File "HL-paper2.html"
Path: /IB QUESTIONBANKS/5 Fifth Edition - PAPER/HTML/Math AA/Topic 5/HL-paper2html
File size: 276.89 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 2</h2><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{{2\,{\text{ln}}\,x + 1}}{{x - 3}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span>, 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 3.</p>
</div>
<div class="specification">
<p>Draw a set of axes showing <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> values between −3 and 3. On these axes</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find the coordinates of the point of inflexion 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">[4]</div>
<div class="question_part_label">b.</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 any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f^{ - 1}}\left( x \right)"> <mi>y</mi> <mo>=</mo> <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>, showing clearly any axis intercepts and giving the equations of any asymptotes.</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>Hence, or otherwise, solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) > {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>></mo> <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>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<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) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>D</mi>
</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(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mo>[</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> to be a 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>Sketch the graph of <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> showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</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>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is an even function.</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>Explain why the inverse function <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> does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function <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> and state its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <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>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A scientist conducted a nine-week experiment on two plants, <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>, of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was given fertilizer regularly, while Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was not.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>sin</mi><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn><mo>)</mo><mo>+</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>27</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>B</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>8</mn><mi>t</mi><mo>+</mo><mn>32</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
</div>
<div class="specification">
<p>Use the scientist’s models to find the initial height of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</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>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> correct to three significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, prove that Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>, find the total amount of time when the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was greater than the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum 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">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</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>Find the total time in the interval 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ≥ 3.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </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>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</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> > 0, use your graph to find the values of <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>Find the coordinates where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> crosses the</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the vertical asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The oblique asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>30</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>30</mn></math>, clearly indicating the points of intersection with each axis and any asymptotes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac></math> in partial fractions.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>3</mn></munderover><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac><mo>d</mo><mi>x</mi></math>, expressing your answer as a single logarithm.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="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{sec}}\,x + 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>sec</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</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>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">[1]</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="f'\left( x \right)"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span>, stating its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The population, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, of a particular species of marsupial on a small remote island can be modelled by the logistic differential equation</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in years and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mo> </mo><mi>N</mi></math> are positive constants.</p>
<p>The constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> represents the maximum population of this species of marsupial that the island can sustain indefinitely.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> be the initial population of marsupials.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of the population model, interpret the meaning of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the population of marsupials will increase at its maximum rate when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math>. Justify your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the logistic differential equation, show that its solution can be expressed in the form</p>
<p style="padding-left:150px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years, the population of marsupials is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></math>. It is known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> for this population model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>7</mn>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \tan x">
<mi>u</mi>
<mo>=</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </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>Sketch a graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}"> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </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>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate(s) of the point(s) of inflexion of the graph of <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>, labelling these clearly on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <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.iii.</div>
</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="\sin x"> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu "><mi>u</mi></math></span>.</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>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>.</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 show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0"> <mrow> <msup> <mi>u</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>7</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>15</mn> <mi>u</mi> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </math></span>.</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>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span>, giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arctan k"> <mi>arctan</mi> <mo></mo> <mi>k</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the polynomial <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) \equiv {z^4} - 6{z^3} - 2{z^2} + 58z - 51,\,\,z \in \mathbb{C}">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>≡<!-- ≡ --></mo>
<mrow>
<msup>
<mi>z</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>58</mn>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mn>51</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {x^4} - 6{x^3} - 2{x^2} + 58x - 51"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>58</mn> <mi>x</mi> <mo>−</mo> <mn>51</mn> </math></span>, stating clearly the coordinates of any maximum and minimum points and intersections with axes.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, state the condition on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span> such that all roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = k"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> </math></span> are real.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A curve <em>C</em> is given by the implicit equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y - {\text{cos}}\left( {xy} \right) = 0">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy = - \frac{\pi }{2}">
