File "HL-paper2.html"
Path: /IB QUESTIONBANKS/4 Fourth Edition - PAPER/HTML/Mathematics HL/Topic 6/HL-paper2html
File size: 298.65 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-212ef6a30de2a281f3295db168f85ac1c6eb97815f52f785535f1adfaee1ef4f.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-13d27c3a5846e837c0ce48b604dc257852658574909702fa21f9891f7bb643ed.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>
<li class="active dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Questionbanks
<b class="caret"></b>
</a><ul class="dropdown-menu">
<li>
<a href="../../geography.html" target="_blank">DP Geography</a>
</li>
<li>
<a href="../../physics.html" target="_blank">DP Physics</a>
</li>
<li>
<a href="../../chemistry.html" target="_blank">DP Chemistry</a>
</li>
<li>
<a href="../../biology.html" target="_blank">DP Biology</a>
</li>
<li>
<a href="../../furtherMath.html" target="_blank">DP Further Mathematics HL</a>
</li>
<li>
<a href="../../mathHL.html" target="_blank">DP Mathematics HL</a>
</li>
<li>
<a href="../../mathSL.html" target="_blank">DP Mathematics SL</a>
</li>
<li>
<a href="../../mathStudies.html" target="_blank">DP Mathematical Studies</a>
</li>
</ul></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>
<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>
</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="images/logo.jpg" alt="Ib qb 46 logo">
</div>
</div>
</div><h2>HL Paper 2</h2><div class="specification">
<p class="p1"><span class="s1">A particle can move along a straight line from a point \(O\)</span>. The velocity \(v\)<span class="s1">, in \({\text{m}}{{\text{s}}^{ - 1}}\), </span>is given by the function \(v(t) = 1 - {{\text{e}}^{ - \sin {t^2}}}\) where time \(t \ge 0\) is measured in seconds.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the first two times \({t_1},{\text{ }}{t_2} > 0\), when the particle changes direction.</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 class="p1">(i) <span class="Apple-converted-space"> </span>Find the time \(t < {t_2}\) when the particle has a maximum velocity.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the time \(t < {t_2}\) when the particle has a minimum velocity.</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 class="p1">Find the distance travelled by the particle between times \(t = {t_1}\) and \(t = {t_2}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(x) = {x^3} - 3{x^2} - 9x + 10\) , \(x \in \mathbb{R}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the equation of the straight line passing through the maximum and minimum points of the graph \(y = f (x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Show that the point of inflexion of the graph \(y = f (x)\) lies on this straight line.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^4} + 0.2{x^3} - 5.8{x^2} - x + 4,{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The domain of \(f\) </span>is now restricted to \([0,{\text{ }}a]\)<span class="s1">.</span></p>
</div>
<div class="specification">
<p class="p1">Let \(g(x) = 2\sin (x - 1) - 3,{\text{ }} - \frac{\pi }{2} + 1 \leqslant x \leqslant \frac{\pi }{2} + 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the solutions of \(f(x) > 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the curve \(y = f(x)\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the coordinates of both local minimum points.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the \(x\)-coordinates of the points of inflexion.</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 class="p1">Write down the largest value of \(a\) for which \(f\) <span class="s1">has an inverse. Give your answer correct to 3 </span>significant figures.</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 class="p1">For this value of <span class="s1"><em>a </em></span>sketch the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) on the same set of axes, showing clearly the coordinates of the end points of each curve.</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 class="p1">Solve \({f^{ - 1}}(x) = 1\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \({g^{ - 1}}(x)\), stating the domain.</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 class="p1">Solve \(({f^{ - 1}} \circ g)(x) < 1\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(2{x^3} - 3x + 1\) can be expressed in the form \(Ax\left( {{x^2} + 1} \right) + Bx + C\), find the values of the constants \(A\), \(B\) and \(C\).</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 \(\int {\frac{{2{x^3} - 3x + 1}}{{{x^2} + 1}}} {\text{d}}x\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graphs of \(y = {x^2}{{\text{e}}^{ - x}}\) and \(y = 1 - 2\sin x\) for \(2 \leqslant x \leqslant 7\) intersect at points A and B.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The <em>x</em>-coordinates of A and B are \({x_{\text{A}}}\) and \({x_{\text{B}}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \({x_{\text{A}}}\) and the value of \({x_{\text{B}}}\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area enclosed between the two graphs for \({x_{\mathbf{A}}} \leqslant x \leqslant {x_{\text{B}}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A particle moves in a straight line, its velocity \(v{\text{ m}}{{\text{s}}^{ - 1}}\) at time \(t\) seconds is given by \(v = 9t - 3{t^2},{\text{ }}0 \le t \le 5\).</p>
<p class="p1">At time \(t = 0\), the displacement \(s\) of the particle from an origin \(O\) is 3 m.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the displacement of the particle when \(t = 4\).</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 class="p1">Sketch a displacement/time graph for the particle, \(0 \le t \le 5\), showing clearly where the curve meets the axes and the coordinates of the points where the displacement takes greatest and least values.</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 class="p1"><span class="s1">For \(t > 5\)</span>, the displacement of the particle is given by \(s = a + b\cos \frac{{2\pi t}}{5}\) <span class="s1">such that \(s\) is continuous for all \(t \ge 0\).</span></p>
<p class="p2">Given further that \(s = 16.5\) when \(t = 7.5\), find the values of \(a\) and \(b\).</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 class="p1"><span class="s1">For \(t > 5\)</span>, the displacement of the particle is given by \(s = a + b\cos \frac{{2\pi t}}{5}\) <span class="s1">such that \(s\) is continuous for all \(t \ge 0\).</span></p>
<p class="p1">Find the times \({t_1}\) and \({t_2}(0 < {t_1} < {t_2} < 8)\) when the particle returns to its starting point.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The following graph shows the two parts of the curve defined by the equation \({x^2}y = 5 - {y^4}\), 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 \(x\)-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\(\pi \) about the \(y\)-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="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int {x{{\sec }^2}x{\text{d}}x} \).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the value of <em>m</em> if \(\int_0^m {x{{\sec }^2}x{\text{d}}x = 0.5} \), where <em>m</em> > 0.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The curve \(y = {{\text{e}}^{ - x}} - x + 1\) intersects the <em>x</em>-axis at P.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the <em>x</em>-coordinate of P.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the area of the region completely enclosed by the curve and the coordinate axes.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The functions \(f\) and \(g\) are defined by</p>
<p class="p1">\[f(x) = \frac{{{{\text{e}}^x} + {{\text{e}}^{ - x}}}}{2},{\text{ }}x \in \mathbb{R}\]</p>
<p class="p1">\[g(x) = \frac{{{{\text{e}}^x} - {{\text{e}}^{ - x}}}}{2},{\text{ }}x \in \mathbb{R}\]</p>
</div>
<div class="specification">
<p class="p1">Let \(h(x) = nf(x) + g(x)\) where \(n \in \mathbb{R},{\text{ }}n > 1\).</p>
</div>
<div class="specification">
<p class="p1">Let \(t(x) = \frac{{g(x)}}{{f(x)}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(\frac{1}{{4f(x) - 2g(x)}} = \frac{{{{\text{e}}^x}}}{{{{\text{e}}^{2x}} + 3}}\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Use the substitution \(u = {{\text{e}}^x}\) to find \(\int_0^{\ln 3} {\frac{1}{{4f(x) - 2g(x)}}} {\text{d}}x\). Give your answer in the form \(\frac{{\pi \sqrt a }}{b}\) where \(a,{\text{ }}b \in {\mathbb{Z}^ + }\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>By forming a quadratic equation in \({{\text{e}}^x}\)<span class="s1">, solve the equation \(h(x) = k\), where \(k \in {\mathbb{R}^ + }\).</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise show that the equation \(h(x) = k\) has two real solutions provided that \(k > \sqrt {{n^2} - 1} \) and \(k \in {\mathbb{R}^ + }\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(t'(x) = \frac{{{{[f(x)]}^2} - {{[g(x)]}^2}}}{{{{[f(x)]}^2}}}\) <span class="s1">for \(x \in \mathbb{R}\).</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence show that \(t'(x) > 0\) for \(x \in \mathbb{R}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the gradient of the tangent to the curve \({x^3}{y^2} = \cos (\pi y)\) at the point (−1, 1) .</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}\).</p>
</div>
<div class="specification">
<p>Let \(u = \tan x\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for \(f’(x)\) in terms of \(x\).</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 \(y = f’(x)\) for \(0 \leqslant x < \frac{\pi }{2}\).</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 \(x\)-coordinate(s) of the point(s) of inflexion of the graph of \(y = f(x)\), labelling these clearly on the graph of \(y = f’(x)\).</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 \(\sin x\) in terms of \(\mu \).</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 \(\sin 2x\) in terms of \(u\).</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 \(f(x) = 0\) can be expressed as \({u^3} - 7{u^2} + 15u - 9 = 0\).</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 \(f(x) = 0\), giving your answers in the form \(\arctan k\) where \(k \in \mathbb{Z}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A function \(f\) is defined by \(f(x) = \frac{1}{2}\left( {{{\text{e}}^x} + {{\text{e}}^{ - x}}} \right),{\text{ }}x \in \mathbb{R}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Explain why the inverse function \({f^{ - 1}}\) does not exist.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that the equation of the normal to the curve at the point P where \(x = \ln 3\) is given by \(9x + 12y - 9\ln 3 - 20 = 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the <em>x</em>-coordinates of the points Q and R on the curve such that the tangents at Q and R pass through \({\text{(0, 0)}}\).</span></p>
<div class="marks">[14]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The domain of \(f\) is now restricted to \(x \geqslant 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find an expression for \({f^{ - 1}}(x)\)<em>.</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the volume generated when the region bounded by the curve \(y = f(x)\) and the lines \(x = 0\) and \(y = 5\) is rotated through an angle of \(2\pi \) radians about the <em>y</em>-axis.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Engineers need to lay pipes to connect two cities A and B that are separated by a river of width 450 metres as shown in the following diagram. They plan to lay the pipes under the river from A to X and then under the ground from X to B. The cost of laying the pipes under the river is five times the cost of laying the pipes under the ground.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \({\text{EX}} = x\).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-15_om_15.01.31.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>k </em>be the cost, in dollars per metre, of laying the pipes under the ground.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the total cost <em>C</em>, in dollars, of laying the pipes from A to B is given by \(C = 5k\sqrt {202\,500 + {x^2}} + (1000 - x)k\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Find \(\frac{{{\text{d}}C}}{{{\text{d}}x}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Hence find the value of <em>x </em>for which the total cost is a minimum, justifying that this value is a minimum.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the minimum total cost in terms of <em>k</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The angle at which the pipes are joined is \({\rm{A\hat XB}} = \theta \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find \(\theta \) for the value of <em>x </em>calculated in (b).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">For safety reasons \(\theta \) must be at least 120°.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given this new requirement,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(e) (i) find the new value of <em>x </em>which minimises the total cost;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) find the percentage increase in the minimum total cost.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The line \(y = m(x - m)\) is a tangent to the curve \((1 - x)y = 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine <em>m</em> and the coordinates of the point where the tangent meets the curve.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a circle with centre at the origin O and radius \(r > 0\) .</span></p>
