIBnLBrRcy4BAgQIECBAIGGB6suf6gkVCS/LdAS8pM41MBICO2Jb+8JonbU6OssjMX+T5ix3x+ab5kZr328znRVL2p+IbL8Vrhy9nevixr7fxFr9Da2zYsmazdGd5R7tt9XVWjdG+xdvjLkTr40NPVkpsic6H74p5rZW+9MaZy+5Lzq6d+y3ynzu6InODQ/Fow+viYlLNkRPHovs7YyHb9x537JgwdIc93lHdG++re963llne3T2ZuXf4hAvvMrz0T7/1Ji1+olopNLqj5i96aabotYzFUNcodMINCTgGYuG+Jw8FIHqG8j+x8LbonsoJ6fmnO3x2IObIi66M7oqL0XXI7Pj2YXvi49veC41KzzYhZS3fSPu/GrE7L7fxLqrpqXnx8UrNmc8ML1aotx5T1z0sbvjgSuvjnt//er96dzSE0+sviqmzvxWnPzlf4/Kr9bHJc8uj1Mvvi068v6fsd0NeTE6V18VH1tzT/zD/3okMtO63euv44vy0/HQnRsiZn86uiqVuO36D8TsXPa5HL2PfT3WxwVxd1d/nR+JqR/fmM+w2Nf6HbH9Bw/HwtVddVwIBz7kzDPP3O/Lnw58tiMIDI+AYDE8rkbdn0Dv5lhx7f+JI2e8eX9HZGJ7+al/jhfOmB2n9/2CoUOi5fT3xNxLIu5b908Z/ab42/jpvxwRFy6aFZPGVe8WqjXNj7++7qzYuOLB2JyZR/Tru3xGTZoXX7n/jvjIde+u74QUHFXe9s345N+sjrhmcfzllMMjxr0p/mzuedGy8ZNx7drOhh79TEF5dS5hbEyad1fcf/f1cd5JE+o8J1uHlX/6TEy4cF7MmDS+b+GlCSfEor++KmZuvD/Wbn4+W8XUWm356djywtviktNb+l46UZrwxpg/97yI+74ZW3J2f7OToRy9HbfHQ0+NiRkttWDsI5BtAcEi2/3L2Oq3R8ftn4+4ckHMOurQjK197+WOOuHMmHr0gN9aWn4ufvb4sbH4wrfFzv8O7H18+m/9pzhh6h/F0XvdIxwSRx0/MXwPTEP3Xown1/19rO4+IWac8V92XWOjYvyp74hLWrpj/Re+G0828rqKNJRoDX0Cr7pviYhRRx0fb8nbP8RRvx9Tpx474PXYL8ezP3s6Ji8+P04fv9cdUT6ujN4tcfstEWdP/4M4Kh8VqYLAoAI5/Nc7aJ02jrhA9dGae+OWuDgum3LEiK+meQuovlZ/Q6xe+ql4csmtsbj6SHLOPl577qkxue9ZjJwVlqlyeuOZrT+JiGPj5ON+d8/KXzshXv/aiFi/OX70by/v2e6r/Am89o/ijMnZfNjigM3o7Yyt3/1a/O2Tl8bnF58e4w54QtYO6H9Q7S/iDWNLWVu89RI4KAHB4qC4HDxkgV2P1lx52ak5+qbxXGxYcnocNnl6zF/+pfjeukfiyVy91v352LJuW1y2YNo+z2QM+Spw4pAFfhPbn90+5LOdmGWBSvRs2RiPX/YXMXPgs6RZLmn32svRs+HaaD1sctz8uXVx7/e+Hd97Mm9vxx/4oFr+Hnja3UpfENglIFi4FBIQ2B4dd26INy67LKbk6pHvI2PaDVui8kpXbLnnkojlc2LqogdjWy5eklL9Zvj5WPP6y+OyHD4Lk8BFbwoCzRF46en4zJojY1muHpTppxkV46ct6/sBGNf/1dy4Ju6JOVOvjfZtOfpJZ71b4s4HjotluXwmpr+PPhPYIyBY7LHw1cEIdLfH3L4fs1r9sZf7+3N2PPzzF6O34+/jgePeG+en/dG2umt6aW+pUS0xZe7H4o673h3dX98SW9P0rMVQa6p+M1z7u7E0K/+ZGWqde3cyxbdeE4cfVePRzpYj4vDfcXee4gYOcWnb4+eb/jmOWHppzh6U2ZfjkJjwhjPi+juui5nd/xiPbM3LsxY7H1Q77vJzPeu7b8vdzq2A70S5be0wF9ZyQaypVKJS88+3Y8YbXo7Naz8b1885PkbvDiCvj+nLOyLWz4/Jo1sb/nneTau07poGe+P52Jh41jkxs2mLadJAQ6mp78dd/jTO/fCfx4lZeYZpKHU2iTiZYcbFMZPfGBG/il++8OKeKX/5dPzwqYg4eWIck5Ve7Vm9r2oK7IhtD/19dP3B9Jh3Yk7fW7FP/aMmvj3eM3Ow+9d9DszKzZ5HY+0nl8acYw7d/QDcgsU3x/Lu7lg//6QYXbowVncO+Peclbqsk0ANAcGiBo5dzRAYt/PlQnsFkF/Gt66ZEjHzrtj6Slesm3figJ8M0ow5R2KMcvQ+82Q8lfU3Ope3xYabvx6HXfSuPaGi90exZvWmjP4Y3ZG4FoZjzrExcdZFMa+lY8CPNC5Hz4+3xMPREjPf8/aY6N58OOBHaMwd0b1hdaw97Lw44/fG7lpDTzyx5r7YmMsfxbqrxN6u2PrUW/PzJvXx0+KGrr0fgFu5YlFc09ISM+/6cbxSWRvzJvX3d4QuNdMSaLKAb0VNBjVcEQRe3vmbw89eEms6uvt+f0C5+1vxiRuej2uvfkd2n/LufSLal86L6YvbYnrrnkfYSofNjM/H63L0pvtsXqOjjp4WCxa/M7rvuz+++ERPlLu/F3et+XJ0T706ll04KQfhPJt9af6qe6Kz/aNx8fS2WDz9mLh8wYJdj3ZPiJM+/9tozckzU+Vt7TG/dVYsWbM5uqvvS6tsjw2fuD2evfayHL5JvflXiREJpFVAsEhrZ6wrxQKjY8KbTosZW5fHpae2xujSyfH++38Vc+65JbsvWag8H+2L3h1zlq8fxP2seM9Zx+frP649G2JJ62ti8