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span> intersects <em>C</em> at P and Q.</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="\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of P and Q.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the gradients of the tangents to <em>C</em> at P and Q are <em>m</em><sub>1</sub> and <em>m</em><sub>2</sub> respectively, show that <em>m</em><sub>1</sub> × <em>m</em><sub>2</sub> = 1.</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>Find the coordinates of the three points on <em>C</em>, nearest the origin, where the tangent is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A large tank initially contains pure water. Water containing salt begins to flow into the tank The solution is kept uniform by stirring and leaves the tank through an outlet at its base. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> grams represent the amount of salt in the tank and let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> minutes represent the time since the salt water began flowing into the tank.</p>
<p>The rate of change of the amount of salt in the tank, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span>, is described by the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 10{{\text{e}}^{- \frac{t}{4}}} - \frac{x}{{t + 1}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>10</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mfrac>
<mi>x</mi>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> + 1 is an integrating factor for this differential equation.</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>Hence, by solving this differential equation, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\left( t \right) = \frac{{200 - 40{{\text{e}}^{ - \frac{t}{4}}}\left( {t + 5} \right)}}{{t + 1}}">
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>200</mn>
<mo>−</mo>
<mn>40</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> versus <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 60 and hence find the maximum amount of salt in the tank and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> at which this occurs.</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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> at which the amount of salt in the tank is decreasing most rapidly.</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>The rate of change of the amount of salt leaving the tank is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{{t + 1}}">
<mfrac>
<mi>x</mi>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span>.</p>
<p>Find the amount of salt that left the tank during the first 60 minutes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>2</mn><mi>x</mi></math>. It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method, with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math>, to find an approximate value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi></math> to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mi>v</mi><mo>-</mo><mn>2</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math>, suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>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> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis to form a solid of revolution that is used to model a water container.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the container is empty. Water is then added to the container at a constant rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></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 the coordinates of the endpoints.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the inverse function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></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>State the domain and range of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</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>Show that the volume, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup></math>, of water in the container when it is filled to a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> metres is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, determine the maximum volume of the container.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time it takes to fill the container to its maximum volume.</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>Find the rate of change of the height of the water when the container is filled to half its maximum volume.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>36</mn></mfrac><mo>+</mo><mfrac><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mn>16</mn></mfrac><mo>=</mo><mn>1</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>4</mn></math>.</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>The curve from point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> to point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis to form the interior surface of a bowl. The rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OPQR</mtext></math>, of height <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo> </mo><mtext>cm</mtext></math>, is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis to form a solid base.</p>
<p>The bowl is assumed to have negligible thickness.</p>
<p>Given that the interior volume of the bowl is to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>285</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>, determine the height of the base.</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) = {\text{sec}}\,x + 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>sec</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question">
<p>Use integration by parts to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\left( {{\text{ln}}\,x} \right)}^2}} {\text{d}}x">
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Two airplanes, <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>, have position vectors with respect to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> given respectively by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mtext mathvariant="bold-italic">A</mtext></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>19</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> represents the time in minutes and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Entries in each column vector give the displacement east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, the displacement north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the distance above sea level, all measured in kilometres.</p>
</div>
<div class="specification">
<p>The two airplanes’ lines of flight cross at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the three-figure bearing on which airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is travelling.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.</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>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the length of time between the first airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and the second airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> represent the distance between airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Xavier, the parachutist, jumps out of a plane at a height of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> metres above the ground. After free falling for 10 seconds his parachute opens. His velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\,{\text{m}}{{\text{s}}^{ - 1}}">
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds after jumping from the plane, can be modelled by the function</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v(t) = \left\{ {\begin{array}{*{20}{l}} {9.8t{\text{,}}}&{0 \leqslant t \leqslant 10} \\ {\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }},}&{t > 10} \end{array}} \right.">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>9.8</mn>
<mi>t</mi>
<mrow>
<mtext>,</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>t</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>t</mi>
<mo>></mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>His velocity when he reaches the ground is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2.8{\text{ m}}{{\text{s}}^{ - 1}}">
<mn>2.8</mn>
<mrow>
<mtext> m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find his velocity when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 15">
<mi>t</mi>
<mo>=</mo>