<p style="text-align: center;"><br><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A point P(\(x\) , \(y\)) , (\(x > 0\), \(y > 0\)) is moving round the circumference of the circle.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(m = \tan \left( {\arcsin \frac{y}{r}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Given that \(\frac{{{\text{d}}y}}{{{\text{d}}t}} = 0.001r\)</span><span style="font-family: times new roman,times; font-size: medium;"> , show that \(\frac{{{\text{d}}m}}{{{\text{d}}t}} = {\left( {\frac{r}{{10\sqrt {{r^2} - {y^2}} }}} \right)^3}\)</span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) State the geometrical meaning of \(\frac{{{\text{d}}m}}{{{\text{d}}t}}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The cubic curve \(y = 8{x^3} + b{x^2} + cx + d\) has two distinct points P and Q, where the gradient is zero.</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Show that \({b^2} > 24c\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Given that the coordinates of P and Q are \(\left( {\frac{1}{2},{\text{ }} - 12} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">and \(\left( { - \frac{3}{2},{\text{ }}20} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> respectively,</span> <span style="font-family: times new roman,times; font-size: medium;">find the values of \(b\) , \(c\) and \(d\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p>Richard, a marine soldier, steps out of a stationary helicopter, 1000 m above the ground, at time \(t = 0\). Let his height, in metres, above the ground be given by \(s(t)\). For the first 10 seconds his velocity, \(v(t){\text{m}}{{\text{s}}^{ - 1}}\), is given by \(v(t) = - 10t\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find his acceleration \(a(t)\) for \(t < 10\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Calculate \(v(10)\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Show that \(s(10) = 500\).</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>At \(t = 10\) his parachute opens and his acceleration \(a(t)\) is subsequently given by \(a(t) = - 10 - 5v,{\text{ }}t \ge 10\).</p>
<p>Given that \(\frac{{{\text{d}}t}}{{{\text{d}}v}} = \frac{1}{{\frac{{{\text{d}}v}}{{{\text{d}}t}}}}\), write down \(\frac{{{\text{d}}t}}{{{\text{d}}v}}\) in terms of \(v\).</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>You are told that Richard’s acceleration, \(a(t) = - 10 - 5v\), is always positive, for \(t \ge 10\).</p>
<p>Hence show that \(t = 10 + \frac{1}{5}\ln \left( {\frac{{98}}{{ - 2 - v}}} \right)\).</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>You are told that Richard’s acceleration, \(a(t) = - 10 - 5v\), is always positive, for \(t \ge 10\).</p>
<p>Hence find an expression for the velocity, \(v\), for \(t \ge 10\).</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>You are told that Richard’s acceleration, \(a(t) = - 10 - 5v\), is always positive, for \(t \ge 10\).</p>
<p>Find an expression for his height, \(s\), above the ground for \(t \ge 10\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>You are told that Richard’s acceleration, \(a(t) = - 10 - 5v\), is always positive, for \(t \ge 10\).</p>
<p>Find the value of \(t\) when Richard lands on the ground.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve \(C\) is defined by equation \(xy - \ln y = 1,{\text{ }}y > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) in terms of \(x\) and \(y\).</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 \(C\) at the point \(\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)\)</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A skydiver jumps from a stationary balloon at a height of 2000 m above the ground.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Her velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\) , <em>t</em> seconds after jumping, is given by \(v = 50(1 - {{\text{e}}^{ - 0.2t}})\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find her acceleration 10 seconds after jumping.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">How far above the ground is she 10 seconds after jumping?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f\) be a function defined by \(f(x) = x + 2\cos x\) , \(x \in \left[ {0,{\text{ }}2\pi } \right]\) . The diagram below </span><span style="font-family: times new roman,times; font-size: medium;">shows a region \(S\) bound by the graph of \(f\) and the line \(y = x\) .</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A and C are the points of intersection of the line \(y = x\) and the graph of \(f\) , and B is </span><span style="font-family: times new roman,times; font-size: medium;">the minimum point of \(f\) .</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) If A, B and C have <em>x</em>-coordinates \(a\frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;">, \(b\frac{\pi }{6}\) and \(c\frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;">, where \(a\) , \(b\), \(c \in \mathbb{N}\) ,</span> <span style="font-family: times new roman,times; font-size: medium;">find the values of \(a\) , \(b\) and \(c\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find the range of \(f\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) Find the equation of the normal to the graph of f at the point C, giving your </span><span style="font-family: times new roman,times; font-size: medium;">answer in the form \(y = px + q\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) The region \(S\) is rotated through \({2\pi }\) about the <em>x</em>-axis to generate a solid.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Write down an integral that represents the volume \(V\) of this solid.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Show that \(V = 6{\pi ^2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
</div>
<br><hr><br><div class="specification">
<p>A curve <em>C</em> is given by the implicit equation \(x + y - {\text{cos}}\left( {xy} \right) = 0\).</p>
</div>
<div class="specification">
<p>The curve \(xy = - \frac{\pi }{2}\) intersects <em>C</em> at P and Q.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)\).</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 \(y = - x\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the curve with equation \({\left( {{x^2} + {y^2}} \right)^2} = 4x{y^2}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use implicit differentiation to find an expression for \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the normal to the curve at the point (1, 1).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">An open glass is created by rotating the curve \(y = {x^2}\) , defined in the domain \(x \in [0,10]\), \(2\pi \) radians about the <em>y</em>-axis. Units on the coordinate axes are defined to be in centimetres.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">When the glass contains water to a height \(h\) cm, find the volume \(V\) of water in terms of \(h\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">If the water in the glass evaporates at the rate of 3 cm<sup>3</sup> per hour for each cm<sup>2</sup> of exposed surface area of the water, show that,</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3\sqrt {2\pi V} \)</span><span style="font-family: times new roman,times; font-size: medium;"> , where \(t\) is measured in hours.</span></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><span style="font-family: times new roman,times; font-size: medium;">If the glass is filled completely, how long will it take for all the water to evaporate?</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Xavier, the parachutist, jumps out of a plane at a height of \(h\) metres above the ground. After free falling for 10 seconds his parachute opens. His velocity, \(v\,{\text{m}}{{\text{s}}^{ - 1}}\), \(t\) seconds after jumping from the plane, can be modelled by the function</p>
<p>\(v(t) = \left\{ {\begin{array}{*{20}{l}} {9.8t{\text{,}}}&{0 \leqslant t \leqslant 10} \\ {\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }},}&{t > 10} \end{array}} \right.\)</p>
</div>
<div class="specification">
<p>His velocity when he reaches the ground is \(2.8{\text{ m}}{{\text{s}}^{ - 1}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find his velocity when \(t = 15\).</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 \(h\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f(x) = 3\sin x + 4\cos x\) is defined for \(0 < x < 2\pi \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the coordinates of the minimum point on the graph of <em>f </em>.</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><span style="font-family: 'times new roman', times; font-size: medium;">The points \({\text{P}}(p,{\text{ }}3)\) and \({\text{Q}}(q,{\text{ }}3){\text{, }}q > p\), lie on the graph of \(y = f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find <em>p </em>and <em>q </em>.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the point, on \(y = f(x)\) , where the gradient of the graph is 3.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the point of intersection of the normals to the graph at the points P and Q.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the normal to the curve \({x^3}{y^3} - xy = 0\) at the point (1, 1).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A curve is defined \({x^2} - 5xy + {y^2} = 7\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{5y - 2x}}{{2y - 5x}}\).</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 class="p1">Find the equation of the normal to the curve at the point \((6,{\text{ }}1)\).</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 class="p1">Find the distance between the two points on the curve where each tangent is parallel to the line \(y = x\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point P, with coordinates \((p,{\text{ }}q)\) , lies on the graph of \({x^{\frac{1}{2}}} + {y^{\frac{1}{2}}} = {a^{\frac{1}{2}}}\) , \(a > 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The tangent to the curve at P cuts the axes at (0, <em>m</em>) and (<em>n</em>, 0) . Show that <em>m</em> + <em>n</em> = <em>a</em> .</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation \(y\frac{{{\text{d}}y}}{{{\text{d}}x}} = \cos 2x\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that the function \(y = \cos x + \sin x\) satisfies the differential equation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the general solution of the differential equation. Express your solution in the form \(y = f(x)\), involving a constant of integration.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) For which value of the constant of integration does your solution coincide with the function given in part (i)?</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A different solution of the differential equation, satisfying <em>y</em> = 2 when \(x = \frac{\pi }{4}\), defines a curve <em>C</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the equation of <em>C</em> in the form \(y = g(x)\) , and state the range of the function <em>g</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A region <em>R</em> in the <em>xy</em> plane is bounded by <em>C</em>, the <em>x</em>-axis and the vertical lines <em>x</em> = 0 and \(x = \frac{\pi }{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the area of <em>R</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the volume generated when that part of <em>R</em> above the line <em>y</em> = 1 is rotated about the <em>x</em>-axis through \(2\pi \) radians.</span></p>
<div class="marks">[12]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">By using the substitution \(x = 2\tan u\), show that \(\int {\frac{{{\text{d}}x}}{{{x^2}\sqrt {{x^2} + 4} }} = \frac{{ - \sqrt {{x^2} + 4} }}{{4x}} + C} \).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined by \(f(x) = x\sqrt {9 - {x^2}} + 2\arcsin \left( {\frac{x}{3}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Write down the largest possible domain, for each of the two terms of the function, <em>f</em> , and hence state the largest possible domain, <em>D</em> , for <em>f</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the volume generated when the region bounded by the curve <em>y</em> = <em>f</em>(<em>x</em>) , the <em>x</em>-axis, the <em>y</em>-axis and the line <em>x</em> = 2.8 is rotated through \(2\pi \) radians about the <em>x</em>-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find \(f'(x)\) in simplified form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) <strong>Hence</strong> show that \(\int_{ - p}^p {\frac{{11 - 2{x^2}}}{{\sqrt {9 - {x^2}} }}} {\text{d}}x = 2p\sqrt {9 - {p^2}} + 4\arcsin \left( {\frac{p}{3}} \right)\), where \(p \in D\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Find the value of <em>p</em> which maximises the value of the integral in (d).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) (i) Show that \(f''(x) = \frac{{x(2{x^2} - 25)}}{{{{(9 - {x^2})}^{\frac{3}{2}}}}}\).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 22px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Hence justify that <em>f</em>(<em>x</em>) has a point of inflexion at <em>x</em> = 0 , but not at \(x = \pm \sqrt {\frac{{25}}{2}} \) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A body is moving through a liquid so that its acceleration can be expressed as</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\left( { - \frac{{{v^2}}}{{200}} - 32} \right){\text{m}}{{\text{s}}^{ - 2}},\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where \(v{\text{ m}}{{\text{s}}^{ - 1}}\) is the velocity of the body at time <em>t</em> seconds.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The initial velocity of the body was known to be \(40{\text{ m}}{{\text{s}}^{ - 1}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the time taken, <em>T</em> seconds, for the body to slow to \(V{\text{ m}}{{\text{s}}^{ - 1}}\) is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[T = 200\int_V^{40} {\frac{1}{{{v^2} + {{80}^2}}}{\text{d}}v.} \]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Explain why acceleration can be expressed as \(v\frac{{{\text{d}}v}}{{{\text{d}}s}}\), where <em>s</em> is displacement, in metres, of the body at time <em>t</em> seconds.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) <strong>Hence</strong> find a similar integral to that shown in part (a) for the distance, <em>S</em> metres, travelled as the body slows to \(V{\text{ m}}{{\text{s}}^{ - 1}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) <strong>Hence</strong>, using parts (a) and (b), find the distance travelled and the time taken until the body momentarily comes to rest.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A helicopter H is moving vertically upwards with a speed of 10 ms<sup>−1</sup> . The helicopter is \(h\) m directly above the point Q which is situated on level ground. The helicopter is observed from the point P which is also at ground level and PQ \( = 40\) m. This information is represented in the diagram below.</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">When \(h = 30\),<br></span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) show that the rate of change of \({\rm{H}}\hat {\text{P}}{\text{Q}}\) is \(0.16\) radians per second;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) find the rate of change of PH.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a vertical cross section of a building. The cross section of the roof of the building can be modelled by the curve \(f(x) = 30{{\text{e}}^{ - \frac{{{x^2}}}{{400}}}}\), <span class="s1">where \( - 20 \le x \le 20\).</span></p>
<p class="p1">Ground level is represented by the <span class="s1">\(x\)</span>-axis.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-29_om_16.39.22.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f''(x)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the gradient of the roof function is greatest when \(x = - \sqrt {200} \).</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 class="p1">The cross section of the living space under the roof can be modelled by a rectangle <span class="s1">\(CDEF\)</span> with points \({\text{C}}( - a,{\text{ }}0)\) and \({\text{D}}(a,{\text{ }}0)\), where \(0 < a \le 20\).</p>
<p class="p2">Show that the maximum area \(A\) <span class="s1">of the rectangle \(CDEF\) </span>is \(600\sqrt 2 {{\text{e}}^{ - \frac{1}{2}}}\).</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 class="p1">A function \(I\) <span class="s1">is known as the Insulation Factor of </span>\(CDEF\). The function is defined as \(I(a) = \frac{{P(a)}}{{A(a)}}\) where \({\text{P}} = {\text{Perimeter}}\) and \({\text{A}} = {\text{Area of the rectangle}}\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find an expression for \(P\) in terms of \(a\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of <span class="s1">\(a\) </span>which minimizes \(I\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Using the value of \(a\) found in part (ii) calculate the percentage of the cross sectional area under the whole roof that is not included in the cross section of the living space.</p>
<div class="marks">[9]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the curve with equation \({x^3} + {y^3} = 4xy\).</p>
</div>
<div class="specification">
<p class="p1">The tangent to this curve is parallel to the \(x\)-axis at the point where \(x = k,{\text{ }}k > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use implicit differentiation to show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4y - 3{x^2}}}{{3{y^2} - 4x}}\).</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 class="p1">Find the value of \(k\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi \).</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve \(y = f(x)\), the \(x\)-axis and the lines \(x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the \(x\)-coordinate of the minimum point on the curve \(y = f(x)\) satisfies the equation \(\tan x = 2x\).</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 \(x\) for which \(f(x)\) 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 \(y = f(x)\) 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 \(f\) where the normal to the graph is parallel to the line \(y = - x\).</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 \(2\pi \) radians about the \(x\)-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="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The acceleration of a car is \(\frac{1}{{40}}(60 - v){\text{ m}}{{\text{s}}^{ - 2}}\), when its velocity is \(v{\text{ m}}{{\text{s}}^{ - 2}}\). Given the car starts from rest, find the velocity of the car after 30 seconds.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A particle moves along a straight line so that after <em>t</em><em> </em>seconds its displacement <em>s</em> , in metres, satisfies the equation \({s^2} + s - 2t = 0\) . Find, in terms of <em>s</em> , expressions for its velocity and its acceleration.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider \(f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)\)</p>
</div>
<div class="specification">
<p>The function \(f\) is defined by \(f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D\)</p>
</div>
<div class="specification">
<p>The function \(g\) is defined by \(g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain \(D\) for \(f\) 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 \(y = f(x)\) 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 \(f\) 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 \({f^{ - 1}}\) 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 \({g^{ - 1}}\) 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 \(g'(x)\).</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 \(g'(x) = 0\);</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 \(({g^{ - 1}})'(x) = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Particle <em>A </em>moves such that its velocity \(v{\text{ m}}{{\text{s}}^{ - 1}}\), at time <em>t </em>seconds, is given by \(v(t) = \frac{t}{{12 + {t^4}}},{\text{ }}t \geqslant 0\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Particle <em>B </em>moves such that its velocity \(v{\text{ m}}{{\text{s}}^{ - 1}}\) is related to its displacement \(s{\text{ m}}\), by the equation \(v(s) = \arcsin \left( {\sqrt s } \right)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(y = v(t)\). Indicate clearly the local maximum and write down its coordinates.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use the substitution \(u = {t^2}\) to find \(\int {\frac{t}{{12 + {t^4}}}{\text{d}}t} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times;"><span style="font-size: medium; line-height: normal; background-color: #f7f7f7;">Find the exact distance travelled by particle </span>\(A\) <span style="font-size: medium; line-height: normal; background-color: #f7f7f7;">between \(t = 0\) and \(t = 6\) seconds.</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Give your answer in the form \(k\arctan (b),{\text{ }}k,{\text{ }}b \in \mathbb{R}\).</span></p>
<p> </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 acceleration of particle B when \(s = 0.1{\text{ m}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A function is defined by \(f(x) = {x^2} + 2,{\text{ }}x \ge 0\). A region \(R\) is enclosed by \(y = f(x)\),the \(y\)-axis and the line \(y = 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Express the area of the region \(R\) as an integral with respect to \(y\).</p>
<p class="p1">(ii) Determine the area of \(R\), giving your answer correct to four significant figures.</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 class="p1">Find the exact volume generated when the region \(R\) is rotated through \(2\pi \) radians about the \(y\)-axis.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">A function \(f\) is defined by \(f(x) = {x^3} + {{\text{e}}^x} + 1,{\text{ }}x \in \mathbb{R}\). By considering \(f'(x)\) determine whether \(f\) is a one-to-one or a many-to-one function.</p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Integrate \(\int {\frac{{\sin \theta }}{{1 - \cos \theta }}} {\text{d}}\theta \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Given that \(\int_{\frac{\pi }{2}}^a {\frac{{\sin \theta }}{{1 - \cos \theta }}} {\text{d}}\theta = \frac{1}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \(\frac{\pi }{2} < a < \pi \)</span><span style="font-family: times new roman,times; font-size: medium;">, find the value of \(a\) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that the graph of \(y = {x^3} - 6{x^2} + kx - 4\) has exactly one point at which the gradient is zero, find the value of <em>k </em>.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A ladder of length 10 m on horizontal ground rests against a vertical wall. The bottom of the ladder is moved away from the wall at a constant speed of \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\). Calculate the speed of descent of the top of the ladder when the bottom of the ladder is 4 m away from the wall.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let the function \(f\) be defined by \(f(x) = \frac{{2 - {{\text{e}}^x}}}{{2{{\text{e}}^x} - 1}},{\text{ }}x \in D\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine \(D\), the largest possible domain of \(f\).</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 class="p1">Show that the graph of \(f\) has three asymptotes and state their equations.</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>Show that \(f'(x) = - \frac{{3{{\text{e}}^x}}}{{{{(2{{\text{e}}^x} - 1)}^2}}}\).</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 class="p1">Use your answers from parts (b) and (c) to justify that \(f\) <span class="s1">has an inverse and state its domain.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \({f^{ - 1}}(x)\).</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 class="p1"><span class="s1">Consider the region \(R\) </span>enclosed by the graph of \(y = f(x)\) and the axes.</p>
<p class="p1">Find the volume of the solid obtained when \(R\) is rotated through \(2\pi \) about the \(y\)-axis.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The vertical cross-section of a container is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-10_om_11.45.14.png" alt></p>