vx/iOi+PqZPOCY9L8kbpAM7Nx0eUxbfGVuuPzLuO2lCjG69LB4/+cbY+uVFOX/9/UCQXT9FaPRJ8bkfvxDdy6fHhFy9nGRHbGu/NqbOuT42Diy77+t8PTM1asKkOHNGVyy/9IxoHV2Kpfd+P7bPuTnunvdmD2K8qvc2EMiOQKlSfZH8ro/qm3AH3Ozf7DOBIQusWrUq2trahnx+Gk9UUxq7Ut+a8ti7fSuv3o9XP/J+X16EXlb7qM59r/Bs3y5KP7PdJauvJXCgrOAZi1p69hEgQIAAAQIECBAgUJeAYFEXk4MIECBAgAABAgQIEKglIFjU0rGPAAECBAgQIECAAIG6BASLupgcRIAAAQIECBAgQIBALQHBopaOfQQIECBAgAABAgQI1CUgWNTF5CACBAgQIECAAAECBGoJCBa1dOwjQIAAAQIECBAgQKAuAcGiLiYHESBAgAABAgQIECBQS0CwqKVjHwECBAgQIECAAAECdQkIFnUxOYgAAQIECBAgQIAAgVoCgkUtHfsIECBAgAABAgQIEKhLQLCoi8lBBAgQIECAAAECBAjUEhAsaunYR4AAAQIECBAgQIBAXQKCRV1MDiJAgAABAgQIECBAoJaAYFFLxz4CBAgQIECAAAECBOoSECzqYnIQAQIECBAgQIAAAQK1BASLWjr2ESBAgAABAgQIECBQl4BgUReTgwgQIECAAAECBAgQqCUgWNTSsY8AAQIECBAgQIAAgboEBIu6mBxEgAABAgQIECBAgEAtAcGilo59BAgQIECAAAECBAjUJSBY1MXkIAIECBAgQIAAAQIEagkIFrV07CNAgAABAgQIECBAoC4BwaIuJgcRIECAAAECBAgQIFBLQLCopWMfAQIECBAgQIAAAQJ1CQgWdTE5iAABAgQIECBAgACBWgKCRS0d+wgQIECAAAECBAgQqEtAsKiLyUEECBAgQIAAAQIECNQSECxq6dhHgAABAgQIECBAgEBdAoJFXUwOIkCAAAECBAgQIECgloBgUUvHPgIECBAgQIAAAQIE6hIQLOpichABAgQIECBAgAABArUEBItaOvYRIECAAAECBAgQIFCXgGBRF5ODCBAgkAaBnujc8FB88ca5MXHJhuhJw5KsgQABAgQI7BIQLFwKBAgQyITAi9G5+qr42Jp74sqr741fZ2LNFkmAAAECRRIQLIrUbbUSIJBhgbExad5dcf/d18d1M1syXIelEyBAgEBeBQSLvHZWXQQIECBAgAABAgQSFBAsEsQ2FQECBAgQIECAAIG8CozJa2HqIkCAQJEESqXSXuX2365UKnttd4MAAQIECAyXQKky4LtO9RvRgJvDNadxCySwatWqAlWrVAIJCFT+Lb5766fiy8deGh85f3KM3TXlggULBp185cqVg263kQCBkRFoa2sbmYnNSqAJAgfKCp6xaAKyIWoL5O1OtBqW1FS752ndm67evRzd7Qujdc4dtblO+GRseeJDMaX/3rr8RBz60GfiW2+ZHfPbpsX4XWf3X5NFeaYiXb2s3cJG9qqzEb30nVvtpw8CeRbo/1aV5xrVRoAAgRQKjImWC26PSuX2FK7NkggQIECAwMELePP2wZs5gwABAgQIECBAgACBfQQEi31A3CRAgAABAgQIECBA4OAFBIuDN3MGAQIERkCgHD0bro3W0SfF/PXd0b18ekwoXRirO18cgbWYkgABAgQIvFrAeyxebWILAQIEUigwKsZPWxZdlWUpXJslESBAgACBCM9YuAoIECBAgAABAgQIEGhYQLBomNAABAgQIECAAAECBAgIFq4BAgQIECBAgAABAgQaFhAsGiY0AAECBAgQIECAAAECgoVrgAABAgQIECBAgACBhgUEi4YJDUCAAAECBAgQIECAgGDhGiBAgAABAgQIECBAoGEBwaJhQgMQIECAAAECBAgQICBYuAYIECBAgAABAgQIEGhYQLBomNAABAgQIECAAAECBAgIFq4BAgQIECBAgAABAgQaFhAsGiY0AAECBAgQIECAAAECgoVrgAABAgQIECBAgACBhgUEi4YJDUCAAAECBAgQIECAgGDhGiBAgAABAgQIECBAoGEBwaJhQgMQIECAAAECBAgQICBYuAYIECBAgAABAgQIEGhYQLBomNAABAgQIECAAAECBAgIFq4BAgQIECBAgAABAgQaFhAsGiY0AAECBAgQIECAAAECgoVrgAABAgQIECBAgACBhgUEi4YJDUCAAAECBAgQIECAgGDhGiBAgAABAgQIECBAoGEBwaJhQgMQIECAAAECBAgQICBYuAYIECBAgAABAgQIEGhYQLBomNAABAgQIECAAAECBAgIFq4BAgQIECBAgAABAgQaFhAsGiY0AAECBAgQIECAAAECgoVrgAABAgQIECBAgACBhgUEi4YJDUCAAAECBAgQIECAgGDhGiBAgAABAgQIECBAoGEBwaJhQgMQIECAAAECBAgQICBYuAYIECBAgAABAgQIEGhYQLBomNAABAgQIECAAAECBAgIFq4BAgQIECBAgAABAgQaFhAsGiY0AAECBAgQIECAAAECgoVrgAABAgQIECBAgACBhgUEi4YJDUCAAAECBAgQIECAgGDhGiBAgAABAgQIECBAoGEBwaJhQgMQIECAAAECBAgQICBYuAYIECBAgAABAgQIEGhYQLBomNAABAgQIECAAAECBAgIFq4BAgQIECBAgAABAgQaFhAsGiY0AAECBAgQIECAAAECgoVrgAABApkQ6InOh2+Kua2lKJVKUTp7Sazp6I5yJtZukQQIECBQBAHBoghdViMBAhkX2BHbHrovvhrvjNu6KlF5pSsemf1sLD31g7GiY3vGa7N8AgQIEMiLgGCRl06qgwCB/AqUt8W/TJgdi2ZMinHVKke1xOmLlsZ1Mx+NFWsfjZ78Vq4yAgQIEMiQwJgMrdVSCRAgUEyBUb8fU6fuU/qoI+P4t7Tus9FNAgQIECAwcgKesRg5ezMTIECgQYHXxblnnLDzWYwGR3I6AQIECBBoVECwaFTQ+QQIEBgJgZ5/inWPnxMLZh4b7shHogHmJECAAIF9BXw/2lfEbQIECKReYHt0fOZL8fpl748p43bejff9pKhSaffK9729e4cvCBAgQIDAMAmUKpVKpX/s6jeiATf7N/tMYMgCq1atGvK5TiSQb4FyvPD4F2LpZ/5v7TKPfFf81UdnxBt2PwxUiZd+/p34+jMnxLlvb41Dd529YMGCQcdZuXLloNttJEBgZATa2tpGZmKzEmiCwIGygmDRBGRDECBAICmB8ravxc3ffEN8cO6bvbciKXTzECBAgECfwIGCxe7HwHgRIECAQLoFyt0b4ua1Y+OiS/pDRTl6n/hirN74XLoXbnUECBAgUAgBwaIQbVYkAQLZFihHb2d7LL34fbF48fRoHb3rt2+XRsdhJ62NaO377RbZLtHqCRAgQCDzAn6PReZbqAACBPIuUN72YCyaOidWdw9S6cxz4qyJYwfZYRMBAgQIEEhWwHsskvU2GwECBAgQIECAAIFMCniPRSbbZtEECBAgQIAAAQIEsiXgPRbZ6pfVEiBAgAABAgQIEEilgGCRyrZYFAECBAgQIECAAIFsCQgW2eqX1RIgQIAAAQIECBBIpYBgkcq2WBQBAgQIECBAgACBbAkIFtnql9USIECAAAECBAgQSKWAYJHKtlgUAQIECBAgQIAAgWwJCBbZ6pfVEiBAgAABAgQIEEilgGCRyrZYFAECBAgQIECAAIFsCQgW2eqX1RIgQIAAAQIECBBIpYBgkcq2WBQBAgQIECBAgACBbAkIFtnql9USIECAAAECBAgQSKWAYJHKtlgUAQIECBAgQIAAgWwJCBbZ6pfVEiBAgAABAgQIEEilgGCRyrZYFAECBAgQIECAAIFsCQgW2eqX1RIgQIAAAQIECBBIpYBgkcq2WBQBAgQIECBAgACBbAkIFtnql9USIECAAAECBAgQSKWAYJHKtlgUAQIECBAgQIAAgWwJCBbZ6pfVEiBAgAABAgQIEEilgGCRyrZYFAECBAgQIECAAIFsCQgW2eqX1RIgQIAAAQIECBBIpYBgkcq2WBQBAgQIECBAgACBbAkIFtnql9USIECAAAECBAgQSKWA8/bW/wAAAR5JREFUYJHKtlgUAQIECBAgQIAAgWwJCBbZ6pfVEiBAgAABAgQIEEilgGCRyrZYFAECBAgQIECAAIFsCQgW2eqX1RIgQIAAAQIECBBIpYBgkcq2WBQBAgQIECBAgACBbAkIFtnql9USIECAAAECBAgQSKWAYJHKtlgUAQIECBAgQIAAgWwJCBbZ6pfVEiBAgAABAgQIEEilgGCRyrZYFAECBAgQIECAAIFsCZQqlUqluuRSqZStlVstAQIECBAgQIAAAQKJC+yKD6+ad0z/lv0d0L/fZwIECBAgQIAAAQIECOxPwEuh9idjOwECBAgQIECAAAECdQsIFnVTOZAAAQIECBAgQIAAgf0JCBb7k7GdAAECBAgQIECAAIG6Bf4/gRQN8XUi4SAAAAAASUVORK5CYII="></p>
<p> </p>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 4\,{\text{cos}}\,x + 1">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \leqslant x \leqslant \frac{\pi }{2}">
<mi>a</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a < \frac{\pi }{2}">
<mi>a</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - \frac{\pi }{2}"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>. Indicate clearly the maximum and minimum values of the function.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the least value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> has an inverse.</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>For the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> found in part (b), write down the domain of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}"> <mrow> <msup> <mi>g</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> found in part (b), find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}\left( x \right)"> <mrow> <msup> <mi>g</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</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^4} - 6{x^2} - 2x + 4">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