<mn>15</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the vertical distance Xavier travelled in the first 10 seconds.</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>Determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mtext>arctan</mtext><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mi>k</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that a finite limit only exists for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></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>Using l’Hôpital’s rule, show algebraically that the value of the limit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> moves in a straight line such that after time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, its velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>−</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mo> </mo><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> has displacement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>; at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>At successive times when the acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>0</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo> </mo></math>, the velocities of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> form a geometric sequence. The acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is zero at times <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub><mo><</mo><msub><mi>t</mi><mn>3</mn></msub></math> and the respective velocities are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>3</mn></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the times when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> comes to instantaneous rest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum displacement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, in metres, from its initial position.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> seconds of its motion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, at these times, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} = {\text{si}}{{\text{n}}^3}\,y">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant y \leqslant \pi ">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>y</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOAAAAGgCAYAAACzAms5AAAgAElEQVR4Ae2dwWscR/bH5z/IWexR9mUxa5vgHHISxPhgBZKL97BmFRNifAhR8M9kjA7GAWHMIhgwWRNYLFj7Eswy2MfNwpCLLwnzw4ccjG5LJAQ/jFmEYI0xw/vx2npST6tnprvndferet8GqWd6eqqrv68+86qrXlV1CBsUgAKtKdBp7cq4MBSAAgQAUQigQIsKAMAWxceloQAANFMGfqP+yiJ1Oh3qLHxJ/Z0373K2N6DuwjJtDP9jJqfIiJ4CAFBPS4WURrQ3WKOFzvvUHbx8l95ohwZrn9JK/zeF9JGENQUAoDWL7PZppbOYAu41bW3eos2t19ZyivwoKAAAFURUTeLtkDYWO7S4MaS3RDTa6dO1tQHtqV4EiVlRAABasYTkYwzA/9Cwd/foeVDOwT4aBQCgOVMeNMas/IO2ht/TWv/fNDKXR2RISwEAqKWkWjoHAC79iT5fe0o7oE9NWYsJAUBzVtmn4cYSdZa+pcHuQVeEuTwiQ1oKAEAtJdXSeUmD7gr6/dT0tJ0QADRlnze0O/ieeoMdPPeZskt9mQGA9WlbMOUR7Q97tLTwNW3+4y51N3+l/YLfxGnhKwAAW7ehRL9coG7/BeBr3R7NZgAANqs3rgYFxhQAgGNy4A0UaFYBANis3rgaFBhTAACOyYE3UKBZBQBgs3qXutp3331X6nycHJ4CANCozZ49e5YMzuU9tngVAIAGbfvf//6Xzp07lwDIe36PLU4FAKBBuz58+PDd1BQ8PUWnQ/1+32AukSUNBQCghoqKabx69WoMvmSOmE6H+Di2+BQAgMZs+s033+QCeOfOHWM5RXY0FACAGioqpcFejgGUP/Z+8pr38IJKQhtKBgAaMkY2KwwgtrgVgIUN2xcAGjaOUtYAoJKQdSQDAOtQ1VaaANCWPcZyAwDH5IjyDQA0bFYAaNg4SlkDgEpC1pEMAKxDVVtpAkBb9hjLDQAckyPKNwDQqFl/+eUXeu+994j32OJVAAAate2PP/6YRMTwHlu8CgBAo7YFgEYNo5wtAKgsqFZyT58+hQfUEtNwOgDQqHF4NDw3wmBUvFEDKWULACoJqZ0MANRW1GZ6ANCmXej69euJB+Q9tngVAIBGbfvFF18kAH722WdGc4hsaSgAADVUrCENfv7jv9///vc1pI4krSgAAK1YIpWP7LQUGIibEieylwDQoEGfP3+eeD/xgltbWwZziSxpKAAANVRUToNnQRP4eI9Z0ZQFNpQcAGzYGKPdZ9RbOXUA2Cla2fgnbe2PLwTP87+kAVxfX284l7hcUwoAwKaU5uvs/0y97vf0c7L2O6+G+y0tdRbowuaLsRVxeTLejz76KIHwD3/4A505c6bJXOJaDSoAABsU+/ilfqP+yiItdAe0d/Dh9vZ2At6tW7eS/R//+Mdkz8exxacAAGzTpnsD6i5+Sf2dN4e5kCDsH374IQHvxo0byR6jIg4liuoFAGzFnCPa3/onbaxcoO7g5VgO0s9+2ddjJ+JNFAoAwMbNuE/DjaXEq70D7BR93v938gzIi7DwMV4bQjzh7du3aXV1NTneeFZxwdoVAIC1SzzhAqNdGvY3aGWhQ52FNRrsjUj6/3ifBrDX6yUA8nFscSkAAFu152va2rxEnc416u++TYYecQsob2kAnzx5kgCIoUmtGquWiwPAWmQtmuiI9gZrtLDUo+H+KFkTUBZhyQJ4+fJldEcUlTWg8wBgi8Ya7f6L1paWaW2wQ/97EH4mK+JmAbx79y6qoS3aqq5LA8C6lD2W7oj2hz1aOhjl0Oks0FL3b/R0uJs0wMgAXFkNNwvg48ePUQ09pmn4BwCgERvys59UPzlLWQD5ORDVUCPGUswGAFQUs2pS6dZPSSMPQFRDRZ149gDQgC3Z80nrp2QnD0D2gidPnqR79+7JadgHrgAAbNmA0vme7WKYBODVq1eTZ0F5Vmw5+7j8nAoAwDkFnPfrAlo22FqOcyQMez75u3//fgKgtJbOe318v10FAGC7+tMnn3yS/GWzMQlABvGDDz6gK1euZL+C9wEqAABbNBpPNcGxnwxbdpsGIHtF/l7Wa2bTwHv7CgDAFm2U7ftLZ2UagOgTTCsV9msA2JL9JjW+SHamAcjVUDTGiFJh7wFgS/aTiZcmzXg2C0BpjOHzsIWrAABsyXbc+MKzX0/aZgEojTHLy8uTksDxABQAgC0YibsQuBFlWldCEQARGdOC8ZQvCQCVBS2SHE87mI18yX6vCIDcGHPixAnieWOwhakAAGzYbtL1MGuy3SIAcjX05s2b6JJo2IaalwOAmmoWSGta10P660UBfPToUQJgNpQtnRZe21UAADZoG1l0pQgsRQFMd0lgEZcGjal0KQCoJGSRZMT7FQGlDIAPHjxIvCDPpoYtLAUAYEP24o53bnjhBpgiWxkA2Qt++umndPr0acIoiSLq2jkHADZki1kd79lslAVQOuZnNe5kr4P37SoAABvQv6z34yyVBVC84NmzZ+EFG7Cp1iUAoJaSU9IR71dmYt0qAErH/LQO/inZxEctKAAAGxB90pi/aZeuAiB7QR4rePHixWlJ4zNDCgDAmo1RJOwsLwtVAYQXzFPT7jEAWLNtqng/zlJVAMULIki7ZsMqJQ8AlYTMS6aq9+O05gEQXjDPGjaPAcAa7VLV+80LoHjBjz/+uMa7Q9IaCgBADRVz0pjH+3Fy83hABhBeMMcoBg8BwJqMMo/30wAQXrAmwyonCwCVBeXk5vV+nMa8HhBesAbD1pAkAKxB1Hm9nxaA8II1GFc5SQCoLKiG99MEEM+CygZWTg4AKguq4f00AYQXVDawcnIAUFFQLe+nDSC8oKKRlZMCgIqCank/bQDFCyJGVNHYSkkBQCUhZcQDe0GNTaMVlMGTP3hBDavopwEAFTSV8X7sAbU2bQDFC2K8oJaFdNIBgAo6ans/zlIdAIoXxKh5BaMrJQEA5xRSvN+0aearXKIOAOEFq1ii3u8AwDn1Fe83aZGVqsnXBWCv10tmUIMXrGoZ3e8BwDn0FO9XdKazMpeqC0D2gjyDGp4Fy1ijvnMB4Bza1uX9OEt1AogZ1OYwuvJXAWBFQWWW6zq8X90AihfkeUSLTBJcUSJ8rYACALCASHmnyCzX2s9+cq06PSADKF6wyDT5kifs9RUAgBU0rdv7cZbqBlC8IK9TCC9YoRAofQUAVhBSvF+dBbcJAGVNCXjBCoVA6SsAsKSQ4v3qLrRNAMhe8OrVq0m3RJ0/JiUldnU6ACxpbm50aaLa1hSAsr7gvXv3SiqB0zUUAIAlVJTVbev2fpylpgBMe8G6GpRKSOzuVABYwuRNeb+mARQviLXmSxQGpVMBYEEhxfs1FcLVpAdkL7i6uppUreEFCxYIpdMAYEEh2fvxApscftbE1jSAjx8/ppMnTxK8YBPWPboGADzSYuIrXlaMG16a8n6ckaYBZC948+ZNeMGJpaCeDwBgAV15oG2T3q8tANkLnjhxgq5cuVJAFZyioQAAnKGiTLTUpPfjLLXhAdNeUGtqjRnyuv8YAM4oAm14vzYBFC+IhV1mFAyljwHgFCHF+7XhDdrygOwFb9++nTwLtnHfU8wR5UcAcIpZNacZnHKZ3I/aBJAh5KWu4QVzTaN6EABOkLNN78dZahtAmcAJXnBCAVE6DABzhJSpJjSnGcy5zNRDbQMoXhCT+U4109wfAsAcCWWqiTZ//S0AKF6w6RbgHJNEewgAZkxrwftxliwAKF4QEzhlConiWwCYEdOC97MEILxgpoAovwWAKUHF+9U10VLqUjNfWvGA8IIzTTXXCQAwJZ94PwsjAiwBCC+YKiTKLwHggaCWvB9nyRKA7AUxma8yeQfJAcADISx5P4sAyjSGaBHVBREAEiVj/Hi0g4VnPzGvNQ8ILyiW0d0DQKJknB+P97Pw7CfmtQggvKBYR2/vHkBrz35iWosAwguKdfT27gG09uwnprUKILygWEhn7xpAq96PTWsVQHhBHfAkFdcAWvV+1gGEFxR85t+7BdCy97MOILzg/OBJCm4BtOz9QgAQ0TGC0Hx7lwCK92tzvN8ss1l+BmQPyH88ah4jJWZZcvrnLgGUwt3meL/pZrHdCCMAihe0rOMsndv+3CWAbc71UtTg8iPBEyRJgbe4Zy+IUfNFrXr8PHcA8q81R71Y/9UOBUB4weNQlTniDsAQvB8bMBQA5VkQM6iVwe7oXFcAhuL9QgMQ84geAVX2lSsAv/jii2SNh7IitXF+SB5QZtO+fv16G1IFfU03ADa9vt+8pSIkALkaipWVqlncDYBNr+9XzRxH3woNQKwveGS7Mq9cALi9vZ20fD58+LCMNq2eGxqA7AWvXr2a6Pzq1atWtQvp4i4A/O6774IrGCECKGvNs97YiikQPYD8a8z9fqEVihABhBcsBl36rOgBlKBrroaGtIUKIIYqlStlUQMoQdeWJlsqap5QAWQviCkMi1qZKGoApeP9+fPnxRUxcmbIACI8rXghihrAUMLO8swVMoDsBbHAZ55Vjx+LFkD2etz4wgU5xC10ACU8zdJUjxbLQbQAhtbxni0coQMoHfPr6+vZW8P7lAJRAihdDyF1vKdskrwMHUB0SWQtmv8+SgAZPK5+hhyREQOA0jEf8g9hPjZ6R6MDMOSuh7RZYwBQuiTOnDmTvjW8TikQHYAhdz2k7BLUgFwGbdIfuiTSVj3+OjoAecyf5dnOjpsg/0gsHpDB5C6JK1eu5N+o86NRASijHjj8LPQtJgBlrGBo4YBNlKGoAAxx1MMkI8cEoDTGhBYQP8k2msejAVAaX+7cuaOpT2tpxQQgV0MvX75MaIw5XpyiATCWxhcxUWwA9nq9pGuI7YTtSIFoAOTIlxAaX0a7z6i3ciopjJ3OAi11H9Fw982RRQ5exQYge8ETJ04QJm4aN3UUAAYT+bL/M22s3KL+1h4RvaHdn/9KKwsd6iz1aLg/GrNMjACurq4GHyAxZiSFN1EAKINubUe+jGhv8FfqDf+TMttr2tq8RJ3OEm0M91PHw5qYd1IfYPa4NMbE0Eo9Zqw53kQBIFc9uf8vvO0t7favUWdhjQZ78XtABvL8+fO0vLwcnqlqynHwAMp8n2E+3L+kQfd9WugOiCulvHEM66S/rEcJ8b0MU0Kf4IG93+3C/S99f9wNEdo22unT54tfUn/HRyMM/2DwMCX+gUGf4LvSGrwHPHfuHAXZ9zf6N/U//xNtjD0THv2ExNgIIx4bfYJHdg4aQBn1Hl71c49ebN6iW4MdGn/yOzJMzABKgHaIc/UcWUjnVdAAcjWGPWBY2xvaHWxQd/NXGm/3HL+LmAFENfTI1kEDyPCF9SyRD99o9190a+Onw4YYNk/MAHJVlKexR2hawNMSSvUznGoMw/ctLeW2cr5P3cHLo59FBwBKaFo49hszj9qbYD1gmNXP4naL3QOyFzx58mRgNZji9it6ZrAAhlf9LGqSd+d5ABCtoYFWQcOrfpaDj8/2ACBaQwMFkKuf3JkbYud7URQ9ACitoZ5nTQuyCsqxn0F2vhelz4kH5OdAroZevHixhDJxnRocgDLvS3id7+UKjgcPyAB6jw0NDsAwhh6Vgy3vbC8AyhAlvl+PW3AA8rCjMIcelSteXgBkL8hDlLyOlA8KQG504cYXDw/tngCUkfIxN6pN+vkNCkDpfvCw5JUnAD1HxQQFoHQ/TPo1iem4JwC5GuqlZpMto0EB6KH7QQzkDUBeV97jVBXBACgzn3HB9LB5A1Cmr7c9sZZ+yQsGQO7342qKl7lEvAF4//79xL6x9+9mEQ4GwNhHP2QN4w1Ar8+BwQDIfX88+7WXzSOA/Bz42WefeTFxcp9BACj9f54mdPUIoDwHeuoPDAJAT/1/8vPvEUB5DvTQzyt2DgJAif/09MvoEUAZnuSpphMEgPzs5yH+U34Vee8RQG6I4eWs19fX01JE/ToIAGOffiKvhHkFkGdLO3v2bJ4kUR4zD6C3DngpZV4B9NYQYx5AWXzF04M5Q+gVQG8NMeYB9NgA4xlAbw0x5gHkuV9CWHpaqo5ae68ekBtieClrjnzysJkH0FsEjBQ6zwByRMzXX38tUkS9Nw8gB2B76heS0uYZQBkhL1rEvDcNoNcWUC5wngGUmdI8DE0yDaDXFlDvAHpqCTUNoHgBD7+E2WqW3Dt7A26Y8PQnUxV6GBtoGkDpgsgWTg/vPQPIPzZenv1NA+i1C8J7FdRTV4RpADkI29Mg3LRn9+4BuSvCw2S9pgHkPsDYF2FJQ5d+DQA/pT//+c9pSaJ8bRpAL88BeSXLO4ASlJ2nTUzHAKBRawLAm0lDjFHzqGXLLICeO+HZugDwHYCxz4JgFkDPnfAA8Al56YwHgGqVCd2EvHtAAKhbnkqnBg/4Y/IM5DEShvsBAWBpZHS/AAABILeCczmIeTNbBZUqmMc4UC5wcv/ePSDPCRvzZhZAz3GgAPBd8LmHfmAAaPTn1bsH9BKQDQABoNmhTvCALRZOVEF9N8LAA7YIH18aAAJAeMAWIQSAABAAtgigLEmNbgh/U1KgI75F8OTS6Ij37QERCSMktLQHgACQq6CIhAGArSjgvR8QHrCVYnd0UXhAeEB4wCMeGn/FAzHZAOwJPG7ePSCmpDBQ6j00Q0+SGQBiSopJZaOx4wCwQ15HQ8ADNobZ5At5XZqMFfHuAXle0K+++mpy4YjkE7PB2KwvJub16wEZwBs3bkSC2eTbMA0gr5J67ty5ybmP+BPvHtDLKrmmAfQcD+odQC/P/6YBlHjQ7e3tiH1d/q15BvDBgwdJFxSWJ8svG40d9dwZ7xlAL1EwDJJpDyid8VwV9bZ5BlC6IGKfFds8gEkGOx16+PChN/5cd0Osrq7S2bNnXdjctAdkC3jtivDsAb2sDZg4GOs/M167IjwDyC2gXmo95j2gFERvI+Plvr2Foj169MhVEL55AL22hHoFsNfrJQDGPhBXap7mAfTaEuoVQGkBlQIa+948gGyATz75xN1a8V4B5AYYD2vDyw9LEADeuXMngVAy7WHvFUAvMaBShoMAUAqjp4YYuWdPjTDeGmAYwiAAlIaY2Jeqkl9F3nsE8O7du0kDjKfY3yAATH4pHPUNeQWQI2C4D9DTFszdckQMj5D3snn0gOfPn6fr1697MXFyn8EAKGMDPQTosmW8Afj48ePE+3kLvA8GQG/Pgd4A9NYBL24+GAA5w55iBL0B6PH5LynTQmIIe0/Pgd4A/OCDD9w9/wUHoDwHeugP9ASgx/4/cXhBVUG5f4iroR7mCvEEIAcbsF099f8FCSBnmqcp5NC02DdPAF6+fJkuXrwYu0lz7y8oD8h34GWAricA2fuxXT1uwQHI4WhssNjD0rwAKN0Psdtz0o9LcADK+MDYpyzwAuDVq1eTH1QvARZZEIMDkG/Aw/AkLwBy98P6+nq2XLp5HySAHmbM9gCgpxmwJ/2iBAkg9wPyc2DM1VAPAEr0i9fqJ0MZJICccR4ZwVNVxLp5AJCrn99++22sJix0X8ECKAU01s5bub9YR8Sj+vmOz2ABjL0aGjuAqH4GDiBnP+bW0NgB5MmXPLd+Sv00WA/INyCtoTF24sYMoHS+e4jpFdAm7YMGUDrlYwxjihlA753vaRiDBpBvRGJDY2vKjhVAnnri5MmTdO/evXQ5dPs6eAAlNjS26kysAMrUgzE+NlT5FQkeQL5pHqLEo+Vj2mIFkKee9zr0KK98RgEgR8RwZExMI+VjBFBGvsccwZQH2bRjUQAofYIxTWkXI4DS9xfTD+U0uIp8FgWAfKOxhabFCCD3/XmbeHcWhNEAGE6f4Ij2t36i/maXljrXqL/7NtdGsQEojS+xNZblGq/EwWgA5G6IIOaL2RtQd6GTPLN2HAHIjS9nzpwpUTR9nBoNgGyucBpjXtPW5iXyAiAaXyb/mEQFoDTG2G9le0u7/WtuAETjixMA+Ta5P5Crora3yQByd8qkvydPnlBofxz5wo0vN27csG2SlnIXlQdkDaUxxvbD/mQApRzE0ggjjS+IfBHLju+jA5Bvj0fK215L0A+APOp9eXl5vNTh3aECUQIoa0jwkmY2Nx8A3r9/P6lOszfHlq9AlADKMCW7U9j7AJCnnD99+jTFNlIlH6VqR6MEkKXgYUp240OlG+ISbW69zrVc6M+A0vUQ41jNXINVPBgtgNIlYa4AvB3SxuJ4S+fixpCy8TChAyiDbhH3OZ3MaAHk25YuiRCrQCEDKINuMefLdPj406gBlHXlQxwlETKAN2/eTKr/dhvBZoPR1BlRA8gicneE/Y754+YOFUDpeMeoh+M2zTsSPYBhdMwfN02oAErHu+1AiON6t3UkegBZWO6YD20a+1AB5I73jz/+uK3yHNx1XQAYohcMEUB4v/L8uwCQZeHnwJC8YIgAsvfDhEvlIHQDoP3wtHHDhQagzHYdYovzuPLNvnMDoIyYD2X6wtAA5BHvZ8+eRdhZSX7dAMi6hOQFQwJQgq7h/UrSF3tHfFaOkLxgSADC+2VLWvH3rjwgyxKKFwwFQHi/4rDlnekOwFC8YCgAsvfDkKM8tIodcwcgyxKCFwwBQHi/YpBNO8slgCF4wRAAhPebhlaxz1wCyNJY94LWAYT3KwbYrLPcAmjdC1oHEN5vFlrFPncLIMtj2QtaBhDerxhcRc5yDaBlL2gZQPT7FUGr2DmuAWSJrHpBqwDC+xUDq+hZ7gG06gWtAsgjHhDzWRSv2ee5B5AlEi9oaRS3RQBlvB/rhU1HAQBIlETwWxsvaBFAeD8d6NKpAMADNax5QWsAivezVEtIF+RQXwPAlOUszR1jDUD2flhkJVVYlF4CwJSQluaOsQSgzPMJ75cqLEovAWBGSPaC/DzIraNtblYAlHk+MdNZPaUBAGZ0FS/YdkufFQDh/TIFRPktAMwRVGbTbtMLWgBQ1njALNc5hUTpEADMEdLCmhIWAFxdXcUaDznlQ/MQAJygJs+e1ub6gm0DKOv73bhxY4JCOKyhAACcoKJ4wbbWF2wbQFnfDyscTSggSocB4BQheYnrtrxgmwA+ePAgue979+5NUQcfaSgAAKeoKKvstjGZb5sA8nCjtn54ppgjyo8A4AyzylrzTVfF2gJQhhu1VfWeYY7oPgaAM0zKXpA75pv2gm0ByN7vzJkzrQcizDBLNB8DwAKmlEDt58+fFzhb55Q2AJSA67aDEHQUDCMVAFjATjJot8nlzdoAEMONChQG5VMAYEFBBYimApLlerdv36YnT57U/ifer6n7Kyh79KcBwBImluFKTYSoNQkgAq5LFALlUwFgCUGbDNRuEkAEXJcoBMqnAsCSgjYVqN0UgAi4LlkAlE8HgCUFlRC1hw8flvxmudObAhAhZ+Xson02AKygaBOB2k0AiIDrCsZX/goArCDo9vZ2EqpVZ7RIEwCK9+P7wdaOAgCwou4SolZX4a0bQIScVTS88tcAYEVB6w7UrhtABFxXNLzy1wDgHIJyQwyPGqgjULtOAMX7IeRsDuMrfRUAziGkhKjVEahdJ4Dnz5/H+g5z2F3zqwBwTjUlUFs7hKsuACXkDN5vTsMrfR0AzimkeEHtQO26AETA9ZwGV/46AFQQVGDR9IKSpmYwtng/zXwqyOc6CQCoZH4J1FZKjrQB5JAz9n6Y4VrLQjrpAEAdHUk7UFsbQARcKxlaORkAqCgoe0GtdSU0AZThRleuXFG8WySloQAA1FDxIA0J1NZoYdQEULxfHf2VivK5TAoAKpud+wQ1vKAWgDLcCDNcKxtaKTkAqCSkJKPlBbUAxPoOYhmbewBYg13EC3K8aNVNA0AMN6qqfnPfA4A1aC1ecJ7hShoAynAjPPvVYGSlJAGgkpDZZOYdtDsvgOL91tfXs1nDe0MKAMCajDGvF5wXQPF+81SDa5IGyaYUAIApMbRfyqDdKhDMA6B4P6xupG1R/fQAoL6mhynKoN0qz4LzAAjvd2gC8y8AYM0mquoFqwII71ezQZWTB4DKgmaTq+oFqwII75e1gO33ALAB+1TxglUAhPdrwJjKlwCAyoLmJVelRbQKgPB+eerbPgYAG7KPRMcUXdilLIAc88kTRKHfryGDKl0GACoJOSsZ8YJFR0qUBRAjHmZZwObnALBBu5TxgmUAxIiHBo2ofCkAqCzotOTKeMEyAML7TVPd9mcAsGH78PJmRWZQKwMgz/Vy/fr1hu8El9NQAABqqFgiDZk7ZtbMZEUBxExnJcQ3eCoAbMEo7AHZE07bigLI3m95eXlaUvjMsAIAsAXjyGza/Ew4aSsCoKzxwOdiC1MBANiC3WQ27Tt37ky8ehEAL1++TKdPn6aifYsTL4YPWlMAALYk/azwtFkASthZ3UtltySPm8sCwJZMLUHakwCaBaBMtlRlrGFLt4zL5igAAHNEaeqQdMznXW8WgCdOnCBMNZinXFjHAGCL9nr+/HkSv5nXJTENQOl64O9jC1sBANiy/bhLgj1hdpsGIC8vffHixexX8D5ABQBgy0aTLonss9wkAKXxpWhQd8u3h8vPUAAAzhCo7o8nNcZMAhCNL3VbpNn0AWCzeudejaug2fjQSQAi7jNXwmAPAkADppP40HRkTB6AvV5vYqONgdtAFiooAAAriKb9FY5k4dHs6ekL8wDkKScQ+aKtfrvpAcB29T+8OsPHy5rJlgcg9/1hsl1RKI49ADRiR+kTlL69LIBS/ZTPjWQb2ZhTAQA4p4CaX2cPKNXQLIBS/dS8HtJqXwEA2L4NDnPAoyOkGpoFENXPQ5miegEADZlTWkO3t7cpDaCM+8sLWTOUfWSlggIAsIJodX1FWkM5yiUNoHS+Y9xfXcq3ly4AbE/73CvzVBX8lwbw/PnzmHQpV63wDwJAYzaU2NCnT58mfYNra2vJHrGfxgyllB0AqCSkVjIyd+hf/vKXBLyVlZVkn46S0boW0mlfAQDYvg2O5fqPOHQAAAO5SURBVICjYi5dupSA9+GHHyb7YyfhQBQKAECDZuTgbIZQ/jDprkEjKWUJACoJqZkMzxMj8PEez3+a6tpKCwDaskeSG+kPFAgRfmbQSEpZAoBKQmomwx3xAh/vs6PlNa+FtNpVAAC2q//Eq6cBnHgSPgheAQBo1ITSCpodKW80u8hWRQUAYEXh6v4aR8OwF5y1iEvd+UD69SoAAOvVt3LqMnW9DE+qnBC+aFoBAGjUPADQqGGUs1UYQK4OYWtOgc3NzaQK+sMPPzR3UcdXmla++cew3EiU36i/spjYb+HzPu2MiEa7z6i3coo6nSXaGO4fKl2YqmkZPEwNL9QUkNEQvMdWvwLTyjd/xgOly9liRPvDHi11LtHm8CfqdR/Ri/3RsRsBgMcksXEAADZrh1kA8uf8x63SxQPjDzzhwpfU33mTe0MAMFeW9g8CwGZtUBRAAZHjdWcHSLymrc1L1Fnp0+6E2ykFoFwc+6NA6Tq1+N3vfpf86tZ5DaRdzZYMIEcsTd8OAFxYo8He8eonf7cUgNMvhk+1FWA4sDWjwDSt0z9SXAUtGps72nlKa93/oZWF96k7eJl7I4UtPC2DuSnj4NwKQPO5JSycwDSt+TNuhCk1KmX/V9rsfk/D/f+jQfdDurD5gt7uPKVbf39BaV8IAAubqPkTpxWK5nMT9xWnac3dELOf9w70Gb2gzQsL1Flao/7WHhGNaG+wRgudU7TSe0a7afrKVEHjlt/m3U0rFDZzjFyVVaCwByybMM6fXwEAOL+G1lMAgIYtBAANG0cpa9UA3N+iQf9v1F06RSv935SygmSyCgDArCLa70e0v/UT9Te7tNS5Rv3dt9oXmJleBQBf0qD7/kH/1CIAnClx9RMAYHXtCn1zb0DdBekHDAbAd7c22tqkCx0AWMjQFU8CgBWFK/W1g87ycDzgwd3t9mkFAJYyddmTAWBZxaqc/5Z2+9eoAwCriBf3dwBgE/YFgE2oHOQ1AGATZrMAYFKdlIfRCftsRDeqoLWXDgBYu8REZAHAKvcJAKuoVuo7ALCUXBVPBoAVhYv/awCwCRsHCuC7boiFJMo7E1/ahGourgEAmzCzdENcos2t101ccOwaFTri92m4sTQ+UHRxg4bNBxGM3UiMbwBgzVZ9O6SNxfE2j8WNITVZlCsAWLMoSP5QAQB4KEW0LwCgYdMCQMPGUcoaAFQSso5kAGAdqtpKEwDasgdy40wBAOjM4LhdWwoAQFv2QG6cKQAAnRkct2tLAQBoyx7IjTMF/h9MpJfyhzWAOwAAAABJRU5ErkJggg=="></p>
</div>
<div class="specification">
<p>The shaded region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is the area bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \pi ">
<mi>y</mi>
<mo>=</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using implicit differentiation, find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the tangent to the curve at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span> is now rotated about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis, through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> radians, to form a solid.</p>
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{si}}{{\text{n}}^3}\,y}"> <mrow> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {\text{co}}{{\text{s}}^2}\,y} \right){\text{sin}}\,y"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </math></span>, show that the volume of the solid formed is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mi>r</mi><mi>x</mi><mo>)</mo></math> . The graph has a local maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mn>5</mn></mrow></mfenced></math> and a local minimum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the area of the shaded region.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> satisfies the conditions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = - 4">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) = 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and its second derivative is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 15\sqrt x + \frac{1}{{{{\left( {x + 1} \right)}^2}}}">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>15</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>A point P moves in a straight line with velocity <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> ms<sup>−1</sup> given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = {{\text{e}}^{ - t}} - 8{t^2}{{\text{e}}^{ - 2t}}">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>8</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> at time <em>t</em> seconds, where <em>t</em> ≥ 0.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the first time <em>t</em><sub>1</sub> at which P has zero velocity.</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 the acceleration of P at time <em>t</em>.</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>Find the value of the acceleration of P at time <em>t</em><sub>1</sub>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l">
<mi>l</mi>
</math></span> be the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x{{\text{e}}^{2x}}">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span> at the point (1, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^2}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>).</p>
<p>Find the coordinates of the point where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l">
<mi>l</mi>
</math></span> meets the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis.</p>
</div>
<br><hr><br><div class="specification">
<p>A body moves in a straight line such that its velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\,{\text{m}}{{\text{s}}^{ - 1}}">
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 2\,{\text{sin}}\left( {\frac{t}{{10}} + \frac{\pi }{5}} \right)\csc \left( {\frac{t}{{30}} + \frac{\pi }{4}} \right)">
<mi>v</mi>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>t</mi>
<mrow>
<mn>10</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>5</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>csc</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>t</mi>
<mrow>
<mn>30</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant t \leqslant 60">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>t</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>60</mn>
</math></span>.</p>