<p class="p1">The curved sides of the cross-section are given by the equation \(y = 0.25{x^2} - 16\). The horizontal cross-sections are circular. The depth of the container is \(48\) cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If the container is filled with water to a depth of \(h\,{\text{cm}}\), show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of the water is given by \(V = 4\pi \left( {\frac{{{h^2}}}{2} + 16h} \right)\).</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 class="p1">The container, initially full of water, begins leaking from a small hole at a rate given by \(\frac{{{\text{d}}V}}{{{\text{d}}t}} =<span class="Apple-converted-space"> </span>- \frac{{250\sqrt h }}{{\pi(h + 16)}}\) where <em>\(t\) </em>is measured in seconds.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(\frac{{{\text{d}}h}}{{{\text{d}}t}} =<span class="Apple-converted-space"> </span>- \frac{{250\sqrt h }}{{4{\pi ^2}{{(h + 16)}^2}}}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>State \(\frac{{{\text{d}}t}}{{{\text{d}}h}}\) and hence show that \(t = \frac{{ - 4{\pi ^2}}}{{250}}\int {\left( {{h^{\frac{3}{2}}} + 32{h^{\frac{1}{2}}} + 256{h^{ - \frac{1}{2}}}} \right){\text{d}}h} \).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find, correct to the nearest minute, the time taken for the container to become empty. (\(60\) seconds = 1 minute)</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Once empty, water is pumped back into the container at a rate of \(8.5\;{\text{c}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\). At the same time, water continues leaking from the container at a rate of \(\frac{{250\sqrt h }}{{\pi (h + 16)}}{\text{c}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\).</p>
<p class="p1">Using an appropriate sketch graph, determine the depth at which the water ultimately stabilizes in the container.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{{{\text{e}}^{2x}} + 1}}{{{{\text{e}}^x} - 2}}\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The line \({L_2}\) is parallel to \({L_1}\) and tangent to the curve \(y = f(x)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equations of the horizontal and vertical asymptotes of the curve \(y = f(x)\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \(f'(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that the curve has exactly one point where its tangent is horizontal.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the coordinates of this point.</span></p>
<p> </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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of \({L_1}\), the normal to the curve at the point where it crosses the <em>y</em>-axis.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the line \({L_2}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The region \(A\) is enclosed by the graph of \(y = 2\arcsin (x - 1) - \frac{\pi }{4}\), the \(y\)-axis and the line \(y = \frac{\pi }{4}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a definite integral to represent the area of \(A\).</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 \(A\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The displacement, \(s\), in metres, of a particle \(t\) seconds after it passes through the origin is given by the expression \(s = \ln (2 - {e^{ - t}}),{\text{ }}t \geqslant 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for the velocity, \(v\), of the particle at time \(t\).</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 class="p1">Find an expression for the acceleration, \(a\), of the particle at time \(t\).</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 class="p1">Find the acceleration of the particle at time \(t = 0\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Differentiate \(f(x) = \arcsin x + 2\sqrt {1 - {x^2}} \) , \(x \in [ - 1, 1]\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find the coordinates of the point on the graph of \(y = f (x)\) in \([ - 1, 1]\), where the </span><span style="font-family: times new roman,times; font-size: medium;">gradient of the tangent to the curve is zero.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the volume of the solid formed when the region bounded by the graph of \(y = \sin (x - 1)\), and the lines <em>y</em> = 0 and <em>y</em> = 1 is rotated by \(2\pi \) about the <em>y</em>-axis.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By using an appropriate substitution find</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\int {\frac{{\tan (\ln y)}}{y}{\text{d}}y,{\text{ }}y > 0{\text{ .}}} \]</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A stalactite has the shape of a circular cone. Its height is 200 mm and is increasing at a rate of 3 mm per century. Its base radius is 40 mm and is decreasing at a rate of 0.5 mm per century. Determine if its volume is increasing or decreasing, and the rate at which the volume is changing.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the triangle \({\text{PQR}}\) where \({\rm{Q\hat PR = 30^\circ }}\), \({\text{PQ}} = (x + 2){\text{ cm}}\) and \({\text{PR}} = {(5 - x)^2}{\text{ cm}}\), where \( - 2 < x < 5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the area, \(A\;{\text{c}}{{\text{m}}^2}\), of the triangle is given by \(A = \frac{1}{4}({x^3} - 8{x^2} + 5x + 50)\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>State \(\frac{{{\text{d}}A}}{{{\text{d}}x}}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Verify that \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) when \(x = \frac{1}{3}\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>Find \(\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}}\) and hence justify that \(x = \frac{1}{3}\) gives the maximum area of triangle \(PQR\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>State the maximum area of triangle \(PQR\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find \(QR\) when the area of triangle \(PQR\) is a maximum.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The particle <em>P</em> moves along the <em>x</em>-axis such that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\) , at time <em>t</em> seconds is given by \(v = \cos ({t^2})\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>P</em> is at the origin O at time <em>t</em> = 0 , calculate</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) the displacement of <em>P</em> from O after 3 seconds;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the total distance travelled by <em>P</em> in the first 3 seconds.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the time at which the total distance travelled by <em>P</em> is 1 m.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows the graphs of \(y = \left| {\frac{3}{2}x - 3} \right|,{\text{ }}y = 3\) and a quadratic function, that all intersect in the same two points.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that the minimum value of the quadratic function is −3, find an expression for the area of the shaded region in the form \(\int_0^t {(a{x^2} + bx + c){\text{d}}x} \), where the constants <em>a</em>, <em>b</em>, <em>c</em> and <em>t</em> are to be determined. (Note: The integral does not need to be evaluated.)</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The shaded region <em>S </em>is enclosed between the curve \(y = x + 2\cos x\), for \(0 \leqslant x \leqslant 2\pi \), and the line \(y = x\), as shown in the diagram below.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-12_om_06.15.17.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the points where the line meets the curve.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region \(S\) is rotated by \(2\pi \) about the \(x\)-axis to generate a solid.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down an integral that represents the volume \(V\) of the solid.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the volume \(V\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the curve, \(C\) defined by the equation \({y^2} - 2xy = 5 - {{\text{e}}^x}\)<span class="s1">. The point A </span>lies on \(C\) <span class="s1">and has coordinates \((0,{\text{ }}a),{\text{ }}a > 0\)</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\).</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 class="p1">Show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{2y - {{\text{e}}^x}}}{{2(y - x)}}\).</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 class="p1">Find the equation of the normal to \(C\) <span class="s1">at the point A</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of the second point at which the normal found in part (c) intersects \(C\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(v = {y^3},{\text{ }}y > 0\)<span class="s1">, find \(\frac{{{\text{d}}v}}{{{\text{d}}x}}\) </span>at \(x = 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If \(y = \ln \left( {\frac{1}{3}(1 + {{\text{e}}^{ - 2x}})} \right)\), show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{2}{3}({{\text{e}}^{ - y}} - 3)\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p>A point P moves in a straight line with velocity \(v\) ms<sup>−1</sup> given by \(v\left( t \right) = {{\text{e}}^{ - t}} - 8{t^2}{{\text{e}}^{ - 2t}}\) 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="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = x{(x + 2)^6}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the inequality \(f(x) > x\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int {f(x){\text{d}}x} \).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f\) , defined by \(f(x) = x - a\sqrt x \) , where \(x \geqslant 0\), \(a \in {\mathbb{R}^ + }\) .</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Find in terms of \(a\)</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) the zeros of \(f\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the values of \(x\) for which \(f\) is decreasing;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) the values of \(x\) for which \(f\) is increasing;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iv) the range of \(f\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) State the concavity of the graph of \(f\) .</span></p>
</div>
<br><hr><br><div class="question">
<p>By using the substitution \({x^2} = 2\sec \theta \), show that \(\int {\frac{{{\text{d}}x}}{{x\sqrt {{x^4} - 4} }} = \frac{1}{4}\arccos \left( {\frac{2}{{{x^2}}}} \right) + c} \).</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The graph of \(y = \ln (5x + 10)\) is obtained from the graph of \(y = \ln x\) by a translation of \(a\) units in the direction of the \(x\)-axis followed by a translation of \(b\) units in the direction of the \(y\)-axis.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\) and the value of \(b\).</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 class="p1">The region bounded by the graph of \(y = \ln (5x + 10)\), the \(x\)-axis and the lines \(x = {\text{e}}\) and \(x = 2{\text{e}}\), is rotated through \(2\pi \) radians about the \(x\)-axis. Find the volume generated.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Two cyclists are at the same road intersection. One cyclist travels north at \(20\,{\text{km}}\,{{\text{h}}^{ - 1}}\). The other cyclist travels west at \(15\,{\text{km}}\,{{\text{h}}^{ - 1}}\).</p>
<p class="p1">Use calculus to show that the rate at which the distance between the two cyclists changes is independent of time.</p>
</div>
<br><hr><br><div class="question">
<p class="p1">A bicycle inner tube can be considered as a joined up cylinder of fixed length <span class="s1">\(200\) cm </span>and radius \(r\) cm. The radius \(r\) increases as the inner tube is pumped up. Air is being pumped into the inner tube so that the volume of air in the tube increases at a constant rate of \(30{\text{ c}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\). Find the rate at which the radius of the inner tube is increasing when \(r = 2{\text{ cm}}\).</p>
</div>
<br><hr><br><div class="question">
<p>The region \(R\) is enclosed by the graph of \(y = {e^{ - {x^2}}}\), the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\).</p>
<p>Find the volume of the solid of revolution that is formed when \(R\) is rotated through \(2\pi \) about the \(x\)-axis.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve defined by the equation \(4{x^2} + {y^2} = 7\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the normal to the curve at the point \(\left( {1,{\text{ }}\sqrt 3 } \right)\).</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 volume of the solid formed when the region bounded by the curve, the \(x\)-axis for \(x \geqslant 0\) and the \(y\)-axis for \(y \geqslant 0\) is rotated through \(2\pi \) about the \(x\)-axis.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove by mathematical induction that, for \(n \in {\mathbb{Z}^ + }\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + n{\left( {\frac{1}{2}} \right)^{n - 1}} = 4 - \frac{{n + 2}}{{{2^{n - 1}}}}.\]</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">A.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Using integration by parts, show that \(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{1}{5}{{\text{e}}^{2x}}} (2\sin x - \cos x) + C\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Solve the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \sqrt {1 - {y^2}} {{\text{e}}^{2x}}\sin x\), given that <em>y</em> = 0 when <em>x</em> = 0,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">writing your answer in the form \(y = f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Sketch the graph of \(y = f(x)\) , found in part (b), for \(0 \leqslant x \leqslant 1.5\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the coordinates of the point P, the first positive intercept on the <em>x</em>-axis, and mark it on your sketch.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) The region bounded by the graph of \(y = f(x)\) and the <em>x</em>-axis, between the origin and P, is rotated 360° about the <em>x</em>-axis to form a solid of revolution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the volume of this solid.</span></p>