</math></span>, <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>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is translated two units to the left to form the function <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>.</p>
<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{x^4} + b{x^3} + c{x^2} + dx + e">
<mi>a</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>c</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>d</mi>
<mi>x</mi>
<mo>+</mo>
<mi>e</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<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">
<mi>d</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="e \in \mathbb{Z}">
<mi>e</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{x - 4}}{{2x - 5}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> </mfrac> </math></span>, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<br><hr><br><div class="question">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>c</mi></mtd></mtr><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>b</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>c</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math>.</p>
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></math>.</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>≥</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></math>, find an expression for the median of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
</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(x) = \frac{1}{{{x^2} + 3x + 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 2,{\text{ }}x \ne - 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<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>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>1</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="{x^2} + 3x + 2"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(x + h)^2} + k"> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>k</mi> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Factorize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </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>Sketch the graph of <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>, indicating on it the equations of the asymptotes, the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept and the local maximum.</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>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {f(x){\text{d}}x = \ln (p)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></span>.</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the area of the region enclosed between the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis and the lines with equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</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{{2{x^2} - 5x - 12}}{{x + 2}}{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \ne - 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>5</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>12</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find all the intercepts of the graph 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> with both the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> axes.</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>Write down the equation of the vertical asymptote.</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>As <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \pm \infty ">
<mi>x</mi>
<mo stretchy="false">→</mo>
<mo>±</mo>
<mi mathvariant="normal">∞</mi>
</math></span> the graph 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> approaches an oblique straight line asymptote.</p>
<p>Divide <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} - 5x - 12">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mi>x</mi>
<mo>−</mo>
<mn>12</mn>
</math></span> by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span> to find the equation of this asymptote.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</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(x) = 2{x^3} + 5,{\text{ }} - 2 \leqslant x \leqslant 2">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}(x)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</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>Write down the domain and range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</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>The quadratic equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 2kx + (k - 1) = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>k</mi>
<mi>x</mi>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> has roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\beta ">
<mi>β</mi>
</math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\alpha ^2} + {\beta ^2} = 4">
<mrow>
<msup>
<mi>α</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>β</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
</math></span>. Without solving the equation, find the possible values of the real number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
</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="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 src="data:image/png;base64,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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" 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>