<p>The following diagram shows the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> against <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>. Point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> is a local maximum and point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> is a local minimum.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The body first comes to rest at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1}">
<mi>t</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>. Find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the coordinates of point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> and the coordinates of point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}"> <mrow> <mtext>B</mtext> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the maximum speed of the body.</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 value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1}"> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the distance travelled between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1}"> <mi>t</mi> <mo>=</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the acceleration when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1}"> <mi>t</mi> <mo>=</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </math></span>.</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>Find the distance travelled in the first 30 seconds.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}">
<mn>0.0008</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V{\text{ }}({{\text{m}}^3})">
<mi>V</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> in the trough in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</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>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{3}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> 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>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mi>p</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mi>q</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>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has exactly one point of inflexion.</p>
</div>
<div class="specification">
<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><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></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><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find 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">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the point of inflexion.</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>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> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></math>, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.</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>Find the equations of all the asymptotes on 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>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering 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><mo>-</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, or otherwise, solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo><</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></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">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> is enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2\arcsin (x - 1) - \frac{\pi }{4}">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{\pi }{4}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a definite integral to represent the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following graph shows the two parts of the curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}y = 5 - {y^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span>, and the normal to the curve at the point P(2 , 1).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there are exactly two points on the curve where the gradient is zero.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the normal to the curve at the point P.</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>The normal at P cuts the curve again at the point Q. Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of Q.</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>The shaded region is rotated by 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span> about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis. Find the volume of the solid formed.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a derivative given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>o</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>k</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is a positive constant.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, the population of a colony of ants, which has an initial value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn></math>.</p>
<p>The rate of change of the population can be modelled by the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow><mrow><mn>5</mn><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in days, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is the upper bound for the population.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn></math> the population of the colony has doubled in size from its initial value.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>, giving your answer correct to four significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when the rate of change of the population is at its maximum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the first three terms of the binomial expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><msup><mo>)</mo><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and the result from part (a), show that the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi></math> up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>x</mi></math> and the result from part (b), find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>x</mi><mtext> arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="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><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></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>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<p>The region enclosed by 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>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn></math> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to form a solid of revolution.</p>
</div>
<div class="specification">
<p>Pedro wants to make a small bowl with a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math> based on the result from part (a). Pedro’s design is shown in the following diagrams.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAooAAAD0CAYAAAAcwwcUAAAgAElEQVR4Ae2dPe8lx3WnDXg/gjdaeG2urTUxmNQrymMsnMiEmHKH4qScQBNaCQfCGmKw5gBeQMw4gaVQDESFIiAGijjAZApEMPUCY+XC6BPcxXOlM6zp6b7/7r79Uqf6KeCi7+3bXVXnqerqX5966T85GSQgAQlIQAISkIAEJNBD4E969rlLAhKQgAQkIAEJSGAhAk+efHH68z//Lye22YJCMVuJmV8JSEACEpCABFIRePjw/bNQZJstKBSzlZj5lYAEJCABCUggDYHnz5+fbt++dbp3793TnTvfSpPvyKhCMUi4lYAEJCABCUhAAgsTePz44xOexE8++enZq/jZZ79YOIV1o1MorsvX2CUgAQlI4GAEnj59eqrp89VXXx2sBOoy9623vnP68svfnDOFZ/HBg+/VlcEbcqNQvAGQf0tAAhKQQC4CfSLt009/dvroox/1fu7efft06XPr1utnTxCTEcZ83nzz2xfju5TWGv+98cY3R+U7bMPem/IxxPLzz3/5ikg+slBl8gpCMQKeRcQi3dFZgkIxS0mZTwlIQAIHIICoCKGH6CgFyQcf/PAVARPiptz2CbX79997Ka4y3j5xE3lg+9vf/scByH9tIvaW9ne/TxXdfUK1T4yWZcL3Mt2sYhNhSNdzhJj9TDd0lrCJUHz27NmLp5nS5fro0YfpXLBZCtZ8SkACEqiBQNzsS9FXCj5EXSnyEBWlN+uSeCDuo4m4Gsp0iTx0xWhZPyjzso5QH7pis6wjHBv1JOpbDcIyJrGU9Tu+M7ElS9hEKAYMRCIu1wgo61I4xn63EpCABCRQP4G4Kf/kJz8+36i///1/Oou8UvyVwi9u5mzjXLZHCHiVuP+VQiHbpIZayun3v//9i/pTCsxSXJbCMjzM1E/qHvWVere2mIxJLF1u7Kce4ETLEDYVigEn+uZxvWZyv2YoUPMoAQlIYCkCcUOOrka6b/HkhNiJG3B4dDhuixvwUvZtEQ/3O8aolRMaSJceNThyXzSsSyAeSoYeaKIbPIYnRD2m/s8NiEAeDPoW2I5e1ixexU2FYkwNBxIXT8aFJ+dWGs+TgAQkUCuBEITcSMMrGGIQYRiemBjLd80NtFYGa+ULMTA0eYF7IJxjRuxaeTDeywSi/vc9EIWI5GEoPJGXYzudyzOuH7al5xjtU/7H93Ce3RTvXv9vKhSBBRQUNk9TWdyuexWO6UpAAhJYgwDjw7gpIgCjiw7vYAhCPDCKwevJIwC55w05RWJig0Owrme9VgwxlpIuazyO5fWCeOQ6an2c7KZCsbxoSoW9VgEbrwQkIAEJ/IEA47G4sXGjw0sS3WyIQsM6BGK41aXuZbyNfAx5CIQHEvEYQzF40MLj2KJo3FQoRr883kSDBCQgAQmsTwAhyM0McYjHkO5jwzYExoxDVChuUxZrpoJw5Lri+sKDzEPY2hNl1rSnG/fmQnHIBd/NmL8lIAEJSOA6AngQEYh4OuxKvo7lnLPHehS77//FKxVdnIh8y24O/X3Ooax4QHjttb84i0bKMbtoXF0oMoGF2V5s+8ZhANWLYJ8KbaoSkEC7BLg5cZOyfd2vjGO41dDs1uhl6zpQwjOFd4oPQt+QhwDjFqPswsOYJ/ev5nR1oXjTExX9+oA0SEACEpDAcgT++Z//9+n11/+7QnE5pLNiilnPfZM3o2u6O+u5KxT/4R/+50tpf/edd07dT3lA9z9+l8H/1+VHeQ0JRcY1IiQzhdWF4iUYPCWFex14BglIQAISWIYAbSoP4rSx2W5MyxCoI5ahdRRjaZy+tYTxBjNkALFBGeoVrqMsx+SCsuIB4C//8r+ey+9v/uYbL7qeGVJAmVK2mcp0N6EIMGAxADTAtThbaEzF8hgJSEACSxNAKPKhjUUs8mHMYvbxUktz2io+xAMTV8LThKex60ncKi+mszwBJo3hCUbXMK40JpGVqwowySXKn2sxS9hNKAISaATA8Z19BglIQAISuJ5ACMWIqbyRIRq5qeFpVDgGIbcSGE+A64leUbQL4rDvQSwEI7FyfIjE2GZxju0iFGmcStcr0HDDss8ukvEV1SMlIAEJDBHoCsXyOMRheZOjDeamhpeDNrj0gpTn+V0CRyQQopDrg+uE6wVhiEjkOhp62CqFYszHCJEY11wGnpsLxT5BCDBCV0BmAGgeJSABCdRI4JJQ7OYXzwZd1JzDzS9uatwMudmxnxsiN8yhm2I3Tn9LIBMB6jX1m7oeghDnFfqE64Hrgv84Zuz4whCKaJtSIJbfia/2sLlQpEECdhlCKLKP/7K4Y0sb/C4BCUigJgJxw7smT303T8QjbXaIyLiBhpCcciO9Jm+eK4GxBBB21Es+XBd8qLcIuVIM8huRyP8ce+1DUQjFMp+l3in31/x9c6HYByMjuD473CcBCUigFgLc6LhRrRV4oCcNvCUhSkmPD206H27CsS9uwByL95Jz+Yz1zqxlh/HmJFCKv/CGU7cYext1LkRgWQ85pqyDa9Y/Hqa6YjOj3lEo5rxGzLUEJCCBiwTWFooXE//jnyEmyUvcoKNbL27mISrZhpcy/otz2JYeyxCZY/LgMfUSKMVelGlZ5t26Et7sqDNRT9jGeWU92bt3sk8U9u2rt4T+kDOFYu0lZP4kIAEJzCBQg1Ccmu3o6ibvpZcIEVB6ihAGMY4yRAPb0nNUiojSmxmCgm3p2QyhUm6n5r+14/uEXMknvMklU753BV6URVlW8T3+i223rMr0ut652nljYzf07eseU9tvhWJtJWJ+JCABCSxAgBssN98jhdKDWQqMrpCJ3yFO+rZ9QjTETXfLsX1x1Lqv65nr2lP+vmRbjE8NnuW25B/f1+zmrbGew7Eb+vZ1j6ntt0KxthIxPxKQgAQWIMDNmZu8YX0CIYSybLN55tYvwXVS6BOFffvWSX25WBWKy7E0JglIQAJVEch4U6oKoJmRwEwCiHE8t92Q8ZpUKHZL0d8SkIAEGiGQ8abUCHrNODgBvMsMPeiGjNekQrFbiv6WgAQk0AgBbkpHGxfWSNFpRnICCsWFCzCjwl4YgdFJQAISWJwAHg1uWAYJSGBbAizTw0z9bsiod/QodkvR3xKQgAQaIaBQbKQgNSMdgZgB3s24QrFLZOTvjOBGmuZhEpCABHYjgEcDz4ZBAhLYlgDXHmKxGzLqHT2K3VL0twQkIIFGCAx5NRoxTzMkUC2BIW++QnFmkWUEN9NUT5OABCSwGYGhcVKbZcCEJHBQAqxh2jc+OKPe0aN40Eqs2RKQQPsEhmZetm+5FkpgXwJDgnBo/765vZy6QvEyH/+VgAQkkJbA0KK/aQ0y4xJIQIAlqYYE4dD+ms1SKNZcOuZNAhKQwJUEMt6YrjTZ0yWwK4FLnvyM16NCcdfqZOISkIAE1iXAa8R8t++6jI1dAiWBTz/92en+/ffKXS++KxRfoJj2JSO4aRZ6tAQkIIF9CAzNvtwnN6YqgfYJXFptIKPe0aPYfp3VQglI4MAEPvjgh73ruR0YiaZLYFUCeBPxKvYFhWIflRH7MoIbYZaHSEACEtidgEvk7F4EZuBgBIaWxgFDRr2jR/FgFVhzJSCBYxG4NLD+WCS0VgLbELgkBi/9t03upqeiUJzOzDMkIAEJpCFwaamONEaYUQkkIXDTklQKxZkFmRHcTFM9TQISkMDmBG7dev3029/+x+bpmqAEjkbg889/eWIC2VDIqHf0KA6VpvslIAEJNELAmc+NFKRmVE/g0oxnMq9QnFmEGcHNNNXTJCABCWxOwJnPmyM3wYMS4KFsaMYzSDLqHT2KB63Mmi0BCRyHwKUFgI9DQUslsD6Bmxa4VyjOLIOM4Gaa6mkSkIAENidw0wD7zTNkghJokMCYiWMZ9Y4exQYrqyZJQAIS6BLgBsWNzCABCaxDYMxSVArFmewzgptpqqdJQAIS2IXApUWAd8mQiUqgMQJMZGE88KWQUe/oUbxUov4nAQlIoBECTmhppCA1o1oCl17dF5lWKAaJiduM4Caa6OESkIAEdiXghJZd8Zv4AQjcNJEFBBn1jh7FA1ReTZSABCTghBbrgATWI8CC9ixsf1NQKN5EaOD/jOAGTHG3BCQggWoJ0Nb6hpZqi8eMJSZw0xtZwrSMekePYpSeWwlIQAKNE2AxYG5oBglIYFkCY8cAKxRncs8IbqapniYBCUhgNwJjZmXuljkTlkBiAmMfwjLqHT2KiSumWZeABCQwhQDrvLFMjkECEliWAAJwzDqlCsWZ3DOCm2mqp0lAAhLYjcCYN0fsljkTlkBSAlMewDLqHT2KSSum2ZaABCQwh4ALb8+h5jkSGCbAkI7vf/+fhg8o/lEoFjCmfM0Ibop9HisBCUigFgJjB93Xkl/zIYHaCTA+kXVKx4SMekeP4piS9RgJSEACjRAYu4xHI+ZqhgRWJ4D4G7vslEJxZnFkBDfTVE+TgAQksCsBxynuit/EGyMwdSH7jHpHj2JjlVZzJCABCdxEwHGKNxHyfwmMI/CTn/x49PhEYlQojuP6ylEZwb1ihDskIAEJJCHgOMUkBWU2qycwZXwixmTUO3oUq6+GZlACEpDAsgQcp7gsT2M7LgGE39jxiVBSKM6sKxnBzTTV0yQgAQnsTsBxirsXgRlogADrJ77xxjcnWZJR7+hRnFTEHiwBCUigDQKMU/S9z22UpVbsQ2DOKzEVijPLKiO4maZ6mgQkIIEqCDBOkY9BAhKYR2DOw1ZGvaNHcV798CwJSEACqQlMee1YakPNvARWIBDDN9hOCQrFKbSKY8eAu3371unx44+Ls/wqAQlIQALXEKDtnTIQ/5q0PFcCLRFg2AYexalhjN6ZGufax6fwKH7yyU/PM4Xu3PnW2jyMXwISkMBhCNy//97oV48dBoqGSmAEAd7tzBjFqUGhOJXYH4+/Cdxbb33n9ODB985i8csvfzMzFU+TgAQkIIGSwNTFgstz/S6BIxNgtjNvZZkabtI7U+Pb4vjqPYoIQzyJz549OwvFR48+3IKLaUhAAhJongA3ulu3Xm/eTg2UwJIEGK4x97pRKM4siUvgEIYxNvHevXfPonFmMp4mAQlIQAIdAnM9I51o/CmBwxC4xhN/Se/UCrB6jyKTWJ4/f37mF2MVP/vsF7XyNF8SkIAEUhGYO9YqlZFmVgILErhmbK9CcWZBDIFDGDI2sQwIx4cP3y93+V0CEpCABGYSmDt7c2ZyniaB1ATmLosTRg/pnfi/xm3VHkUmsQC1+0EsGiQgAQlIYBkCtLEuk7MMS2Npm8C1D1YKxZn1ow9cTGLpRvnkyRdn4Yi30SABCUhAAtcTuKYr7frUjUECeQhcO1SjT+/Ubn21HkW8iTGJpQuRWdAxE7r7n78lIAEJSGAaAQbnIxYNEpDAZQLXTv5SKF7mO/hvFxxdy+zj010Op9sd7cSWQaz+IQEJSGAUAbqdu+3wqBM9SAIHIsByUgjFa0LG66xaj+I1BeG5EpCABCQwjQCvI2P8lUECEugn8MEHPzzR9XxNUCjOpJcR3ExTPU0CEpBAlQR4Hdm1N8EqDTNTEliIwBIPUxn1zq4eRaaZMzYGcHw3tEHgBz/4welP//Q/nf7sz/7z6d///f+9YtR3v/vu6ec///kr+/t2cGxfHH3Huk8CEphPYIlutfmpe6YE6iZwzdtYSssUiiWNEd9R50Djw3dDfgKIxG984xunX//612exyLYM//iPb57+9V//b7nr4ncEJYKzG8/Fk/xTAhKYReDagfqzEvUkCSQgsNSEL4XixMIOkXjTlncq3r379ksfukn4UHhPnz49f/RKTiyAhQ/H84cn8d/+7ce9MT948OCEkJwafvWrX53F59TzPF4CEphGgDFYfAwSkMDLBNAgn376s5d3zvilUJwIjafXEImXZhLh8g0xyJYB1yEUGVMTIjLiwjvJPho8juMculUM6xL427/9H4OCDo8gnsHf/e53szJBF/QUT+SsRDxJAgcnQNt6qS0+OB7NPyiBa9/GUmJTKJY0RnxHACLmAMf3pUKISkQi8SMaQ5QiIlkvjP9oFJdMd6n8XxsPnjs8e3yiyxaBxu81xvsh4CI9tnQvdwN56tsf5yIiCeQPwVnmnf14KenSNkhAAusSoAfHB+t1GRt7LgJ4EpdaZ1ShOLPstwSHiKTQEYoISBpF0g8PJP+10EgiEBFb4YVDKCLA6MZdI4QQjfS6aSDyhv5DHCIUEYOIyT6vI/nuisduGv6WgASuJ0AvDUN6DBKQwB8ILPnmoi31zlLlt+us5zBib3DRtY14pEKU4jE8jxnHP5ZCEdZ49boiLMYVcuzQZ2jMYZQfWyadDAm5m0Qk58dM6SGPZ8QxJi9lvvwuAQlMI0BPi5MLpzHz6HYJRLfzUr2Pe+udOSWlUBygRqWgwYyuawqX7muetvE6LlVpBpJfZDdePMQhAQE25NFbIjGEXnQfd+MLkXcpff7D4zkUxsQxdK77JSCBaQR4WM7Qxk2zyqMlMJ0A9/slH5wUitPL4HxGFnB0W8cUeRrS2oUjQpFJIAS2iK21AiIv0uqmMUbk4SnEIzmUx4hDj2KXrr8lsDwBu5+XZ2qMOQnQy7jkUIwseqcsLT2KJY2J3xnLWLNwZLwfH7qFhwTWEl3PEcclj+GlMYqIQPKJ2BxaiDvGXMbknIlF5eESkMAEAnY/T4Dloc0SWLrbGVAKxZnVJSO4PlNL4YhNuKvpuqbR3SPg4Su7n9fKAwIRb+DQ+ELSHZr1zH/kEwHIluP43hWdCF1sMUhAAtsQoA2z+3kb1qZSJ4Glu52xMqPe0aO4Yv2kqxqhiGCkcoQLe6vGF9HFuMFLAm4J8xF4l8YXkgbij7yUXcshMMPbGRNi8C52A2nMWay7G4+/JSCBcQSivRp3tEdJoD0Ca1wDCsWZ9SQjuKmmIg55OmHsT4xvXNvbiFAc6sqdmv9LxyMAux7AvuPJD5+pwTezTCXm8RK4nsAa3pTrc2UMEtiGwBrdzuQ8o97Ro7hNnXslldLbiHDkyYWGealleBCIW3jgEHFdT+ErxhY78BZOEYvYQfyOTSwg+lUCGxCgLXL28wagTaJKAms9KCkUZxZ3RnAzTe09DW9jTIqBBV3V/J7TRR2TRqaIsd5MXdgZM5ARiXQ5j/EmltHRjTzW08mxa3edl3nzuwQk8DUBZz9/zcJvxyKwRrczBDPqHT2KldV9nuLLLuqYEDPmbTEIOLxvfWP8ljSTdBCkTGCZKhKXzIdxSUAC6xJw9vO6fI29TgJrdTtjrUJxZplnBDfT1Mmn0VB3xzWOEY2TE/IECUhAAj0E7H7ugeKupgms1e0MtIx6R49iouoe4xpZ6JsPk2EUjYkK0KxKICEBu58TFppZvorAWt3OZEqhOLNoMoKbaepipyEQEYqKxsWQGpEEJNBDgF4N2hmDBI5AILqd2a4RMuodPYpr1ISN41Q0bgzc5CRwMAJ0P9t7cbBCP6i5dDvjUVwrKBRnks0Ibqapq58WopGGPbqn58yeXj2jJiABCaQhQPczPRgGCbRO4O7dt88TSteyM6Pe0aO4Vm2oIN5yIsw1S+5UYIpZkIAEdiTA+Gi7n3csAJPehABOFYTcWt3OGKFQnFmUGcHNNHW30xCNuNNhHU9Ma14MuxlqwhKQwCoEEIp2P6+C1kgrIRDrGa+ZnYx6R4/imjWiwrgRhzEGgwqLeEREGiQgAQlcIkDXM13QBgm0SoCet7XvhwrFmbUnI7iZplZ1WrwRhouDMY3cBPQYVFVEZkYC1RCgbaCdMEigRQLcD7eo3zfpnS+//M3p3r13T48efTgJ87Nnz87nEP/z588HzyX+Bw++d+5dZDsm6FEcQ+kAx8QkGLqX+Hz00Y9mvULwAKg0UQKHJUDbsLbH5bBwNXxXAlt5zG8Sirdv3zqLuKlC8c6db53PuyQUHz/++IVARFiODQrFsaQOdBw3AryLPF05nvFABa+pEriBAGO47H6+AZJ/pyTAQxCTttYONwlF0kcsThWKnBdCsM+j+MknPz2LRI6ZGhSKU4kd6PgYz4hYpHJzg9CbcKAKoKkS6BDYYlZoJ0l/SmB1AvSoIRS3CHsIRYQj4vPhw/dnmahQnIXteCdxg6A7mouJj13Tx6sDWiwBCEQvgzQk0AoBnCBbrRM6VSjSRcxYQj54BaNr+rPPfvEK/iGPYuzHSxnnjx2fSCIKxVdQu+MmArjn7Zq+iZL/S6BNAqyagFg0SKAVAgyz2moi51Sh+NZb3zn36DEGMcQh+/jdDSEIu13PTI7heCayEOK4sd3bCsUuaX+PJhBd0+Ws6S3GeIzOoAdKQAKLE+C652ZHL4NBAtkJMJyKe9hWYapQJF8Iw9IDGJ7Bbp5DAHaFYvd8zgvx2I2j77dCsY+K+yYT4KaB6z66phn07o1kMkZPkEAKAvQocI0bJJCdAGsJb1mX9xCKiEI+ZRgSm+Ux8V2hGCTcLkaAJ7R4C4wLei+G1YgkUA0BrvGtBv9XY7QZaY7AHt7xPYQinkbGJpaBiS1d8Vj+X35XKJY0/L4oAS5CntTCy4jHUS/jooiNTAK7EdhyXNduRppw0wS4P+HM2DKMFYrlDGXGF5aijv+6wg8bouu5u0YiXdHEEd3XjFXk/CdPvhhlukJxFCYPupZA3wSYa+P0fAlIYD8CPPhtNVN0PytNuWUCjE1kctaW4ZJQRNDxf3wQdjFLmX385hP/l2Kx3M//MfElbEM8lhNjmEE9NigUx5LyuEUI9E2A2Wq22SIGGIkEJHAmwHW7xSvPxC2BNQjQu7VH/UXEZQsKxWwl1lB+udHgkeBijSc7hKRBAhLIQYDr1kX4c5SVuXyZAPceJmVtHRSKM4lnBDfTVE8bIBBrs1EXuHhdZmcAlLslUBGBPcZ4VWS+WUlMgLHze/RmZdQ7ehQTV/QWs053QPkGGG5EehlbLGltaoEA1ys3Pq/RFkrzODbsOWtfoTiznmUEN9NUT5tAwGV2JsDyUAnsRGDrdeh2MtNkGyJArxUOiT1CRr2jR3GPmmKakwjgtSiX2fE905PwebAEViXAsJEt32yxqjFG3jwBvN+INe4rewSF4kzqGcHNNNXTriQQy+xQZ3jf7NZLG1yZfU+XQJMEXFOxyWJt0ijuGVuvnViCzKh39CiWJej3NAR4KgxPBjcpZrDt9YSYBpoZlcBKBLj++BgkUDuBWGFjr3wqFGeSzwhupqmetgIBZq4x5sRldlaAa5QSGEHANRVHQPKQ3QngTOA+sWfIqHf0KO5ZY0x7UQLhZaRLmsYA8bjH8geLGmVkEkhCYG9PTRJMZnNHAnutnViarFAsaUz4nhHcBPM8dAcCPDnSKISX0WV2digEkzwUAddUPFRxpzSW+8HezoOMekePYsrqbqanEIjBy1ygNS7mzQvay/d08vJ23sPJh/8MEshAYO/ZpBkYmcf9COy5dmJptUKxpDHhe0ZwE8zz0EoI4GUsF/OuYZkdXtxO/b93790TL22PEMLxyZMvYpdbCVRPgAcxPIsGCdRGoJb1PjPqHT2KtdVm87MJAZ4uualx0dKA7LHMDsLw9u1bZ5HYZ/TDh++fEJIGCWQhUIvXJgsv87kNAZwEtPV4vfcOCsWZJZAR3ExTPa0yAjQceEAYiM/4FcY1bjWGBSFI3R/yGiIk6X42SCATAd6h67vaM5VY+3mtafxsRr2jR7H9a0QLRxJAIOJljAkweBnXfAJlLCKNxvPnz0fm0MMkUD8BhnRwHRkkUAsBHl7wdtcQFIozSyEjuJmmeloSAohEltmhbnLTW9pDgjgkbrqeDRJoiUBN3XwtcdWWeQRouxGKtYSMekePYi21x3xUSWDNCTCIRIVilcVupq4k4Os1rwTo6YsR4EG/prcGKRRnFm1GcDNN9bTEBMoJMHEjvKZrmpnO1P2hMYqJUZn1gxPAI8+4X4ME9iRA+8xQIh74awkZ9Y4exVpqj/lIQ4DGJ26ENEJzu6bLpXH6jGcyi7Oe+8i4LwMBro2tJoZl4GEetycQQ4i2T3k4RYXiMJuL/2QEd9Eg/zwMAZ5U6dZgDAyfqWszPnr04dmriHexXFyb2c7MijZIICsBrouauvyycjTf8wnU+FrJjHpHj+L8OuiZEniJQHRNT501jSiMbmgaEcYtPn788Utx+0MC2QjgTeRaMEhgDwK11j+F4szakBHcTFM97QAEomuahbyp23st6H0A1JpYOYEaPTqVIzN7CxHAm13jMk0Z9Y4exYUqpdFIoI8AXdPlgt40XLWs59WXX/dJYEkCNY4RW9I+46qXQK1jZBWKM+tMRnAzTfW0AxMol9qhEVM0HrgyHMR0vOu07zXNOj0I+kObGZMNa4SQUe/oUayxJpmn5gkwfiYmwZSikQaOhoQP3w0SyE6AByImeRkksBWBWL5sq/SmpKNQnEKrODYjuCL7fpXAVQQQjdxIGc8VIjG2V0XsyRKogEBtb8aoAIlZWJEA3mvaT7zZNYaMekePYo01yTwdlkAIxNgeFoSGN0WgpnftNgVWY14hUPu7xhWKrxTZuB0ZwY2zzKMkMI1A2fVsd900dh5dLwEmdDH73yCBtQnwUIIXu9aQUe/oUay1NpmvQxPI2JgcusA0/iIBJ7VcxOOfCxFgRQmEYs0hY9uuUKy5Rpm3wxLI2JgctrA0fBQBPIp4Fg0SWItAhjqWsW1XKK5VY41XAlcQyNiYXGGupx6AQAZvzwGKoVkTa5/EEuAztu0KxSg9txKoiEDGxqQifGalUgK1jx+rFJvZGkEgyzjYjG27QnFEBfQQCWxNIGNjsjUj08tHoPYZqfmImuMgkGVmfca2XaEYtcytBCoikLExqQifWamUQJbuwUrxma0BApnW6szYtisUByqeuyWwJ4GMjcmevEw7D4Ga35qRh6I5LQlkevtPxrZdoVjWNr9LoBICGRuTStCZjcoJ1Pwe3srRmb0eAtmWXsrYtisUe8+MXjYAAA7BSURBVCqeuySwN4GMjcnezEw/DwHeb86rKw0SuJYAk1jwUmcJGdt2hWKW2mU+D0UgY2NyqALS2KsIfPDBD090FxokcC2BN9/89gkvdZaQsW1XKGapXebzUAQyNiaHKiCNvYoAk1rwKtJtaJDAXAJ4palHmULGtl2hmKmGmdfDEMjYmBymcDR0EQLZPEGLGG0kixLAK413OlPI2LYrFDPVMPN6GAIZG5PDFI6GLkKA7sJMY8sWMdpIFiOANxpvIt7pTCFj265QzFTDzOthCGRsTA5TOBq6CIGsN/pFjDeSqwlknT2fsW1XKF5dXY1AAssTyNiYLE/BGFsnkLHrsPUyyWJf1vU4M7btCsUsV4X5PBSBjI3JoQpIYxchkHEywiKGG8lVBDJPhsrYtisUr6quniyBdQhkbEzWIWGsrRNwUkvrJby8fZmXV8rYtisUl6/DxiiBqwlkbEyuNtoIDkmABZPv33/vkLZr9DwCmRdsz9i2KxTn1VPPksCqBDI2JqsCMfJmCWR7BVuzBZHEsKyTWAJvxrZdoRil51YCFRHI2JhUhM+sJCPApJaPPvpRslyb3T0I4H3GC501ZGzbFYpZa5v5bppAxsak6QLRuFUJPH369PTGG99cNQ0jz0+ASSy0jZnf6JOxbVco5r92tKBBAhkbkwaLQZM2JIBQ/PzzX26YokllI4DXOfs7wjO27QrFbFeK+T0EgYyNySEKRiNXI9CCCFgNjhGfCfAwgfc5c8jYtisUM9c4894sgYyNSbOFoWGbEGihW3ETUAdNBG9zC8MTMrbtCsWDXnSaXTeBjI1J3UTNXQYC2ScqZGCcNY+tTHjK2LYrFLNeNea7aQIZG5OmC0TjNiHQitdoE1gHSqSlJZQytu0KxQNdbJqah0DGxiQPXXNaM4HMiynXzDVz3lpalD1j265QzHz1mPdmCWRsTJotDA3blEDm17NtCupAibU0Iz5j265QPNDFpql5CGRsTPLQNac1E2BSC17FzGvl1cw3W96++uqrc33Ilu+h/GZs2xWKQ6XpfgnsSCBjY7IjLpNujMDdu2+feFWbQQJMYsHL3ErI2LYrFFupfdrRFIGMjUlTBaAxuxLI/j7fXeE1lDheZbzLeJlbCRnbdoViK7VPO5oikLExaaoANGZXAi0KhF2BJk28xQeGjG27QjHpBWS22yaQsTFpu0S0bmsCrXU5bs2vhfRaHIKQsW1XKLZwNWlDcwQyNibNFYIG7UqgtUkMu8JMmHirk5oytu0KxYQXkFlun0DGxqT9UtHCrQm0tCzK1uyyp9fqMkkZ23aFYvaryfw3SSBjY9JkQWjUrgRaWmh5V5AJE+ch4enTpwlzfjnLGdt2heLlMvVfCexCIGNjsgsoE22aQLy6zTUVmy7mV4xr+VWOGdt2heIrVdQdEtifQMbGZH9q5qBFAvfvv3fCs2g4DgEmMrVa5hnbdoXica49LU1EIGNjkgivWU1EoGXvUqJi2Cyr4UVuae3EEl7Gtl2hWJag3yVQCYGMjUkl6MxGgwRYdJlZ0Ib2CbQ+LjVj265QbP+608KEBDI2Jgkxm+UkBFqdAZsE/6bZfPPNbzf9+saMbbtCcdNLwMQkMI5AxsZknGUeJYHpBFxTcTqzjGccoZwztu0KxYxXk3lunkDGxqT5QtHAXQm07mnaFW4liR/Bc5yxbVcoVnKBmA0JlAQyNiZl/v0ugaUJtD52bWleGeM7wljUjG27QjHj1WSemyeQsTFpvlA0cFcCrc+G3RVuBYkfZXZ7xrZdoVjBBWIWJNAlkLEx6drgbwksTcA1FZcmWk98RynbjG27QrGe68ScSOAFgYyNyYvM7/wFzxPeCUN7BI7idWqv5C5bdCRvcca2XaF4uf76rwR2IZCxMdkFVE+iLNTLWKe7d98+tbpob4/Zh9l1hHFshynMPxp6pPGnGdv23YUi0+Ezgjvahay92xHwmrieNR4KXgNG2/LRRz+6PkJjqIYAM2P5GNohcKQZ7Rn1zq5CMRpywPHdIIGjE/CaWLYGPH369PTGG988cSPiuyE/gSOstZe/lMZbED0APNwdITQhFDHCjwysA9aBFuvAp5/+7Aj3ouZtRPw7DrWNYj7C2ollSdGuZgu7ehQZaxI3I74bJHB0Al4Ty9eA0quIN8qQnwBj2uyFyl+OWHA00a9QnFhveSKkSwhwPh1OhOfhTRKIa+Jojecahek4xTWo1hEn3ZXcN47SXVkH9eVzwYMbbd2RgkJxZmlnBDfTVE+TgAQ2IIDgxjvrzOcNYO+UxJEmQOyEePVk8QofbWJSRr2za9dz1MKM4CLvbiUggboI0NWMSKR7shtoa/g8evTh+a/Hjz8+/3748P3uof6unADjTVmk2ZCXANfp0YaDZNQ7iwvFL7/8zenevXfPjS9Abt++9aJRHqrOGcEN2eJ+CUigbgIhDtl+8slP686suRskQLcz9w67nwcRVf0HXv+jdTtTIBn1zqJCMRrgeFoHCsLxrbe+c/48f/68t+JmBNdriDslMJEA1wT1v+/DQxbXlGFZAsG8bKeWTcHYtiKAR9GZ7FvRXjYdup2PuMZpRr2zmFB88uSL882ur/F99uzZ2bP44MH3emtaRnC9hrhTAjMJcN1wHZQPU1wv7LNbdCbUC6fR69HXVl04xb8qJIBIZByqIR8Bup2P+OakjHpnMaEYNzVEYV+I7mg8jN2QEVzXBn9L4BoC4Y0vhSLxcd3gWTQsR+Czz35xunPnW+dejuViNaY9CET38xEFxx68l0ozVndYKr5M8WTUO4sJRW5ml25ocSPs60rLCC5TxTSv9ROI66NPKHJ9dPfXb1GdOeRBFvHNA2tw7WuT6sy9ueojQPdz38SlvmO33MckDSZWjf1gA12xN304bmycHFfjZBG6nWsss7XqB7bS3vR9EM21h0WEIjcxACgUay9u81crgT6hGPvsel6m1GijGC8dvR54FWmzGDZjyEtg6dnPpcAj7lK4IUrp6i4/fTd/9jFRozzupu8sE1OmNfQdkXVTXOX/5GMoj+VxfI9xg5E2IiZE6ZJe2yN2O/eVA8wzhEWEIoaO9Sj2zTKkEhskcGQCIQq7DTpixiABCQwTiO7nMbOfETsIn/Dehegqb+KlwEMYhmhii3AM4RTbMekO536/f8h32BDb4BI2l0IScRftE2tYlsIyuIwRk0ftdsbu4BfbGr29fTVyMaEYYxSHns7xigAnnubLzLDfIIEjEwihWHYxs48HMMRi33VzZF7aLoGSQHf2MyKIGzOCh//iDWAhAsNzFgInyw27tHnP711hGYI7xCS8Q2RzbBk49miLbIf9iOsQiXDIEhYTijHrua+bjJsfNzxnPWepFuZzawJ9QpE8xP6+62rrPJqeBGolgCcsxCJbbsaIFQRJiMFa896XL4QudiC8MooqxCFlghgKb+077/yv09///d+dXnvtL07/8i//p8/s5vfhcaVuUq6ZPNGLCUVKOG5q5bITzDBkXBADyEtvSVkj9CiWNPx+RAJx7XSvEYZqcH0oFI9YK7R5LAFuwH/1V6+db8AIw0w34T4bEVhc9/HBpsyB8gnBGDYd1YuL8EdEZwqLCkUMRxjGUjhUCDyJ3AQvBY4zSODIBPrWUYxlXLg++paVOjIvbZdAlwAeKwRWdpGIXeEVDVF1587fvWTud99559T3KQ/q+599ZZh7TBkH3/vi6R7z13/9314IX+zqdkl3j2/1d8b6ubhQnFO4CsU51DynBQJ4EONm0LfloQvBaJCABC4T4AYcAostXpusYoR8R3tAN+WYSSKX6Wz/L+WBHYwTjTGiYRO/DXkIKBTzlJU5lYAEJCCBGwggquiqLcfHlZMrYsxi7eKL/CG0avdAxXJCCHNEIRM2opsZ7pRDDAcIm24oQv+ujIBCsbICMTsSkIAEJLAsAQRXzIJGuJRiBo8dv/nwXywNE4KSc482ng6b4wOHYBLsYFV6B/nN2DuOg/NYXozBLoeq8Z1hNuU8h2VrgrHNIaBQnEPNcyQgAQlIoAkC0UWKMApBxDbEI9vwkIU4Ylv+H99LoVnGFd9L8RlCbMltePUive6WLvnIa2y73cLYxr74n3MinujOJ8/XejpjXHY5US/enMTcBkM9BBSK9ZSFOZGABCQggQQESnFZCr2bhFrpkQshtuR2rlDdGnms5jA00ZWVUrorQGydR9P7moBC8WsWfpOABCQgAQlIYGUC8frMoWQQkr5kYIjO9vsVitszN0UJSEACEpDAIQkgAOnexmtoyEFAoZijnMylBCQgAQk0TADhVI6BLL8zjq+VrliW+8K2oTe1NVzEaU1TKKYtOjMuAQlIQAItEQgRVa6dGm9tasUDx6xmhCIznA05CCgUc5STuZSABCQggcYJhIgqhSImh1h88uSLJggwq9mZzXmKUqGYp6zMqQQkIAEJNEzgJqHYFZBZUcTSOEOznrETFoY6CCgU6ygHcyEBCUhAAgcn0CcUEU3MEm6l6zmKOBba7o6/RDwOCcg41+22BBSK2/I2NQlIQAISkEAvgRCK5USW+N7icjEIQkRw2IgYZmkcQ10EFIp1lYe5kYAEJCCBgxIIoVh2MfM9ZkSX+w+KSLN3IKBQ3AG6SUpAAhKQgAS6BPqEIsfEfrxvBglsTUChuDVx05OABCQgAQn0EAhB2PUcxiLVCsUeaO5anYBCcXXEJiABCUhAAhK4mQACkfF6pVBEJMbEDyd53MzQI5YnoFBcnqkxSkACEpCABCYRiHGIMbGj3OJJVCROwunBCxJQKC4I06gkIAEJSEACEpBASwQUii2VprZIQAISkIAEJCCBBQkoFBeEaVQSkIAEJCABCUigJQIKxZZKU1skIAEJSEACEpDAggQUigvCNCoJSEACEpCABCTQEgGFYkulqS0SkIAEJCABCUhgQQJVCMUF7TEqCUhAAhKQgAQkIIGFCCgUFwJpNBKQgAQkIAEJSKA1AgrF1kpUeyQgAQlIQAISkMBCBBSKC4E0GglIQAISkIAEJNAaAYViayWqPRKQgAQkIAEJSGAhAv8fo+VvX/e5k0AAAAAASUVORK5CYII="></p>
<p style="text-align: left;">The vertical height of the bowl, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BO</mtext></math>, is measured along the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. The radius of the bowl’s top is <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext></math> and the radius of the bowl’s base is <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext></math>. All lengths are measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cm</mtext></math>.</p>
</div>
<div class="specification">
<p>For design purposes, Pedro investigates how the cross-sectional radius of the bowl changes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of the solid formed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>15</mn><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi></mrow><mn>34</mn></mfrac></math> cubic units.</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> that satisfies the requirements of Pedro’s design.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext></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>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By sketching the graph of a suitable derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, find where the cross-sectional radius of the bowl is decreasing most rapidly.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the cross-sectional radius of the bowl at this point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="question">
<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) = {\left( {x - 1} \right)^2}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 1 and 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) = {x^2} + 1">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is bounded by the curves <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>, <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> and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 9">
<mi>y</mi>
<mo>=</mo>
<mn>9</mn>
</math></span> as shown on the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The shape of a clay vase can be modelled by rotating the region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> through 360˚ about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
<p style="text-align: left;">Find the volume of clay used to make the vase.</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{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve <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>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>6</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of the minimum point on the curve <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> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x"> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is a decreasing function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <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> showing clearly the minimum point and any asymptotic behaviour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> where the normal to the graph is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </math></span>.</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>This region is now rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> radians about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>f</mi><mfenced><mfrac><mi>y</mi><mi>x</mi></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>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> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> has a gradient function given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>.</p>
<p>The curve passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi></math> to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mi>f</mi><mfenced><mi>v</mi></mfenced><mo>-</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>C</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> is an arbitrary constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the result from part (a) or otherwise, solve the differential equation and hence show that the curve has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mfenced><mrow><mi>tan</mi><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The curve has a point of inflexion at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup><mo><</mo><msub><mi>x</mi><mn>1</mn></msub><mo><</mo><msup><mtext>e</mtext><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msup></math>. Determine the coordinates of this point of inflexion.</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>Use the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> to show that the points of zero gradient on the curve lie on two straight lines of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi></math> where the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> are to be determined.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4{x^2} + {y^2} = 7">
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>7</mn>
</math></span>.</p>
</div>
<div class="question">
<p>Find the volume of the solid formed when the region bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \geqslant 0"> <mi>x</mi> <mo>⩾</mo> <mn>0</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 for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y \geqslant 0"> <mi>y</mi> <mo>⩾</mo> <mn>0</mn> </math></span> is rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis.</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><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></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><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><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>
<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 even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering limits, show that 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 horizontal asymptote and state its equation.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></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">[6]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and the result <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfenced open="|" close="|"><mi>x</mi></mfenced></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <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>, justifying your answer.</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>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">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><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>, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> is defined by equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy - \ln y = 1,{\text{ }}y > 0">
<mi>x</mi>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>></mo>
<mn>0</mn>
</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="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> in terms of <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>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the equation of the tangent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> e</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Differentiate from first principles the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 3{x^3} - x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>By using the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = 2\sec \theta ">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
</math></span>, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}x}}{{x\sqrt {{x^4} - 4} }} = \frac{1}{4}\arccos \left( {\frac{2}{{{x^2}}}} \right) + c} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mi>x</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mi>arccos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> given by</p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable columnalign="left"><mtr><mtd><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></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"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>y</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"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at the point Ρ is parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of Ρ.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^3} - 3x + 1">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> can be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Ax\left( {{x^2} + 1} \right) + Bx + C">
<mi>A</mi>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>B</mi>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
</math></span>, find the values of the constants <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>.</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>Hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{2{x^3} - 3x + 1}}{{{x^2} + 1}}} {\text{d}}x">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A particle moves along a horizontal line such that at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≥ 0, its acceleration <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> − 1. When <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 6 , its displacement <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> from a fixed origin O is 18.25 m. When <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 15, its displacement from O is 922.75 m. Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>x</mi><mi>y</mi><mo> </mo><mi>ln</mi><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the equation of the tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math>.</p>
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> has a local maximum between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math>. Determine the coordinates of this local maximum.</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>Show that there are no points of inflexion on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></math>, show that the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> is</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>5</mn><mn>12</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></math></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating the series in part (a), show that the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> .</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> as far as the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Two boats <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> travel due north.</p>
<p>Initially, boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is positioned <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> metres due east of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<p>The distances travelled by boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, after <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> metres and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> metres respectively. The angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is the radian measure of the bearing of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> from boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. This information is shown on 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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>50</mn><mo> </mo><mtext>cot</mtext><mo> </mo><mi>θ</mi></math> .</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, the following conditions are true.</p>
<p style="padding-left:60px;">Boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> has travelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> metres further than boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.<br>Boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is travelling at double the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.<br>The rate of change of the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> radians per second.</p>
<p>Find the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A small bead is free to move along a smooth wire in the shape of the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>10</mn><mrow><mn>3</mn><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi></mrow></msup></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At the point on the curve where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>, it is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>x</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac></math> at this exact same instant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>An earth satellite moves in a path that can be described by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="72.5{x^2} + 71.5{y^2} = 1">
<mn>72.5</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>71.5</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = x(t)">
<mi>x</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = y(t)">
<mi>y</mi>
<mo>=</mo>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</math></span> are in thousands of kilometres and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is time in seconds.</p>
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>7.75</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</msup>
</mrow>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3.2 \times {10^{ - 3}}">
<mi>x</mi>
<mo>=</mo>
<mn>3.2</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>, find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span>.</p>
<p>Give your answers in standard form.</p>
</div>
<br><hr><br>