<div class="marks">[17]</div>
<div class="question_part_label">B.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A rocket is rising vertically at a speed of \(300{\text{ m}}{{\text{s}}^{ - 1}}\) when it is 800 m directly above the launch site. Calculate the rate of change of the distance between the rocket and an observer, who is 600 m from the launch site and on the same horizontal level as the launch site.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 23px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A particle moves such that its velocity \(v\,{\text{m}}{{\text{s}}^{ - 1}}\) is related to its displacement \(s\,{\text{m}}\), by the equation \(v(s) = \arctan (\sin s),{\text{ }}0 \leqslant s \leqslant 1\). The particle’s acceleration is \(a\,{\text{m}}{{\text{s}}^{ - 2}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the particle’s acceleration in terms of \(s\).</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 class="p1">Using an appropriate sketch graph, find the particle’s displacement when its acceleration is \(0.25{\text{ m}}{{\text{s}}^{ - 2}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> has inverse \({f^{ - 1}}\) and derivative \(f'(x)\) for all \(x \in \mathbb{R}\). For all functions with these properties you are given the result that for \(a \in \mathbb{R}\) with \(b = f(a)\) and \(f'(a) \ne 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[({f^{ - 1}})'(b) = \frac{1}{{f'(a)}}.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Verify that this is true for \(f(x) = {x^3} + 1\) at <em>x</em> = 2.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(g(x) = x{{\text{e}}^{{x^2}}}\), show that \(g'(x) > 0\) for all values of <em>x</em>.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the result given at the start of the question, find the value of the gradient function of \(y = {g^{ - 1}}(x)\) at <em>x</em> = 2.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) With <em>f</em> and <em>g</em> as defined in parts (a) and (b), solve \(g \circ f(x) = 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Let \(h(x) = {(g \circ f)^{ - 1}}(x)\). Find \(h'(2)\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A family of cubic functions is defined as \({f_k}(x) = {k^2}{x^3} - k{x^2} + x,{\text{ }}k \in {\mathbb{Z}^ + }\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Express in terms of <em>k</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({{f'}_k}(x){\text{ and }}{{f''}_k}(x)\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the coordinates of the points of inflexion \({P_k}\) on the graphs of \({f_k}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that all \({P_k}\) lie on a straight line and state its equation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that for all values of <em>k</em>, the tangents to the graphs of \({f_k}\) at \({P_k}\) are parallel, and find the equation of the tangent lines.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A triangle is formed by the three lines \(y = 10 - 2x,{\text{ }}y = mx\) and \(y = -\frac{1}{m}x\), where \(m > \frac{1}{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>m </em>for which the area of the triangle is a minimum.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A particle moves in a straight line such that its velocity, \(v\,{\text{m}}\,{{\text{s}}^{ - 1}}\) , at time <em>t </em>seconds, is given by</p>
<p class="p1">\(v(t) = \left\{ {\begin{array}{*{20}{c}} {5 - {{(t - 2)}^2},}&{0 \le t \le 4} \\ {3 - \frac{t}{2},}&{t > 4} \end{array}.} \right.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of <em>\(t\) </em>when the particle is instantaneously at 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 class="p1">The particle returns to its initial position at \(t = T\).</p>
<p class="p1">Find the value of <em>T</em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{a + b{{\text{e}}^x}}}{{a{{\text{e}}^x} + b}}\), where \(0 < b < a\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(f'(x) = \frac{{({b^2} - {a^2}){{\text{e}}^x}}}{{{{(a{{\text{e}}^x} + b)}^2}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence</strong> justify that the graph of <em>f</em> has no local maxima or minima.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Given that the graph of <em>f</em> has a point of inflexion, find its coordinates.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Show that the graph of <em>f</em> has exactly two asymptotes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Let <em>a</em> = 4 and <em>b</em> =1. Consider the region <em>R</em> enclosed by the graph of \(y = f(x)\), the <em>y</em>-axis and the line with equation \(y = \frac{1}{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the volume <em>V</em> of the solid obtained when <em>R</em> is rotated through \(2\pi \) about the <em>x</em>-axis.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Points A , B and T lie on a line on an indoor soccer field. The goal, [AB] , is 2 metres wide. A player situated at point P kicks a ball at the goal. [PT] is perpendicular to (AB) and is 6 metres from a parallel line through the centre of [AB] . Let PT <span class="s1">be \(x\) metros and let \(\alpha = {\rm{A\hat PB}}\) measured in degrees. Assume that the ball travels along the floor.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-03_om_11.38.31.png" alt="M16/5/MATHL/HP2/ENG/TZ2/11"></span></p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The maximum for \(\tan \alpha \) </span>gives the maximum for \(\alpha \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(\alpha \) when \(x = 10\).</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 class="p1">Show that \(\tan \alpha = \frac{{2x}}{{{x^2} + 35}}\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>Find \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha )\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise find the value of \(\alpha \) <span class="s1">such that \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha ) = 0\).</span></p>
<p class="p2"><span class="s1">(iii) <span class="Apple-converted-space"> </span>Find \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha )\) </span>and hence show that the value of \(\alpha \) <span class="s1">never exceeds 10°.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the set of values of \(x\) for which \(\alpha \geqslant 7^\circ \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A lighthouse L is located offshore, 500 metres from the nearest point P on a long straight shoreline. The narrow beam of light from the lighthouse rotates at a constant rate of \(8\pi \) radians per minute, producing an illuminated spot S that moves along the shoreline. You may assume that the height of the lighthouse can be ignored and that the beam of light lies in the horizontal plane defined by sea level.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 27px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVoAAADKCAIAAABrIh/WAAAMAUlEQVR4nO3cT2gc5x3G8TnFlzRxVZ/6J7tkTxIIEopAJ5lAmio6SMYEE1BJC5JACgR80cUrBC31wSmsidRYUFJWrlJIAu4sNb3YCYNB7EFmkQ6mSAM9mJUQRSxmWTrGh+HtYXZnZ2VZu1rNzPvOO98Pe4icoBlLO8++M783jyEAQAghhBHXgdy6tTycyebar+l1+3lcRwfQXWxx4HlWKVzNDV4vHbyI97gAuos5Dp7bxenc4LJVd+M9LoDuiAMATcQBgCbiAEATcQCgiTgA0KRIHDy3//nAJiIAqdSIg8bWSuFRPd5TAXCMlG1IHXHgHm5t5H+3ZB3FeyYAjpO4SZkNy4BaYl4ddFer1VZXVxcXFxcXFzc3N2WfDpAiasVBtVodGRkxDGNxcXFqasowjNXVVdknBaSFWnFw8+bNkZGRarXqfbm6umoYxt7entyzAlJCrTgwDOPu3bv+l47jGIZx7949iacEpAdxAKBJrTg48Wbhr1995TiO3BMD0kCtOAg+SvT+4dcffJDLZD++ds22bdlnB2hOrTgQQtRqtbt37158482Biz/2Bo07OzvvXb6cy2TX7qyxTACio1wceLztSf6XjuOs3VnLZbLvXb68s7Mj8cQAjSUjDjy2bX987Vouk83fuFGr1aScGKCxJMWBEMJxnK83NrxlwsOHD+M/MUBjCYsDz/7+/qcLC94yYX9/P84TAzSWyDjwlEzTe8RYMs14zgrQW4LjQAhRq9XyN24wiQRCoWgcnEm5XPaWCV9vbDCJBPqmQxwIIRzH+fzWLW+ZwCQS6I8mceDZ2dnxJpGf37rFMgE4K63iQDCJBM5Btzjw2LbtTyLZsAT0SM848JRM05tQMIkEeqFzHIjAJPLThQU2LAGnUzQOetx30KOHDx8yiQS6SkUcCCFqtZo/iWTDEnCitMSBx69OYBIJvCxdcSA6J5HlcjmKQwAJlbo48FCdALwspXHg8SaRbFgCPKmOAxGoTmASCaQ9Djz+JLJkmjxiRGoRB01UJwDEQQdK3JFmisaBRH51AiXuSBvi4GT+JPLzW7eYRCIliINXojoBaUMcdEGJO9KDOOgJJe5IA+KgV0wioT3i4GyoToDGFI0DWfsOekGJO3RFHPSJEnfohzjoH5NIaIY4OC9K3KEN4iAclLhDA8RBaPb39ylxR6IRByFjEonkIg7CR4k7Eoo4iAol7kgc4iBClLgjWYiDyFHijqRQNA404zgOJe5QH3EQH0rcoTjiIG6UuENZxIEEVCdATcSBNP4kkg1LUARxIBMl7lAKcSAfJe5QhKJxoNO+g15QnQAVEAcKocQdchEHyqHEHbIQByryJ5GfLiwwiURsiAN1UZ2AmBEHSqPEHXEiDhKAEnfEgzhIBiaRiAFxkCSUuCNSxEHy+NUJTCIRLkXjAKejxB1RIA4SjEkkwkUcJBsl7ggRcaADStwRCuJAE47jrN1ZozoB50EcaIUSd5wHcaAbStzRN0XjgH0H50SJO/pAHOiMEnecCXGgOUrc0TviIBXK5TIbltAVcZAWlLijK+IgXShxxymIg9ShOgGvQhykFCXueBlxkGrehiVK3OEhDtKOEnf4FI0DxIzqBAjiAD5K3EEcoAMl7mlGHOA4JpGpRRzgZFQnpBBxgNNQ4p4qxAG6oMQ9PRSNA/YdqIZJZBoQB+gVJe7aIw5wNn6J+9qdNZYJmiEOcGaUuOuKOECfmETqhzhA/yhx1wxxgPOixF0bxAHCESxxl30u6BNxgNBQ4p50xAFCRol7chEHCB8l7gmlaBxAA5S4Jw5xgAhRnZAsxAEiFyxxZ5mgMuIAMaHEXX3EAeJDibviiAPEjeoEZREHkIASdzUpGgfsO0gDvzqBEndFEAeQKTiJLJfLsk8n7YgDyEd1giKIA6iCEnfpiAMohBJ3uYgDKIdJpCzEAVREdYIUxAHURYl7zIgDKI0S9zgRB0gAJpHxUDQOgGMocY8BcYAkocQ9UsQBkqdkmpS4R4E4QCIxiYwCcYAEo8Q9XMQBko0S9xARB9ABJe6hUDQO2HeAs6LE/fyIA2jFtm1K3PtGHEBDlLj3hziAnihx7wNxAJ1RnXAmxAE0FyxxZ5lwOuIAqUCJey+IA6QFJe5dEQdIF6oTTkEcII0ocT8RcYCUosT9ZYrGARAPfxJZMk0eMRIHSDuqE3zEASAEJe5CCOIA8FHiThwAHdI8iSQOgONSW+JOHAAn80vc8zdupGQSqWgcsO8AikhViTtxAHSRnkkkcQD0JA0l7sQB0CvtS9yJA+BsdnZ2/BJ3zZYJxAFwZrqWuBMHQJ/0K3EnDoBz0anEnTgAzkubEnfiAAiHBiXuisYBkES1Wi3RJe7EARCy5Ja4EwdA+BJa4k4cAFFJXHUCcQBEK0El7sQBEDm/OkHxEnfiAIiJ+iXuisYB+w6gJcWrE4gDIG7KlrgTB4AEapa4EweANKpNIokDQCalqhOIA0A+RUrciQNAFdJL3IkDQCFyJ5EqxsH29vbFN968NDBQrVZlnwsggVfi/vZbmd9+8slnn3128+bNeK4F5eJge3vbaBkZGVFqKgvExnGcK1NThmH86PXXDcOYmpqK4aCGEMJbnCjyuvjGm4ZhZH7+i0sDA4ZhvPXTn0k/JV68pLwuvHbhwmsX3n4r410UUV8Ltm0bQohyuVwyTUVev5meNgzj0sDApZ9cMgzjzpdfSj8lXrykvH71/vuGYXhZYBjGd99+G+nharWacjcL1Wp1ZGTE+/vPzs7KPh1AmuC1sLm5GcMRlYsDIYTjOHt7e3t7e7JPBJDMuxZi27CoYhwAkII4SCi3YT/6R2H+o+KuG+0hZn+Zt+rNP3hamh+bKGw1ojoiJIs/Dp7bxenjTzUHZ2/fe2Q3Intj66duLQ1mc5mhK9HFQfMQ2WHiQCnuYeVv+YlMNpcZmsgXLftZfXv7PyG9CSStDtzd9cl3W2/lum0uT2SywzMbuyRC7zp+htEdYqgdB5DvWaVw1b9S3MOtjfxkbrJoaxQHorVkGFuyjuScTxIRB+nj2sUrxy4T92np999oFgdu3VoejnTpqx/iIH1cu3glM9R5v+ZqdrMgWB10499PFUpb5cfbNSH8n+G/67a5ND6Uy4zOFZ+03yUN2yp6d5jZ4ZmVH+x6+8/vFebeWf5hd3NlZrS17Kzb36/MDb70H3fEwbEni3Xb+u72zPy6/V/v3HKD8+u7gdxo2D8UZocz2VxmdK7wgAdD4XCfluZHW08Nwk9piXEw1HovvjjcWpsbzObGVyq8aU7i2sWP5s0DVwgh3IMHG4+OhGjGwcT8nza2Dl3x4sC8PpyZXrefC+G9aT5spkNjt5SfbF2rR1Z+zHt2u7J1UK+sTIyvVBq1yhd/WNk69J7j7BbnhweXrbrbOkQrDjqeLHqruWwu8+FC4ZvHhy+a39m/iW1sreT/8vjwhRBCNJ6sz4yyxAiN9wttPkr8e8X7IYdEZhwEhgujcwUz3L+YRp7bxenhmUKpctgRlp0rrMBdpVu3ltuXtBCisXV73P+QP7LyY7ngv21d54FXa5l27Gahc7FQt5bbAeSt75rf1g+L4PAocEScV922ikvj3hU0uWTuhjXrUeRmAadxD8yFQS80v7M6VvInxkHnB7UQnRFwLA7curU8/KpH033GwZGVf59fbvT8W7yrtyvPQvmOxEEyuIeV+81nAaML5lNXdIuDjk/jbnHQvqqPHbXvOBgLcfoFn2s/uN/5m3IPzIXB0J7BEweJ4h4+/mK2eSNw+s1CxyfGkZUf63KzMJ7f8G9GGk+siv+0su+bheCdrdvY3qxws3Burr2xcGwbWKiXkqQ4aGzdHn+X/W29ceuP/vWoeS29ODCvN9f2dWsp8LHQMZFuPFmfGc2NL5fsuhBuY3dj7p3rpQPvynx57fCsUrjacZ/vP9PtMw6aTysC33OI33UoXLt4JfigzduhOP5HK6TnbtI3Kb9imYo2t75t3f/eXM9P+g+TXbt4pfUzHM5/X20/umsFRHvOFxhKtZ/gjk2Mz7Z/8u19r4FBY/AR42TRfhb88s+lr6YDzx3/3RxYBH6hzQ1zDBpD5doP7tv/a9jWejTDBf4XJgBN/weg+Dselu2uLgAAAABJRU5ErkJggg==" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When S is 2000 metres from P,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) show that the speed of S, correct to three significant figures, is \({\text{214}}\,{\text{000}}\) metres per minute;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) find the acceleration of S.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Using the substitution \(x = 2\sin \theta \) , show that\[\int {\sqrt {4 - {x^2}} } {\text{d}}x = Ax\sqrt {4 - {x^2}} + B\arcsin \frac{x}{2} + {\text{constant ,}}\]where \(A\) and \(B\) are constants whose values you are required to find.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A particle moves in a straight line in a positive direction from a fixed point O.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The velocity <em>v</em> m \({{\text{s}}^{ - 1}}\) , at time <em>t</em> seconds, where \(t \geqslant 0\) , satisfies the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{{\text{d}}v}}{{{\text{d}}t}} = \frac{{ - v(1 + {v^2})}}{{50}}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The particle starts from O with an initial velocity of 10 m \({{\text{s}}^{ - 1}}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Express as a definite integral, the time taken for the particle’s velocity to decrease from 10 m \({{\text{s}}^{ - 1}}\) to 5 m \({{\text{s}}^{ - 1}}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>Hence</strong> calculate the time taken for the particle’s velocity to decrease from 10 m \({{\text{s}}^{ - 1}}\) to 5 m \({{\text{s}}^{ - 1}}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Show that, when \(v > 0\) , the motion of this particle can also be described by the differential equation \(\frac{{{\text{d}}v}}{{{\text{d}}x}} = \frac{{ - (1 + {v^2})}}{{50}}\) where <em>x</em> metres is the displacement from O.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that <em>v</em> =10 when <em>x</em> = 0 , solve the differential equation expressing <em>x</em> in terms of <em>v</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) <strong>Hence</strong> show that \(v = \frac{{10 - \tan \frac{x}{{50}}}}{{1 + 10\tan \frac{x}{{50}}}}\).</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the gradient of the curve \({{\text{e}}^{xy}} + \ln \left( {{y^2}} \right) + {{\text{e}}^y} = 1 + {\text{e}}\) at the point (0, 1) .</span></p>
</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 \(\theta \) 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 \(0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water \(V{\text{ }}({{\text{m}}^3})\) in the trough in terms of \(\theta \).</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 \(\frac{{{\text{d}}\theta }}{{{\text{d}}t}}\) when \(\theta = \frac{\pi }{3}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points P and Q lie on the larger circle and \({\rm{P}}\hat {\text{O}}{\text{Q}} = x\) , where \(0 < x < \frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYcAAAERCAIAAAAfQ/x3AAAgAElEQVR4nO2db2wb53nA/c1B1zpBm2b2svXYMf8kVLEVzanSbGa1uLGkZJriNheuFrJmtmyf3LoaknQxqSiQESM+zpA6b+AVaBMZo4q0TZMj8mcBch4IfTkUHojcgGIQXiDDipxAFMHBE4RdIATEuw8PeTpR/Cvx7t737vmBX2yT9PHI+93zPO/zvu8eiiAIwhJ7gj6AKEIIIYRompZX1byqplMpeCRFMS7EOno4r1WyCrwbvLlt20F/SgTZIWglbzFN0zCMvKoqWSWdSg0lEnEhlhRFt0cMwwCVmKbZ0Zvbtk2qgOOWcjmQFDhrSpLSqVReVTVNQ1UhvIBW6jKmaWqaBg5yBOR4wbIsPw+GEAJOzMjylCTB8WRkGUIqlBTCJmil3WLbNlz5joZYvuxN09R1HUIqOFolq+i63mmYhiDegVbaIYSQpVwOApApSVrK5QzDYFBDzYHULyPLQ4mEYyifAzoEqQGt1AGWZem6DtcwvyZqBOSeGVl2PIsBFBIIaKXWWJalaRqERRlZ1jQt9NGEYRhLuVxSFIcSCSWrGIYR9BEhEQKt1BBHRnBl6roemrCofUzTdOsJoyfEB9BKdYA0DSKjaMpoO46ekqKYV9XQR4tIgKCVNrEsaymXg5qRpmkoo7oYhqFkFWjg1HU96MNBQghaiVJKdV2HFkdMUtrEtm1d1yGzw9AJ6S6RtpJt25qmQVaCmdrOIIRA6IRCR7pFRK1kWVZeVYcSiXQqRQgJ+nC4B88n0kUiZyXLsvDe7hFO7IklJ2Q3RMhKbh9hHcRToOSUFEVsdEJ2QCSsZNv2Ui6HPvIZcBPmdEinhNxKtm1DvQN9FBSapkG9CfNlpE3CbCVd17H+ygJ4b0A6IpxWIoRMSRLWNZgC6nrQ3xT0sSBMEzYr2baNP32WIYTAWsAYwCKNCJWVIGXLyDKmCYwDxaaMLGPnKrKdkFjJsiy8A/OFbduwUhV2NiE1hMFKcONdyuXwxssdhBDoHsDwFnHg20oQIk1JEoZI/ALdZBg0IQ4cWwmqSBgihQMImrDShFBOreSUJDBEChP4tSIAf1YyTRNvqiEGQmBs7IgynFlJ07S4ENM0LegDQTzENE3Y9RdvPNGEGytBeJ8URZxOFQWcbljM5iIIH1aCmydmbVEDsjkMjaMGB1YyDAMLDZEFyohKVgn6QBD/YN1K0CGJnSxRxrbtKUmakiSMlCMC01aCygIWkhBKqZJVsKoYERi1km3b0LSNt0fEAQJnrH+HHhatBBE71raR7ei6HhdimNGHG+asBMNtWN1EGmGaJpYaww1bVoIfHCoJaQ78TpZyuaAPBPEEhqyE90CkffAGFmJYsRIqCekUqD+imMIHE1ZCJSE7A8UUSoK3EioJ2Q0opvARsJVQScjuQTGFjCCthEpCuoVt2zhdLjQEZiVUEtJd8BcVGoKxEobciBegmMJBAFZCJSHeAWLCndy5JgArZWQ5nUr5//8iEQHWisPVBfjFbyspWQVXAkC8BsSEPzNO8dVK+FtBfAPvf/zin5UIIRhXI36STqUyshz0USAd45OVLMvCwRHEZ2BcBVd85w4/rAQ/Dlx3AvEfHJLjET+spGQVHHRDggL2yLEsK+gDQdrFcytpmpYURSw6IgGylMth5ZsjvLWSaZpxIYYVbiRw0qkUNu7ygodWggmTuPEpwgK2bWOBiRc8tBKWkxCmwAITL3hlJfgFYCaPMMVSLod3SvbxxErQnYTRcvuYpkkI0TQtr6p5VU2nUi0fSlaBJxNCcOPGNsEOJi7wxErpVAq7k5pg2zYhJK+qGVmekqS4EEuKols0hmGQVsAz4eYPbzKUSMCZ13UdRxgagSMw7NN9K2ErQF0sy9J1XckqSVGMCzFHH10Mc0zTNAwDQq2hRAIklVdVvAJryKvqlCQFfRRIQ7psJcjdMKFwME0TmmXiQiwjy5qm+XZy3B6ErdN0Xce7BYB5HMt02UoZWcbcjVZllBTFpCgu5XKBl9hM09Q0zZEjTkiEPA7H49ikm1bCcTfbtuHih9iEwdTJsixIsZk9Qt+AVDfoo0Dq4LbSp2TxRFyIbXn0nFp4c5msl1u+UcS71CzLUrIKlHK4iERM0+TrgLsOdPlG87MzzrZYqbxybax/fHGlTCmla0SdHRVifSdzK63EFNlOEEIIVJeVrMJdRgDBHWSamqZFLc6FNb+i9qnZp7mVaDWAOjJT+KTJu0QzS3d8lFdV3n/Zuq5D4hmCz9IRWAllkJZWKq8VZvuEXtff1AFGoD06RAYxTTM0PnJDCJmSpEjlNTBqHOX6GoN0IVYyDCM6DUq2bUM5JmQ+cqPrelIUpyQpIh0eWPZmjbpW6q0WkjZKN5TJnlh8+GqxcV0pOrdWTdOGEomMLEchV82rKnzYsMrXAcrekR2oYZD6VnINww1OzqvF0kaj12uaFoX7DKRsSVGMSPgAWJaVkeWhRCL0y9FAeBj0USAVWmZwzYBugNBfqHlVjQuxSBXO3ECGnk6lwh0hTklSREJ+9tmJlQzDWMrlfvXLX4a+G8CyLJj7GvFqqFNNC/F1i10C7NCZlSzLevLb37n3rrv3f/mOO/cfiAuxK1eueH+QwQCt6jhs7AAnRMkqYb10ozaUzCzbrLR+Y2G4f3T+xnq9Z58+NXnn/gN7quz7/Bd67rk3lGXCpVwuyq3qjbAsa0qSwho8YrjECE1mnJy4Rj51P9U0zYNf69uzldu/+KW/n/57nw/aU2zbhqwt3GWU3RBiZWO4xAIdzM4lhHzlj+6ssdJt+2596snwDF6YppkUxRAnKd1C1/W4EAvf2ByGSyzQmZV6771ve6z0zPe+593x+Qnss4q3yjaB0xW+7YzSqVT4bMsXna1kMpRI3HH77Y6Sbtm796t/8pVzU1MhSHbg5h/iMSYvME1zSpJCJiZCCPYuBUtnVjJNs//gwTv3H7j9i1/a/+U77op99dGjjz4xfjwuxC5fvsyvm3RdD2uhxGtgff6Q7UwbnekKbNLxqm+FQuHx0dEnv/2dhwYfmp5+du7ipbmLly5cmJmYePqBQ/0/PH+eu6ZKUFIoB5X8IXxighUUgj6K6NKxleA2klfViYmnQUnO48XZuYmJp/vvPyid5WZiJyqpK4RPTFGbXcQUnVnJSbnrWsntpocf+oZ0lvUWflRSF3HEFPSBdAfYFyvoo4gonVkJNiyjTa3kPE6fkR5+6BuPP/bY8vKyNwe/K6BTGZXURUBM4Sh+27YdwYUMGaEDK1mW5XxP7VjJcdPgg18ffvQYU26CUW3GQzkeCZOYnHsw4jMdWCmvqs6vrX0rwePcufMjx0YGDz/46zfeCLz0AMsPopI8IjTGxxaBoOjASu6VsTq1Ejymp58dOTZyeGDg9ddfD8pNcDPHObeeAuu4h6DTAleDC4R2rVRz39iZlRw3PTb6+OGBgZ/99Kf+5+0ZWQ736iuMEI6RBE3TsObtP+1aqSbH3o2VnBYnaL+8cuWKb26CXYYCTyEjgpJVeO8VgJo31x+BR9qyEqw56XbH7q1U4yYfWsNh4iXvd2++CEHlOyPLOC3OZ9qy0vZW125ZyXETtIbPpGc8al2DFePx5+UzMLDAdWkG+7z9py0rbb9ddNdK8ID2ywcO9XvRGp6RZSwQBAL0hXHd+MP78XNHayvVbSfzwkpuNx3qu186K3XrHgsXBlYHgoL3EQYlq2CU7SetraTr+vaflHdWch5Oa/guO1+gKMZ1EsE78BXwe2EbhoFJnJ+0tlLdap8PVnLctMvWcCWrYO4WOLyHq5jE+UlrK9WdDeSblRw3jRwbGX70WKet4bjgKTtwXdrDkTg/aWGlRrGrz1aCB7SGP/hnh9tvDcdxN3aA8ThOlwepW8dAPKKFlRpNUAzESo6bjj7yLZi20jyo1jQNywFMkVdVTr8RbKf0kxZWSopi3bbDAK0ED6f98scLC3XdFJHdxrmD38Vnp6SuDQojzWlmJZj8XfefArdSjZu2t4bnVRVDbgbRdZ3TifjuNTMQT2lmJU3TGn0NjFjJcdNTYtLdGg6BEk4uYRNOi32wV2DQRxEJmlkpI8uNgm2mrAQPd2u4klXwtsYssJxA0EexE7A/wB+aWanJd8CglRw3PSUm40IsnUphUYlZOK0uNblPI12koZWax6vMWmnu4qWJiad/eP481C84/fWHHk6rS01qGkgXaWil5l8Ay1Z6+KFvOFESuolZePxSsLTkDw2t1LyZlVkrnT4jTXz3RM3RQgtcUhTzqootJ4ygaRqPg6S48YkPNLRSo04lgFkrjRwbaTRjjhCSTqWGEgl0EwtwOk6aTqWwa8lr6lsJNllq8jI2rXThwszhgYHmxkE3sYOSVbjb1iGvqrgdk9fUt5JhGM2jazat9JSYvHLlSjsf27IsJavEhZiSVTAgD4ombbrM0vLSQHZPfSvlVbX5TYxNK/3Fw3/eUUaAbgoc7rY2gjnGQR9FyKlvpXQq1Xx8hEErnTt3fnRkZAenANw0lEgoWQVbnHyGx62NsJfSa+pbqXmpmzJppSfGj//spz/d8YmwbTuvqkOJBLZf+glUMPkq8GHB22vqW6l5qZsyaaX++w/ufkAH3eQ/3M3FX8rlsODtKXWsRAhpuQgOa1Z67vkXdpa+1cW2bWy/9A3u5uLzmHXyRR0r6bre8qSzZqWJiad/ovyk62cH3eQD3I3EQXNJ0EcRZupYqZ2ODNasNJT4pndZAOxTCOtv8FUB4QW+2ilhXcqgjyLM1LFSOxOjmbLSi7NzPlRMsf3SO7jbcI27Cj1f1LFSO4Vepqx07tx56axPq0Gjm7ygnaIBU+BgiKfUsVI79wGmrORRUakJpmli+2UX4a60hAsteUp9K7V8GVNWGjk2EsjQMraGdxG+Sks4G85Taq3UTlsAZcxKwS4u4W4N5+i6Yo2W0wmYouWULGQ31LFSO6Oe7FjpwoWZwcMPenNyOgDbL3cJX9EHNgd4Sq2V2qw7smOlc+fOf//c9705OR2DbtoxfM3FRyt5Sq2V2rxlsWOliYmnc4zF0rZta5qWFMUpSeIoKwkWvgreuHKAp3BvpSfGjzPb6oKt4R3BV2siX0fLF7VWarOfjR0r/dXjf8V4ruR2E7Y4NSEpiox/lW7QSt5Ra6U2CyLsWOmBQ/1cjMpj+2VL+CrG4SpL3sG9lfi6ZaGbmqBkFY6G4fhyKF/wbSWYAefNmfEQQojT4oT3Wwe+mgPQSt5Ra6UpSeLIStPTz5585u+8OTOeg63hNfDVmohW8o5aK7UZerBjJd/m5XqE2007aw0vl4rvLKZHhVhciMWFwcl5tVja6Ppx+gBfLUtoJe/g3ko/PH/eg9PiN43aL23bfu/dd3/0/I9+9PyP3nv33W2lqI1S4eVRYWzmX2+UypRSStdJYTE9Koy9VFgt+/kBugFfrYm4erd38G2l02ekmfSMB6clGGrcZNv2k9/+zn1333P7F790+xe/dN/d9zz57e+4xFReL14dFQYl9XdbBXSzOH88LhxfKN70/RPsCr6sxFcVjC/4ttLExNOvvfaaB6clSJzW8KN/+cid+w/scfHHB/7o4txc5XnllWtjvfGe2cLatqhorTDTE+s7q/IVL6GVEACtxC5xIXbL3r1uK/3B5z7nfEFlsjguxPrShbXtrwRhCSeukU/9PeRdgVZCALQSu8SF2J5tVL+g8lphtq+RlegnhfSRuHBkpvCJv4e8K9BKCIBWYpdD99//B5/7nFtJ+z7/hf6DBymlaKXAQSt5B1qJRSzLWsrlDn6t70+/IjhJ3C179957191OR0/rDK5uyYlh0EoIwL2V/uWf/9mD0xIYNa2Vr7zySs899x74wz+8c/+BnnvufeWVVzaf2rLaXT+MYhe0EgLwbaXQ9CvRxq3epmnCBbCtx7JRZ8DGqjrd1zOdX+Wsl5IvK/G1pC9foJWCxz1lt8OpJ2tEna3povz3+VN9PaeuVv+CI7izEvZ2e0StlZKi2M68B3asxO88ONqdJQTK62T5rflTfZUZJ2Mzi+9xOuNE0zSOdoVDK3kH32sGPPf8C8OPHvPmzHiLruvpVCopirikiQNflRq0knfwbaU53tZXorhsbmPQSgiAVvIP9FFz+Koft1nrQHZArZXa3KqYHSuNjowyfstyz7nFWeZN4Cv64Oh2yB3c73Fy9JFvMXup4/ZwHRHsHsidglbyDu6t9JSY/PUbb3hzcnaOZVngo4wso4/awbZtjq5zvo6WO2qtpGmaklVavowdKz3zzKnLly97c3J2Aq57uzMIIVMSN8uK8tVaxR21VmrzdLNjpenpZ/8mmfTm5HQG+mg3tHk7ZAS0kqfUsVI7tyx2rMTCNieEkIwsDyUSS7kc+mhn8LXtEl8O5Y5aK7WZMLNjpbmLl/7i4T8PqnaD+7t1izY312EEvlqruKPWSrS9wQWmrPTE+PF2diHvLrAhB/qoK8C9kKPTmJFl/39y0aGOldq5azFlJZ8L3tgM2XX4KnVT3lqruKOOldpp9mPKSr7NhkMfeQRf+1NSSocSCSwgekcdK7WTMzNlpbmLlx441O9d+z80QyZFka8pERwxJUnMtsLWJfABlnBTx0rtjC+wZqXHRh9//9/e7/rZweZsH+CuqMRdvskddazUTi8Ga1Y6fUa68MKFLp4XpzkbfeQ1uq7zdZHrus7ROlA8UsdK7TQHsGYl6Frqyv0WmyF9RskqfI1nYVuA19SxEm2jmMealeYuXhpKfHOXRR/0USAMJRJ8LQmCaz94TX0rtTzvDFrpmWdO7TiJI4QoWQWbs/3HMIykKAZ9FJ2BKyt5TX0rtYxRGbTShQszO0jisDk7WPiaaEJxtQBfqG8laFxu8jIGrTR38dLIsZHl5eU2Pzn6KHDgCucrOMV5uT5Q30qWZQ0lEk1exqaVTp+RJr57ouVndjdDoo8ChLvRN8phwyeP1LcSbVXwZtNKL87OPXCov8lhY3M2U0xJEndfBHbS+kBDKzU/+2xaae7ipSfGj1+5cqXmaG3b1jQtKYo8XgZhhRAylEhwF6viXBMfaGilvKo26fBm1koXLswcHhhwfuvYnM0sSlbhLhUyTbN5ZQPpCg2t1LytnlkrzV28dPSRb/36jTfQRyxjWRZ3dW6KXd1+0dBKlNImA+0sW+n0GWnw8IPQDIl9JWyiZBUe13Lkrg2dU5pZqUkvJZtWunBh5onx43EhNv7Xf42/HmaxbZu7fm6A08PmjmZWalJaYs1K09PPPjF+/PDAwI8XFizLIoRw1zEcHfKqymPLDxaVfKOZlUzTbHRts2Ol6elnjz7yrcMDA6+//ro730ynUhguMQhUlHiMOHAHAd9oZiXaeByUBStNTz87cmxk8PCDUNuuOUJOB55DD6cVJdr2ZvfI7mlhpUblvWCtdPqMNPjg14cfPba8vNzEO+lUiq85VqHHNE0eh94Afo+cO1pYSdf1uiWAoKx0+oz08EPfGB1pa74bp8PPIYbf+4RhGNxNjuGXFlZqtHqpz1Z6cXYOfDTx3RMdRdFLuRyPhdVQous6vzk1d2sbcE0LK9EGU098s9KLs3MTE08f6rtfOruTXQxhEBrX6Aoc+CL4rctgT4CftLaSpmnb+1l9sBL46IFD/T88f343zdmGYfB7iw4NSlbhN2jlcWk6rmltJajO1FzVnloJmiEfONR/+fLlrtygMrKMEwUCBMZD+S3w8Thlj2taW4nWW3HCIys5zdmXL1/u4o8Y87gAgZPPde8Ypm8+05aVtidxXbfSc8+/AD6C5uyuf07I4/i9XfNLRpb5zd0opm9B0JaVto/EddFKjZqzuw6Ox/mPpmm8F/Vw9M1/2rISpTQjy+4gvCtWcpqzvfYRYNv2lCRhgcA3oGeS62VkIP3EENtn2rVSzRLLu7TS6TPSUOKbw48ee//f3vfzRgoTLLHA5AO2bSdFkfcoo1EXMeIp7VqJbq357dhK0AwJk0W8+UQtgAITFi+9ZkqSQjDuiat0B0IHVlrK5Zx5lZ1ayWnOfvyxxwL/mmENb66LHYyjZJUpSeL9DMNOP7x/Ch7pwEruL6l9K+2yOdsjwnHZsAksTByCWoz7Noz4SQdWoq6Ath0rOc3Z3z/3fXZ85JBOpUKQYrAGTHYLQYLM74KZIaAzK8F+arSVldzN2Qz6CIAhObwZdhFd13kfdHPAOneAdGYlSmlSFAkhjazkUXO2R6CYuggoKfCiYbfArUwDpDMrWZb1vb/924FD/d9MJAa/Pnjhwsx2H3nUnO0RKKauEDIl4brvwdKBlUzTPPi1vjv3H7ht36237bv1zv0H7onf9dzzL/jWnO0RKKZdEjIlUVz0PWg6sNLpU5P7v3zHHhf7v3xH7733Pfhnhzn1kQOIKZ1Kcf0pAgHK22FSErTa4i8hQDqwUlyI3bJ3r9tKt+zd22QnS76wbTudSmG7QEcs5XLhG6jCiW+B05mV9mwjLsS8Ozj/UbJKUhRDdpl5gW3boTxX2DnJAi4rlUvFt1+dGe6NC7G40DuafvWdYqnseurRv3zktn23upV0275bw7dvH0xzx7lyTbAsa0qSQhlX8rsxVJioWKlcuv7ScP9o+ufF0gallNI1cv3qZM/g5D/pjpnee/fdu776p/s+/wVQ0r7PfyEuxN57991gDtxLYK4cri5QF1hYMpSXLm6Kwwh7KKV0/cbCcG/fWXXVHRrR8nrx6qgwOLn42/XqX/3ql7+MC7H77r7nvrvviQuxEH+FEA6kU6mwfsCdAbNJwlTbdoMr4TLCHko/JYsn4sKJa+TTbf/6SSF9JN4zW1jb1JVt24QQQghUFkJ5z3SAai5mc9SVtYWskOSAQ2/ssIeWV66N9daopwoI68hM4ZO6L45CxAvZXEaWo/x7hVrbUi4X4pPA7w6a4WMPXSvM9MQaWKm8VpjtE2J96cJag9fnVTX004Vs287IcjSDJtM006kUTDMK+lg8BIplIXYuX+zWSjC1Otw/WQBWlY9Opcm27byqxoVYuEMkYEqSsJmbHdrJ4HrHF1e2/5uDpmkR2YI9OhcqLA6RTqXCWkVy46yEgTDCnqbFI6h2T+dXN5q/S6QmWFuWBQldXlXD5yZCyJQkRecLjU6wzxF7KKXlVVXqiW3rDIC/7x2dv7Fe/7WbRDAtJ4SkU6kwucn9iYI+Fv/A/bgYBLooy+srucme3i1dlIXFmeH+0dnrpSbJm4uMLEew1yMcboINbHj/FDsA9obyplBYXifL+TfnJ3t+kC99Rju5xyObM07KpWJ+/lSfEIP2yHjPqYXrpP0zCBOIohkJE0IyshwXYkpW4agQA2WypChCvhYpHwEedgOU1Em4jsYWSZlS+llJ/UFciMUPzRc/8+Q/DBM1s3PL60SdGe6NC73tR0kO0Sl718WyLOi6hAEdli9ywzBAoxlZjmC7AwBb3Xj29hur6nSfa6QoSrFSeX3lFwtvf9zgX9dWFpV3Ss3cXG/NgHKp+K/pUSEWH05fe7vYkZumJClSVYm66LruXPNMxSCGYShZZSiRgP0jI9LiUBfvQ/uP8yf748LIQjECFtrCGlFnR+vPFaF0fSWfHqvGjw1pvJJJuVR8W31r/lRfq84AN5Coc5TFeIdt246eYG3DQE6LZVlwGCCjpVwOvx3qWRm0XLqRS4/Fhd7Rf/iHyc02wI1S8f38YnrUlb5Vnzkovaq80BNzXagbpaK6cHIwLsTiwtiMurJptXVSeHN+8tBsYe3myuLZvs1uno1S8eczw73xnh+89upzrgu2vE6W84vp8XRhrSIL+KdqSjR8tbhe98quHPB4urDmvPnWJ5dLxXcW06OV9UVUsl6ulqerJSChUlCrsrayeHazQHRSLTV8n853E2hJXlWjnMdtB/QEqxHB5JW8qnpagDNN0/kfg3Uim0BDbNdj2HLp+kvDvfHhlwulNbJ4Il5pP660Isdd3cjVZ14trv++kD4SF2IDFV9tlAovjwqxeM/ZaytrlQ5nAez2P/mT/XEh1pe+vlq8OirEqnNX4SW9o/M31ivPrwZoldpW/6T60ao6DccwMF/cWFWlnrriqPBZcX5AiMWF3vHF/1yF49ny5Ip9+s6qq2X4dI4Hoc3I+Tg1J2jl2livK35s+D7dtxLFPK4xELnAHplxIQbbXudVVdd1mPC8g/ckhBiGkVdVJaukU6m4EPPHfZziVe5W/l3+7GC17w9q2861Cn+stgRueeZ6cX7E9U9w3VZfCFpxYqKSOin0jr/6q9fGeuNCpZUH2noqz/msuHDIPU8D3jy5sDg/s3ijCG2J6ttX0z9fWb0+syVAq/kscBgnXrv+i5fmrpduFtxPrv6P0MZYreJD7FN5Yf2Zs2WyOO76OE3exxMrYR7XJoQQXdfzqpqRZRAKPNKplPMAv+RVFTprnAfEQaC2dCq1lMuBhqJcLWoHOFfdftdqQFRJcz4ppI+41uGAP0KsAQsEOc+E8lM1DNkSTVTfs3IZwx+T/zifljYXF7pZnD8eF6pFdLBYNTmqvHnPyOR8ofQZvHP/6MmrhdIGREONZpJV9CEcGZ99b1s0BJ+lwR87mL7W7H08sRKt5nHsFHo5wrIs4gK0BQ/336P0dwCMu3nws6x3cTpVpDV3rPGJO2WrxAuuotKqOt0nHF8o3oRVzzaNA8lRz9Hx9Nub3c5bUrbaUb8tjqg88/hC8WartUAcG0IUAwdcfXLlfarC3fLH6gvrh2D1TlH99/HMStSrmxKC7BAPQ3hInSqXLsQvsb709Y8//PCjmlijEg31T6ofO88cmNd/t6x/BPlR6UZuPj0JC1X3nFpQq4PglfmqW6Z/VYOaH+RLn1UtNrJw47+Wl/+7TN3/79Y4pdnUV9oi/IFwrCLcyjzZqjdrwp+t1Gio2ft4aSVI4CPbDoMwBWyu5VW5c9NKq6XiB9fSR+NCrO97F9/68KYr1viILH9YzaSOzBR+v04+gLG26jMhuRubWXwzr9/kApsAAAboSURBVC6TraNjIKCanMtVqblJrl+d7InFhcFn0r8yarPILdWryqvGXi2u/KY6l8P9ppWiEuijUvk++SYhvymWNlyS+qw6IaQycOZKP1fJ8oc1HUWV//TQfPEmWS6Wyk3ex1Mr0eqSaVjpQAIHhgI8e/tK9jSa/nmx9Pvi/HHXoD7YYXBy/gOyXnY9c7FAPiqkj7ieWTOyHosPz+YJWAiiCYiwXFQK52MziwUCBWznJe4YZ6toKmnj5ptvfcuamjT8cfPJG6UbSuUgh9PXCq7pH5WDcT7p1ret1LadGSON38drK1FKl3I5bBRAggW20mS8ylkuFfPzsy+pvykW3n7LmfvVquEwlHhuJUop7peNBAiUk5hvkvikkD6yxUFQJ4rkvDk/rARL2ERkvR6EKWzbTooiD+tMQo622cxd2QwtEvPmavHDShQ7mJCASKdS3MTp5VJRnd+sK22rtkQHn6xEOcntkTABPfT4k+MO/6xE8VeC+AjeBfnFVytRSmEKhc//KRI1oCUFKwac4reVoJmNm1Qf4RDYBRdHV/jFbyvR6pAcD8MiCH/AjAJUEtcEYCWKdzPEGzASDwfBWImimJBug0raKRul4vtvzZ/q21wCJWACsxKlVNd1HppuEQ5AJe2U6nJrnSyE7TVBWolWh29xrATZDaAknG65U2BFgQbr/wdBwFaiKCZkdzhKwtaknfJx/mQ/UxPugrcSRTEhOyXCSoI9S2JxYXBy/hdv/YtqbO6e4uwaMjj5T3p1naNG+6BU1jzZunLTRqn43rX0WNzZ18BfmLASRTEhnRNhJcGyR2CWjVLh5dFKobrsWkHt01Lh5dHKiuCN90GpLFPpWi23vFqYHYsPv1wo/d/WPRH8gxUr0aqYcFQOaQdQUjqViqCSaHWJyL6TuZX1MqWWUfxdeXM5t6vF9cpKb23sg1Kz8QEs2nt8oXizuopbAPUmhqxEq6NyKCakOTjiVlVGLD78cqGyyu3WZf83aboPypadDir7r/SlC/9b2URz7KXCqv8Dc2xZiWIfE9IK0zSjriRg/bfXYNlvKAk12vWo2T4odXdVilW27V18r86q3r7AnJVoVUz4s0O2A7+NiO+dUyYfvANZlXu9yi27hlBKabn0W6O00XQflK1bMG3fY3L9v94p/DfGShWc+2E0qwZIXWAlgMjH0eW1wuzASeU/ShuQx1VipYpxBicXf7tOy+tk+Z1iqdx8H5TNLZg+LRU/qO5uUMn1yqXiO28XS0E0VjJqJRrtERZkO5qm4UZelFJKP/1oWV9ZgV2b3BuKlNeJOjPcGxdifSfn845PmuyDUi1O9Z28+u9kzf0OW7ai8x12rQQoWSUpitgxEHGUrIKNI9GBdSvR6k0y8nF7RMGQOYJwYCVaLShEvMYZQZxxD1RSpODDSrRa/45s11wE0TQNm9eiCTdWopTatg31BVz8JNzYtp2RZawnRhaerATAxJS8qgZ9IIgnmKaZFMWMLGNQHFn4sxJ1ZXOWZQV9LEg3yatqXIjhmu4Rh0srAUu5HI7NhQbLstKp1JQkYdaGcGwlSikhBKP9EADNH0u5HH6PCOXdSpRS27YxaOIXCJGSoogjGIgD91YCIGjCShNf5FUVQyRkOyGxEnUFTTg8xz6EEOjYxhAJ2U54rATA8BxmBMziNJ3hQBvSiLBZCYCepowsY0LHFFDVxu8FaU44rUSrCV1ciGHZggUMw0iKIqZsSDuE1kqAZVkZWYZiE7opEAghMMqGg6RIm4TcSgBcGNg94DPOacfxB6QjImElAG/avuH2EYaoSKdEyEoAXjCegqcX2T2RsxLgXDxLuRyOB3UFXdehJwN9hOySiFoJME1TySpxIaZkFRwb2hm2bedVNSmKmBoj3SLSVgIsy4KpD1OSpOs63ufbhBACTs/IMjod6SJopU10XYe0DkOnJliWpWkaBEd5VcX8F+k6aKVaLMtayuXgqsOqk4Nt27quZ2QZgiPclw3xDrRSQwzDgBlbU5KkaVo09eSWEWa4iD+glVrj6AmipygslmhZlltGkZUyEghopQ4wDAOSO6g96boesmuVELKUy01JEqRp4fuACBeglXaCaZqapkEokRRFrg1FCMmrajqVgrBoKZfDmhESLGil3eJc1UOJBCzTkVdVQgiz9RfTNHVdd2Iix0TMHjASNdBK3cS54CH0gEV786qq63pQnjJNE7ypZBXQEOy/wLg6kSiDVvIQ0zQNw8irakaWwQhxIZZOpUBVeVU1DIMQsvveKFAPIUTTNBAQzEN2zLiUy2mahhpCuACt5Cu2bRNCQFWQ98EmaCAst7ZaPkA68AD1pFMpJavkVRUEFIWxQiSUoJXYArTVDigdJKz8Pw/MdlYn3FbnAAAAAElFTkSuQmCC" alt></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Show that the area of the shaded region is \(8\sin x - 2x\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find the maximum area of the shaded region.</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">An earth satellite moves in a path that can be described by the curve \(72.5{x^2} + 71.5{y^2} = 1\) where \(x = x(t)\) and \(y = y(t)\) are in thousands of kilometres and \(t\) is time in seconds.</p>
<p class="p1">Given that \(\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}\) when \(x = 3.2 \times {10^{ - 3}}\), find the possible values of \(\frac{{{\text{d}}y}}{{{\text{d}}t}}\).</p>
<p class="p1">Give your answers in standard form.</p>
</div>
<br><hr><br><div class="question">
<p class="p1">Find the equation of the normal to the curve \(y = \frac{{{{\text{e}}^x}\cos x\ln (x + {\text{e}})}}{{{{({x^{17}} + 1)}^5}}}\) at the point where \(x = 0\).</p>
<p class="p1">In your answer give the value of the gradient, of the normal, to three decimal places.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sand is being poured to form a cone of height \(h\) cm and base radius \(r\) cm. The height remains equal to the base radius at all times. The height of the cone is increasing at a rate of \(0.5{\text{ cm}}\,{\text{mi}}{{\text{n}}^{ - 1}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the rate at which sand is being poured, in \({\text{c}}{{\text{m}}^3}\,{\text{mi}}{{\text{n}}^{ - 1}}\), when the height is 4 cm.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">By using the substitution \(x = \sin t\) , find \(\int {\frac{{{x^3}}}{{\sqrt {1 - {x^2}} }}{\text{d}}x} \) .<br></span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A cone has height <em>h </em>and base radius <em>r </em>. Deduce the formula for the volume of this cone by rotating the triangular region, enclosed by the line \(y = h - \frac{h}{r}x\) and the coordinate axes, through \(2\pi \) about the <em>y</em>-axis.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Two non-intersecting circles C<sub>1</sub> , containing points M and S , and C<sub>2</sub> , containing points N and R, have centres P and Q where PQ \( = 50\) . The line segments [MN] and [SR] are common tangents to the circles. The size of the reflex angle MPS is \( \alpha\), the size of the obtuse angle NQR is \( \beta\) , and the size of the angle MPQ is \( \theta\) . The arc length MS is \({l_1}\) and the arc length NR is \({l_2}\) . This information is represented in the diagram below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAiQAAADdCAIAAAAEk/7fAAAgAElEQVR4nO2d/2sc193v58+oflrB9hep3fYxZoy5EGNoELICGmFqaGKi9sbjViHxc2PIigYpcm7q4bkgkIinIVDYriampFyYrHDIbTdYLIYhT6po7z7Q8FizNFB3Ii3FDL7L8owQZjj3h8/uaPartLszZ86c+bzID7G02j27O/N5n8+X8/kIBEEQBEFCRoh6AQiCIAj/oNggCIIgoYNiExD10urUZDo1mU5NL+T33V4Pcav5hRQ85vpW9Yj2ChEEQaIDxSZAjg8Kb2VSk+nZe+VGt9w8La1cGiBFCIIgHINiEyBuvfT+wo1fzKUuvl540qEo7kHhX2/I/30KxQZBkCSCYhMgbr30/k9zn3+6dDE9n6+2Scqz8sbbmw//9yqKDYIgiQTFJkDceun9n+b/8/+V1jKpS6ulpye/OCj865uFg2clFBsEQZIJik2AgNjsu+7+1vy0z7k5quZXNsvPSB3FBkGQhIJiEyAtsSFuvbSWSV3dLD8jhJB6afVqvuoSFBsEQRILik2AeGJDSGN3c3Y6s1Kqk6Nq/lYzpIZigyBIUkGxCRCf2DTLoK9v7X7edGsIig2CIMkFxSZA/GLjHfP0VQqg2CAIklRQbALk+KDw9onYkKNq/nrbAU8UGwRBkgqKTUCctKuZTE+tleouIcQ9+NOHD+B0J7QPaD0A29UgCJIwUGwQBEGQ0EGxQRAEQUIHxQZBEAQJHRSbTizLMk3TMAxd13VdVxQlexqKoui6vrCwcO3atdu3b6uq+rvf/a5QKJimaZqm4zhRvycEQZCISbrYWJZlGIamadlsVpIkoQtJkk4Vm55/2I1fmXRdNwwD1QhBkISQOLFxHKdSqYC6+JVAlmWQAdAAy7JGeHLbtsErEgRhdnZWlmX/S/zoRz+am5t74YUXUI0QBEkaSREby7J0Xfdbf1mWNU0zDGM0XRmA4ziCIOi67r10t7zJsry5ubm9vW2aJkgLPGCwn4RqhCBITOFcbCqViqqqoij6BaZSqYRqnU3TFATBNM1+v/XLniiKqqpWKpXuR0L2aAQ1ymaz8CfFYhGewbbt8N4vgiDIqfApNpZlaZrmaYyiKIZhUDO4EEY71WFyHMcwDE8LB6hOP0ZQI1mWUY0QBKEPV2ID5ttzGkBj6IeYdF0XhOE+2EqloigKLFuSJF3XA7H+/dSoI5mEaoQgSNhwIja2bXsuQoDGejTAtxjhDzvEUlXVfrG4oICKBtM0i8UiqAuqEYIgYRB7sTFNU1VVz5UZKgwVEmCOx3kGy7K8N5XNZsOWnAGMr0a6rsMzBF6IgSBIjIix2Jim6dV3qarKzs5a8JWijYNt27qug7sWreQMYAQ18o4uaZqGaoQgCSGWYmPbtpfh0DSNHZkhhNi2LQhCsVgM6gkdx2FfcgaAaoQgCImd2DiOo2kag96Mx+C655HxS46iKAy+8ZFxHAe0pLtFEKoRgnBDnMSmWCyyb22LxaIgCCEtz6+1uq4n5CxngGoUO78QQbghHmJjWRZEXWRZZtxegBiE+hK2bYOdlSSJ8U+DDiOokSiK8ABVVeFPKpUKqlGicKv5BRhm6B+nSwghbr20lmmNOsyslOqRrZErWBcbCB+BdQgwERIekI2g8EKVSsXz8xLi4owMqhHSh3q1sDaXmswsFQ46prW7T7bfXNk+OI5mXTzCtNh4Dk02m2U2btaBJEmKotB5LS+qJooiCzXf8cVTo0qlAtKiqiqIjdeHAtWIT9z9rfnpdGp6bmO30faLp6W1dZjvjgQCu2ITL4fGQwio7vnsmKYJ/Wk0TUMXJzxQjfjE3d96c+PT3FImdfFm/huf3qDYBAyLYuM4DsQ3YuTQAJZlCYJgGAbl13UcB2rBZVnGiqyoGFaNBGzjzQLu/tab96vPD0pr8+nU/J2SF05DsQkY5sTGNE24OSn7B4EQUt3zGTEMQxRFURTpqx1yFlCNWATExiWk8c3WjYvpqaWtfSgIQLEJGLbEBuqGRVGMaagBQn8RGgIvy6VpWlRrQEamQ41wxBENPLEhxK09vDM7nZ69W6odo9gEDkNiA93AZFmO730CbyHaNXghtWw2G99PEunJyCOOUI364hMbQtzG/v2bU1AM/U8Um2BhQmy8JI2qqlGvZSzgxo56FYS0fCxZluOV9ELGBAfuDU2b2BBCjmulu3OpyczSb3+/gmITJNGLjeM4EPmJY5KmAxiAFvUqmsAMN1EUsWQA8cCBe510ig0h5Pig8FYmNZmeWkOxCZCIxcayLMiO8pHTZk0y4eNFvUHOTuIG7jV2N+ff7xKVZ+WNqyg2wRKl2HimkI8DiVCKxtp78T5kPuQciRyOBu4dVfPX0822NNML+f02YXGfbL/5v1BsAiQyseFv012pVITo6p4HYFkWBElQbxAK4MA9pCfRiA1/SkNaOfmoV9EbLzGGeoNEDo44SiYRGEculYYQoiiKKIpRr6Ivnt4w6HshiB9UIy6hLTa2bXOpNISluud+cPzhI4kCB+7FEapiw/fmWojDuX1e3UoE8bAsa2dnR1XV27dvv/LKKy+++GI/BQIuXLjw5ptvwolXvC/Cg6rYcJw2cBxHYKzuuR+gN5Ik4RlyhA8cx6lUKpqmdcTZvMbbeheffPKJqqqqqr7yyiuCIHz/+9/3/yE4QJVKBe+RAKEnNtDKJV7zAs5OtC04hwUK5+gMeUOQkLBtu1gs+kNnsixrmgYtec74JP5O7ZAlUlXVL1qKohSLxairtHmAkthAh012TtcHDrzBGPng3H8jCMcYhuFpjCRJqqqO7IX02yaCt6SqqteTW1EULqMy1KAhNvB18r2PZrnuuR98+5oIf9i2rWkaWH9oDTX+9u4sndpN0/RURxRFXdfR0RmB0O2jVwHFd/QTuktFvYqhgXBBjBwyJJnYtg17I0ioBOhhDNWp3e9RqaqKkjMUoYtNQswZHIGOehVDk5CtABJf/DKjqmrgadERTiyYpglTPGBJeO+ckXDFRtO0hARq4lKK1g0EORVFiXohCNKJrusQvFIUJSQ3YuRO7bZtg5cDgbXAF8YfIYoNWLE47veHxbbtWGtqcvYESFwwTRMa+smyHGqR55jbRNM0IXgjSVJcilGjIiyxcRxHkqSExGfiVffcE1mWRVHEGDQSOY7jwO5HFMWwN0BBdWovFovggWmalgSLNxphiQ1cLqz12w8JKGiJtaWG0wZJcEMRlrEsCxwFRVEoWG0YMBjINtGLqsmyzH2KejRCEZukpQFAWaNexbiAZGIwDYkKwzCgXIXaRRj4iQU4vobjo3oSiolMTgANgH60Ua8iADCYhkQFlJxRdgsURQn8xII3Por9TomUCV5sYLOQKGGXJIkPNy5pLinCAo7jQACKTujMT0id2iN8RywTsNjAuY2khf7jW/fcDYQEY13sgMQIrxN8JJ2TQvU/PF8N9QYIWGzgrFOi8mP+Rn4c4DiOKIp8RAURxol2gCycWAh1mwgpHNQbIEixgSBM0no7QgdlnlwBuEO4kU+ETSIfVU7nxAIUvKHekGDFJpvNJqouADhLI7/YIUkSDrxBwiNypSGtTRWFchjUGyAwsYFtAjepi7MD7WCjXkXAwO2RwG8ToUPkSkPodmr39IbOy7FJYJ91Mt0aElpBS+Qk9gtFwgYy55FvZSifWIDSm6RlGfwEIzaJdWvIGI38GCfJ3ykSHuBPsHDLSJJEeZuY8AlSwYhNknfBHFtkOOMZ9SoQfmBqjiL9O9fLVPFUT3R2AhCbJG+Bg2rkxyYQaMayNCQQoKqekV0pnFig72QkeYJUAGIDZ2sS+NmRQBv5sQmUpUW9CoQH4FA9IzdLhJ3a4bBEAvt0jCs2cDCKhQhsJNAsaIkEKA9lxEAg8QUuJHbahUV7YiFRTfE9xjWU8KkltnVjGI38mAJCHwnchSEBArEjpk5uRdupPVHjvjzG/bgT2AnND691z36ghCax+wlkfCDSzpR/HPmdm8B+K2OJDQQfk5xAZioyEBKQSk1mAQgyPmxaVRb8dQY1OFTGEhtFUZJcGus4TkKssCzLfEcLkfCAAnrW4kUs3LlJ65E/utiAqWVtw0KTCAtaKAPZ3UQ180YCgc2+R+ycWEjU9K/RxQYuIxa+sKhIjgmGmkPuA4ZI4LCZBmenUzsU4CQkbDC62HBfiEUIsW27Uqnouq5pWrYdVVUhcx71GimBkTRkWNh0awhjJxZgMUnYtY/4ifMdQ6tUKqqqwiBxjw6xgf0II403KJAcNw4JCjazNYSQ5eVldpLN4NwkIXMzothwGUNzHEfXdU9jstmsruumafa7W2RZTsIlAmAkDRkKlrtYwQ0uy7KqqiwYMTj0w/1ObkSx4WyIC8iMKIpwCRqGcZbtmKdJmqZVKhUGd3DBIstycjw5ZExYPp4lCMKtW7egfY4gCKIoapoW4VIT0odlRLFhoUo9KCqVCshMNps9e84Qro833njDu2Sh3xELWceQSHi3COTsQJidTRPhP7HgOI5hGN4trGlaVFtGOEbC94Z1FLGJqmFq4DiOA+eqJEkaViQ66p4hzTOCaMUIeMsJKdNExoHlMHvPEwu2bYMpEEUxkmUn4YD8KGJDbXZ3qFiWBemZ0bYzUEPS8SH4w3GKovC3T0mCs4+MD8vHvQeUupim6dkE+gvjKVzUk1HEhoOiZ8uyYKrEyFuJAY38HMeB30a1SwqPbDYb968eCRvGS1UH1z170Q7664csF387VI9RxCbuCuwpzTjlH6c28vN2SRzEGz2ibcyOxAKWY2iEkGw2e2qdC9h9ynoDkTRmP7fxGVpsIDEeXwMaiNIQQiRJOlVxvSmwzO7yhoWdPh8IszB+2PmMJxbo6w3jHuH4DH1NsNPpYQS8wbTjl7Sf/QwBXLVsHjgYluT0HkVGRpIkls+fDXvn0txY8x2mHlpsmOr0MCxQ4zj+xrxvXZZ7uLe1fFkQBGFmWX/caH9dPkpNzuLSIYkFIh/MbkeGraSFJgjUiqF6lh1xw9CycZaIJ5tAFUogt0Fv967x19xrb2z85dAlxD384p3LC+t7z+A3EE+jedWGBwflIUh4QMKG2cjHsJ3aQZyoOWp8h6mHFpuYbmxhdERQMtkrT/5sb31xeedp65+NvfXLE8s79da/KV+14RFr1xYJG8YvjxEqXOBP6Mgn32HqoS+LmH4WEH4NqvtQd7ce18zNzORM1/tBY2/9srCoH/oew8fsilgn7ZCwYTzyMeDEQj8oN8pkPOM1DsN97jE9Qx5466GuuucjM3ctvb73/OQnT3eWz3eIDSFEkqS4x6D49vSRMWE88nHqiYWewDaRTqNMjmsERhGb2O1qA+8JKIpim3S5j3MzE0IX7fJDSCuiHTu17iCm3i1CAcavjdHOCNJsec54HHIchntXcTzTF7hb0yOueqgvCr/SD0+UxTVzM0J6Uf9H959z4NxE1c8DYRxITLK8lxpZC6kND+SjGVhPRhGbkJYSEvDlBeiNdceRnu+tpyfe2al7GZsjM3dNEK7lzKN+64n17IrRYhEI9zAe+TglAuwe7unrixChmFiEslIPakXJjH+G4zCccrDcX68fgW9Juos7n++tp/0hM/dxbmbCX4rmh4MpZCg2SE8YLx7585//3H95z/bWFy4v62bDJeT4O/2NibbtI71UJYpNk9hZGQh5BWvZu90718zNnIiN29jbuDzxhv7dcb9niPsUsjjuORAKMBtmNwwjk8lAJvUnP/lJt2a4Zm7Gry6H+qJweX2v4X8MnXRUn2Onzw/1X0EiuGdwPhZ0iM0/9MW0IAhCewbCI3ZiE8ZWq8epRvdxbub8a/rfXeI2Hm8tTsy8s/Od2+fPyUj1l0wRx2gqQgE2LwwwAh2024QjM3fNXzvaqT2EEIrzNfqpmmvmZvoE52NB95VxZOauCW1HRk44Yw87dgjj6u+luG7D1JcvT0Cw9wuzZ/zsBMZPWZ8KmzYFiRw2L4yZmZlusVldXfU95B/64g9nco+bNs/9u/7ate79IrWtdh+xGWSZY0HXleE+zs2kTz73jkezXdfYTbBHzAzDgFkXr7766jjR27iHZdm0KUjkeIftR8YwDH08FEXJttOtNMDJuus7yxPnl3eeumZups+hBRK52AzMBMeCTpPhmrkZ4byv7Ur7o+MmNkG5YpZlvfzyyx0X6+3bt0cLT8e9KQWzoflosSxrHDtbqVTGtLOqqmbHA4bMxho4gX+q2GQyGe+Lc83cjD9xcBIVbyNisWkpIoUFhESH2Lj1nXcmuoKVJ4+Om4kMasGvvvpqz0v2vffei3ZhkdDtmY1pZ03THNPOapo2pp2FSXexBrqqjIOqquN8C2DZx7kMwjgS8N5773V/Vr7sy/ND/Vft4annh/qvutt/UBsaKfQwDm59552JOCdsSJfYPN1ZPj8gLNjrU2CaUxbcMHdy7RMBem0fIMXSj7/97W8j3FRwucP/F4vFMU3tmCYGJrzFnTE/BEVRxvwWDMMYx86asQ2reuhMxlcty/rxj3/sv1Refvlln1/+dGf5fHt46sjMXesOWAm0rF+vF2rsrV+OdcKGdIrN87319ES/hA3hTGwaf80tXl7cMA5d+P/Xcubh3vpLE6/pHZlB6HbDMhAtHIehrKo3xND7SbFYHNPOYlCOA3RWx7HYtq2qKkTCf/Ob37RdbO7j3Ez68jtfnJzhdB/nZv5bx46TWnOE3i8U/4QN6RCbwQkbwpXYPNtbf8n/5bnmx8v/dve1iZe8ITQe2f45RkEQ3n333THtLIM352DMmBc4ICHB/oXRwyDUd5YnBEH4YTNJ03isL7/Upj2EEIqNP3p+hqda5ljgF5sjM3dN6J+wITyJTX1neaI9APp8bz0t9Nw7wLGYfrDcCSok2LcpSCREfmG4tfJn+ZW51GQ6NZlOza/mPy/X2s5Wd7bQbUZzlvV/zzUb1Vxezu2YDdIJtYPMvT5DHhI2pF1sTknYkHiKTc9zWF3dzAYd+4evvyc/+MEPEhj/4bhXIDIO0Irp7EOXA+W4Vro7l5pf/Xi3Brd1o7qzIWem5HutH5A+w0G6awE6gPJROic69R6lnk3L/Pi7/1sy64QcH/7lt4sTgiD8cDH3125dZJaOYvOJmdx/fLf372bj7/piWoioYUOAdF1bTTq7mTX+sn55YsA1t7GxgW6NB5t5YIQFIjIRbqN8by518fXCk/at8rPyxtX01NLWfjNgAelG3wPaj3P2AUqE6Mxw6tFepBl0+eMXuT/sNVziPt565+PHjeeNvY3LsXJ3Tt7V8731tHB+Wf8slwOF+Ye++FKH2FAr/guKvs5vfWe51VTGPTQ2Fs9ffmcr92+/22scHZb+vNcrkNgRTMtkMi2lOarmr6ebnnvrvyl5U39UbcS5dqQ/KDZIP6IZNOnub81Pp+fz1e4brl5anZrMrJRAbTr9hu5wei9ojgXJdp9Dh6zSxGu5x+0x/l6FDCzjmQwICwoTi1uPmyayh9j0cxSYpX95zPHh3sfQYEaYWFzX9w7d4+/0NyaEQW3NHMepVCqyLGcymc7ombu/NX9uIb/vEkJIvVpYm0tNZm7c3+dRb1BskH5koxg06VbzC6kTRWn/3f7W/HR6aq1Ud0lnRsSt77wzcVo9MeWBh0Ns6N3HuZf+x4CGv6wxwGTwIDZmCHOse5vaNrEhLXfn0mopNvuOsxP3RqJIePRKOYSNWy+tZfqJDXlaWrnk3YnDVjA7jiOKIjX5hKTX2eKQbmPvg8X1v8QzZ9NJb7GJXW/8wDN7vTtJd4oN3ADTvp/wQ+z2HAg14O6gk95oMYTYkCGzStALkVp93dm71LuHOxsbO4exMi7DiU0c4yeB1yz29paS5Nmg2CD9iKTv3+lhtNT1rWozMXP2rBIE0GjOOYSYwel+YeOvWxufQ8J5r9SjUJtN+BebMEKuPW4nd39rfrqVpDmu7X50c2oyPXuvzGPOpvuwAoJ4SJJEO/7RnphpAwoElgoHrd+cMTwDdkOWZZohwbOMVXQPjY3FH7YKlQY1fGGNfsrR2Fu/DGec/HrD+NjXfoiiGOzVL0lSZxKvuYHyCtIu3twodBwo44bYVcAjNDnr9jxIjg8Kb2V6lD7Dz69ulk/agpxlxxyJ0oQxVpgphnNTIj8hPBp6a8xGUE/YI47UGUbjlj5jaxGkCWxJqR9Bq+/nlzLthzpL+ZW51Pyd0oH/rjz1SDI8gLLSELqneSJhOLGJ6SAWKCkJMM3QoxwrMWIT0w0HQhNBEKI4kHdcKxc2b1z0AgyZG/d2qp15nAEXsOM4UBFAX2kIxY44UTF0AoZa24ZgAecmqP14j+M7iRGbKGpbkZgBJjuii6R5yi3d5dMA/cqLi8UizI6LJJBFsyNOVAwtNvGtRJIkSRTFQDp69dgcNXY3Z8/NbezGpTJkZFRV5Xv/hYxPRJG0E9za7v2V+XRqem4l91m51iE5fkWxbVvXdRidJ8tyVC479zE0MoLYxPdAHyhEIErZnrfoaFdzUmTJJfHdbSA0CTZwPRpurfxZ4Y+bNy523JVQ9KXrujc2UJblaPscyrLM/R5uaNmgNtchDCAEFIivGsfcVSDwXTCDBAXsStlsDQ5RPkEQJEnSNC1yawabV+5vq6HFJowGMDSB62z89cNwzECWFCPi/u0j1IDUCJtJCNbyjtCLmk1hDpBRAmKxFmHHccB3HtNixrFzz/jE2q9FKAMjbtmx6R6QIGHkMgZVjlc3/dEYRWzibmc9vRknDhbHZgrjg9UByNkBP5jBaDMsjJFsPLg1SThLMIq5jOKEcMB4eqMoymhvJJl7/B6tExCkP9lsVhRF1mwFO+cFwa1JSEB+FLGJorFrKMCeQpKkEbYVCTzbGOnQXySWwG3CYOaGkf5+yXFryGhiw1MPn0qlAie5VFUdKkE3zOQJTkimM4eMCWRuWMt+s1DBD0VoyQkVjJh1iGQeX0g4jgP7i2ElhxvFPSPct9NAwoDNSBEL2Uc2ZTg8RhSbU5vZxQ7TNOG7hxvDMIxTA80sbI5owmY8BGEfyPIyFXiPvPoZTGiiQiMjig2vrX9t24Ytj3euWNO0YrFomqY/rmpZlmmav/zlL7lx706Fm0QdQh/HcaBZFDuVAtFOS/GmTbPzgVBg9OLdCEYkUaRSqWia5vk6A4h6pZSASGOi7g0kQKBSgJ38BGyXozqeDIYlIXUBHqPbSpbbUQQL+DGVSkVvYRiGaZqFQiE5V4woiuxYCiSOgMVgJxwSVRQLAmiJSvcCo4sNr5G0s8PU0bBQibyJL8IB3uE2RgoaIzk0BmaT45jQAMaKAkmSlJykRTfsHA0LG6hDwxgaMiaWZbGTq6DfCQVSNaIoMiK3lBlLbPDgBSNHw0IlCWOdEGpAXzIWtvaUp6V4jl0SYiE9GeuzZrmxKx2SUP0MW4qEpKYQCgQ46WMcKJ/fgBKbJOcdxhX2hAdYWDgaFjYJD5YiYQCWN1q9odlxioX3Gznjig18YYmV68iPhoVNwr9fJDwit7/Uev1F/k4ZIYCQZZJ3vtEeDaMAm117ET6AHEaEVphCCTIqjUcAYgMZv2TWxUZ7NCxsMCeHhIrXljCqayzsebuoNH6CKcYQRZH7PHk/OK5+Tsi0WiRa4DLLZrP0HWhFUUKKyni1Z6g0HsGIDaQuOI4mDYDXeWLo1iDUgCpkWZYpn6MIad6uZVmgNAlsEzCAYD5oOKyUTOeG1+pndGsQmkA0XhRFmsdQ4EWDVTjDMODkJq/R9ZEJTNUT69xQPhpGB3RrEPpYliVJEjgEdEJqwXac8lJQkiQl+ah7PwKzkl7T7KCeMC6AynLmAaBbg0SC4ziKoow8rH2Elwsq52qaJiiloihYvdmTILfkcCI3ac4jzaNhdGB2bjySEIrFIsyUouDijN9xynNoKMcAY0fA8R/WRiRRgNrRMGrg2RokcmzbBhdHFMVQb64xc66eLqJDcyoBiw0ccuS1FLgfPL1lSJly83aQWOPFpiRJCilkMnLHKcMwvLXxFNgIj+Az2zCELlHh/rCPhlEjsYk3hGX8Zt0wjGAdiGE7TjmO07GeABfDN8GLDYSV+DC+Z4T+YIyQgMo63KYhrOE38aIoapoW1Hb27B2nbNvWNA2CZigzIxBKzS5sFnhKYwwmpKNhlGFtSjyCdGMYBsRO4BBosVgcU3VO7Thl23axWIRDmrCNRpkZjbBMpCzLoigmJJjGwRA5x3ESWNyBxBRwMsDRAdXRdX1kj7xnktI0TU3TPI2RJClAdyqZhCU2sF9ISDCNg+pnCKBh4SYSLyzL8ksCCI+macVi0TTNM+6cJElaXV01TbNYLKqq2v1ssd5HskOIwZ/kBNMCPBoWCRhAQ+KObduGYWia5gXZPCRJynZx69YtWZZ//vOfX7hwoePx2WxW0zTDMNCPCZZwMw2wR0jCviC+TfegAg0DaAhPWJZlmuYnn3yiqurt27dh5tYLL7xw4cKF733ve0IXqqqappkESxUh4YqNbdtQSsu9IYtvO07YCcY6BogkHNM0IQim67qqqtls1kvndLs4iqLoug45HrzsaRJ6DRVUFnIfolEUZbSjYdECoc74BgCR5ADOimEYuq5DuMyfXPGA9vPZbBYUpVKpnD15g4QKjYJdSD7zbdHiWP0M+4CYOmQIl9i2bZpmpVIBqQDZ6FYUuG4huaLrumEYpmliioVxKNlHuGI4LnaCLi8x8soty0pIhBNhDcdxIIQFiqIoCrTj61YU6M2hqqqnKJhWiS+UxMY7xsHrtRLsYIyw4f7rQFhghFQKlCzHaNOGnB16kR/YSnNc9RSXUKE3HT0u0oiwzGipFFCUKE1BvbQ6NZlOtf83JW8WyjU3skXxDdU0A2z/ZVnmUm/GH4xBB+jcji03kLMzWioFkvMMp1Lcemktk7q+VT0ihBC39vUHciY1mVkqHKDehADtnDbkNrjUm1hUP8OUp5geCUJCJXmplHaxIYS4T7aXLtmvwrkAAAuzSURBVKb9P0GCI4ICKl71Bux41KsYBKwwFu4XEh6YSmnRJTbkaWnlUjp1abX0NMp1cUo0xpFLvRl2MAZlUGkSRVxTKVTp9mz2t+an01NrpTrG0YInsp04dErmSW/OPhiDPqg0XMJpKoUafrFxG9XS1sp8OjV/p4Qpm1CIMuzDmX9z6mCMqECliTXJS6VQA8TGq0abnlv5Q7l2HPWquCXiHANneiMwVv3sOA7UnqHSsA+mUqjj92yODwpvZWbvlRvo1YRF9Alt0Bs+DhhKksROFzjvPA3WnrEDplJYoqP0+cn20kWsew6P6MWGEFKpVOC8Z9z3aGA7ol4FIYRYlgX7YgbDetyDqZSY0FUg0NjdnD03t7HbiHRZvMKE2JBWfwEh5sPWoOVo1Ks4EW/sERAemEqJP92lz26jfG8OkzfhEL1l9PDCPqqqxjRiANXP0W5OYQ2SJKFFCwR/KgUUBVMpPNDRrmY+X21Gz44PCm9lOn+IBABDYgNA6ZQsy3G0ldCPJyor4zgOhGuy2WxM1Toq/KkUSM5jKgVBgoU5sSGEGIYBUaDYhdRs244qEmiaJsRwmCqHYwp/KgWS85hKQRBqsCg2hBDLsmBrqShKvHaO9M294ziQK+KgwmJ8OlIpIBuYSkGQyGFUbADPhsYo0Q0mjNrLmaYJKYTYqfKYDJVKAUXBVAqCRAjTYkPajWksQhlg9Si8kN+hiZEYD8VQqRRIzmMqBUHYhHWxIe1Wlf2EBBSDhf0qkNbiw6EZNpUCioKpFASJFzEQG8CyLLBBkiSxHAmBhgjhRf9N04zF59DBsKkUSM5jKgVBuCE2YgN4O/psNsumqQ2v+tm2bagLZ9nDw1QKgiA9iZnYEEIcx9F1nVnJcRwn8II0T2YENk68YioFQZBhiZ/YAI7jePaXNckRgut92SEzNFMUmEpBECRA4io2ANhiz8thpOkkGN8xn8Q0TZgOEKrMYCoFQRA6xFtsAH9gTZIkXdej3VkriiKK4mh/6zhOsViEPIcoikHJDKZSEASJFh7ExsMwDC95oCiKYRiRZAhGq36uVCpexEySpGKxOOziMZWCIAizcCU2gGVZXmwNYlCUVadSqQhnLkgDjYHVgisz+A8xlYIgSBzhUGw8/HYcfJ1isUgh0wDVzwNO9du2bRgGRNt6+mGYSkEQhDN4FhsPUB0vSyGKIghPeAkJoav62bIswzA6lvHuu+9+8sknDx48wFQKgsQbt1Z+8MfNG0tb1SNC3Mb+/ZtTkzhk2k8ixMbDsqxiseh3KcA5ADseoFtw7ty5d999t1gsapq2tLTkV44LFy68+OKLmEpBEI54Wlq5lE5Npv9lo/yc8Cw2bu3r395/VO/xrtzalx/mv6z3/9NkiY0fcDUg7dERoQJPwkt4+P2JDqC+C3jvvfdeeeWVCxcu9MygYCoFQTjGreYXUpOZldIAaxt7GvvbK/O9Bpi6jWphdfbcQn5/gLYmV2w6gMS7vzi4ZynXAK5cueL9/+zsrCzLH374IaZSEIRfjmu7H92cmkzP3t3a+Fk6dWm19JQQ4tbKnxVyq7O/2KoewePc2pf3blxMT936fe7tTGryPPg/pF4t5Vdnp9OpyfTs2na1pVPNiNz8aunb/fxSZmqtBJ6EW/v6AzmTuvh67qNfT3le1HGt/KdPN5Z+mt93QQzg8c0HX90sP+tet1srf6Zv3Lyar7rHtfIfVmen252wk4Vlbnz0de2YkHq1sDbXnKLdfJtN4EXhV95SezwDis3ZACnqpkNCMJWCIInh+KDwViZ18fXCE7deWp1qmdrn5c1/mfSF1Ih7UHh9anpu7WHtWWl1yjPW8OeTmRv39xvP66W1TGoynZpeyH/99cYVMNw7u7mF1GQ6dWu79py4T7aXLqZn75ZqBxCyAy/qeXnjPAjAw93tpYvp1GQ6dWVzd7+0BgJw7maha5vbWmFm5eFB+V5LQm5t15rLLa3Np6eWtvbrpF5anWq5a+7+1vy0T05O6PTq+jwDig2CIMjQuAeF172sDIiNF1+qFW56xhdEYuqt7YPjppVveh77W/PTLeFx66W1TGq6FYaytm+cS8++vVmo1Mr3FpYKB65P2EijvHHF5140yhtX0lNv/z7/0f3//Gprfjo99fbvc3ffL327n7/e6YU0Oarmr6dTl36d++jO/2xJYHPx8EKtldQKN1uqCYqSvlGo9X42T9X6PgOKDYIgyLBARUArblbNL5xIBShHU0Ua5XtznvD4NckvPK36gmZ4rV5anZqeW/pt6eC4+WqN3c3Z6fYHX2/G6EC0XvzZnQdPXLDsL/5s9eNvGv0dEW/xC2ufH7jtOud30ZpvBASmQw67n621nr7PgGITNG5t934zgjm/mi9VG0//o/yEs4IUBEk6bSa13drCP9t+1dN9qbfyMc8bzVgW5FeaD5vb2G00X6xpsnsolhfCWiq0ZKPpbA0qWGgu/q3tg+PWCq9slhudL9T8FbhTHe+x69ma6xnwDCg2wdLY3Zy9dDP/TYMQQiDzdkqFBoIgsaOZKblRqLXiaSfWH8JrTWtrbd8415Z0OYlruY3qF/c23v91M4u+sV2uuYR0SRch5HmtcKsVp2pleprPDyEsUCn4Q/h//590MMCVaQuINd/X7L1yw21/WPezdayn1zOg2ATKUTV/veP7cA8evP8xig2CcMVJAqNRLeVX5lKTmZU/7Ze/qjb+q9rMlHxbfVSpuZ7Y2K3qrFuflncfVZ81yvfmUvOref2zpsa06Ej/EOJTjieNamlrZT6dOndT/6r8qNoWK4O4XLMqoalYvy99+aja4duAHjTFDFSztfjnJzrU+GbrBtQjHJOTEoCHB9WvyrXjrmfz3q/b7xkIik2g+HcZHjaG0RCEN5ppf3nzYbUOhrhZ4NtMj8+tFKoNt+2fBw9XpybTU/K93ZrbOvXZrBg+eXzTUWjVRrderekirG1Xv4XA1M0Pvqy1x8ra42YgNvOrhf1G58r9uZxWSqlVndwqmPayAE3T1RRXf312EzB63vr7PgNBsQmW5jXR9SkjCIJ4uLXy9sbaHf1h6UHh0w25VfTMeQgExSZY4CTtdLNA4OPdGt+XD4IgQ/O0tHLJX0MMfkOHN8MfKDYh0Arjdh4MRhAEIc/KG1db8TRCyHGtdHeuz1F/nkCxCY1GdQcc5FYxBoIgCCHt+9G2UjSeQbEJkKPqgy/aW9RBerBXcTqCIEiSQLEJkKNqfqXDF3ar+QUUGwRBEg+KTYDAORt5s9B0id3a7v2Vq3NrD/n3kBEEQQaCYhMgEEbztQ3HgjQEQRBCCIoNgiAIQgEUGwRBEK5pTmNbijZ5jGKDIAjCMc35Bb3GOVMFxQZBEIRr3P2t+enIOxSg2CAIgjCKW/vy3o2LcPDzU/3z8j9benFyLPRia6aJ1wTz4uu5j3495ZtLXc0vtI3sdBvVR82ebDC8mQooNgiCIGxyMtvGrT28c/VeUzxqD+/MTmdufPR1rVEr3Z0DUYFG1LN3S7UDiJu1zZg5GX1SrxbW5mbvlmp2eeNKn5k3oYBigyAIwibNSZc38980iFuvVL51PVG5V264xN3fmj83t/aw5kKzEpiJ2ShvXDmZ0ubub81Pt4TH14etOWsA5nXSAMUGQRCETdzWxOj5O6UDOCnePlu6RWN3c3a655hqt5pf8NwXeNh8vlrf316Z9z0tDVBsEARBmAV8kcmeQtKifTZz26xP/1zO5sOa583zhRLdhvQoNgiCIAxy9O2jrw5c0urne2Wz3CDEmzPtryzz5kZbrQe3hAdiZU3hgYdFNqUNxQZBEIRBnpZWfrFa2G+AR9Lm2UzPbew2CCGNaqnwp3Ltv1pi86RRLW2tzKdT527qX5Ufwcjq6YX8/vNa+f+Uv3sGDtBS4cAlhNSrpQefURxtgGKDIAjCHu4/q9X98scrc+0zGL1i6PSUvKk/gvHzzYH0s2vb1W+bNQUffFlzWykfb1BbA1I1k+nU9NxKHsNoCIIgCG/8f9t2eabTPzDLAAAAAElFTkSuQmCC" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The radius of C<sub>1</sub> is \(x\) , where \(x \geqslant 10\) and the radius of C<sub>2</sub> is \(10\).</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Explain why \(x < 40\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Show that cosθ = x −10 </span><span style="font-family: times new roman,times; font-size: medium;">50</span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) (i) Find an expression for MN in terms of \(x\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the value of \(x\) that maximises MN.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) Find an expression in terms of \(x\) for</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) \( \alpha\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \( \beta\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) The length of the perimeter is given by \({l_1} + {l_2} + {\text{MN}} + {\text{SR}}\).</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Find an expression, \(b (x)\) , for the length of the perimeter in terms of \(x\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the maximum value of the length of the perimeter.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) Find the value of \(x\) that gives a perimeter of length \(200\).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A straight street of width 20 metres is bounded on its parallel sides by two vertical walls, one of height 13 metres, the other of height 8 metres. The intensity of light at point P at ground level on the street is proportional to the angle \(\theta \) where \(\theta = {\rm{A\hat PB}}\), as shown in the diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \(\theta \) in terms of <em>x</em>, where <em>x</em> is the distance of P from the base of the wall of height 8 m.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate the value of \(\theta \) when <em>x</em> = 0.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Calculate the value of \(\theta \) when <em>x</em> = 20.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(\theta \), for \(0 \leqslant x \leqslant 20\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{{5(744 - 64x - {x^2})}}{{({x^2} + 64)({x^2} - 40x + 569)}}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the result in part (d), or otherwise, determine the value of <em>x</em> corresponding to the maximum light intensity at P. Give your answer to four significant figures.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point P moves across the street with speed \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\). Determine the rate of change of \(\theta \) with respect to time when P is at the midpoint of the street.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">An electricity station is on the edge of a straight coastline. A lighthouse is located in the sea 200 m from the electricity station. The angle between the coastline and the line joining the lighthouse with the electricity station is 60°. A cable needs to be laid connecting the lighthouse to the electricity station. It is decided to lay the cable in a straight line to the coast and then along the coast to the electricity station. The length of cable laid along the coastline is <em>x</em> metres. This information is illustrated in the diagram below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 24px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaMAAAHdCAIAAAA/34ESAAAgAElEQVR4nO3d/2sj6Z3g8fwnJaYuHDYxdG+H4J+scW4Jbl9W19t3VHTXu7DgNljeGxiSyxckn4cbBm7lcPaNyVl3cwSZkWe57WxSugnLwqiHojNBTNwRUdieQRSZhaGlFaHR9AmxakxTee6HR5IlWbb1pVRf3y/0w4xsy9UN8556VJ969CUBwFZ7e3tfu/VVVYnIx5dfeeXf/EmsUCi4fVyh9iW3DwAIji+++OJrt7769VdfffDgwRdffCGf/Ox3v0slk19+5ZX/uP2X7h5emFE6wDZff/XVP7/3ZyO/9Otf//rLr7zy4MEDhw8JEqUD7CFbJk/l8rq+nUjI1eteOm2aphDif7/zztdufdXtwwwpSgfYI5VM/vm9P2u329uJxHYi8Ytf/KJQKPzqV7/K6/pqNFosFoUQqhJx+zBDitIB9kglk6lk8iSXSyWTpmmuRqOr0aiqRLYTid/97ner0ahpml9ZWPzoo4/cPtIwonSAPWTpVCXSaDRSyeRqNFqtVovFoqpEisViXtczR5mvv/oqF2FdQekAe6SSyb/c3t5OJOQ/l8tl+fxJLpfXddM0U8kkpXMLpQPsMVQ6eRVCCJHXdUrnOkoH2COVTH7/e9/rrV6HSpfX9b10mtK5hdIB9pDv02WOMnvp9FDpTnI5eUWC0rmF0gH2kKWTUyZr3/jG48eP5fN76fRXb948yeWEEJTOLZQOmFW73c7revxb34p/61t5XX/wN3/zx1//V/9S/Rfd+17VH3z/+3IB+9U/+qM333yzd7ECjqF0wKzke3C99+PkI3N0JB/y2mv/I65p1WrV7aMOF0oHzKpara5Go7Fv/sn/zGT+bz5/xeMH3/u+qkRSyaTbhxw6lA6whzx3u/p7+q9UwEmUDrDHOBWjdG6hdIA9xqnYOOd9mAdKB9hjNRptNBpXf4+8IuHM8aAfpQPsMc6OTIVCIXOUceBgMITSATZot9vjlE7e/erA8WAIpQNsMGbCKJ1bKB2u0zKNn+5v3tg1mpZ8wqrpiYW7B6XnY/3g3axpXfs7mqbx4GBj69h8MfvxumLMhI156gfbUTpc7ZmRWlGViLowaem6Pxi7tnRW09hdUiKqcs+/pRv/DThK5wpKh2u9MLP3+ks32Q9eXjqr9vGjzzpps8zsHT+XbvyLquNcooXtKB2uNafSPS/tp3pp83vpxh+UY3jYFZQO15pH6axW6XC9L21+L934/aJ0rqB0uNZQ6eTVg9iO8az7DU3z4eHmQkRVFtdTutmyBn4wljWblXwqpiqRpY1cpWV1MxeRmxrJV+6W7vemvruuRNSFreNK8/wQWqaRTckfWdo4/NBsik4cI6qyeCdbsc7f7FvpHpjVMvWdtUV14f6B/stHp582h4/2/KVmN36/uE3CFZQO1+ov3YigNI3dpbW3jPqZEM1KdmtJ9ihlNOUPrr128O7HdUtex5BVEuLCSZxlZu8otxP7/+dx/UxYn+e3ls9PBq3P81u3N7NPWkKIViWfip13sGnsDL9m98CsyvHd/5SvnclX+Pm7xaYQQjwvvf3m4WndEqJztNOcq44w/rtv3CbhCkqHaw2vXgeCIp4ZqZWllNE5NWoaO+eXZQdXr1blOLbY+85Rpev964W29veodXqw1n2dy0tnmdk7C/cP9FK9v2NNY2ehey7Zeaz0nZxOb/wrqtwm4QpKh2tNWrpeemwp3TMjtTL4Zt8zI7XS+epV53Sf57eWVSWytLH/M8NsCdGJ5vVTLxObaEqO4WFXUDpc6+rSCaumJ27I5eRZ3XhrfShYtpRuYI05XumEEFa99P6Pd9YWVSWytKXXLLn0tv+6x0TxonSuoHS41jWlE6JZeXf/je/HVCWirqVy5+tF+1avSv+gct9Z5NWl6zirn2Y2F1Z2jGed1Wv/QbaeGKVZp9smihe3SbiC0uFaV5fOapV+lOi2ZtDgwnNE6VZ2jN+3fvOBUXsxeLY1+BtbT443ltW13bzZFMJqVXKbN17vXmqoHMcWl7b0miWseunnneuzi3eyFav5cf5R31nngnzx56X9uwPv060dllqzrmYnfeuN0jmP0uFq3Zu6Ondr/XP32mtvvKP/GzrPr6d0s/npcWzxfI7k6cPzSwGyfVbN2I2pSuwN47NK9l7f9YFPB3/jCyGEaJkf7t9f6rx41jgfDbFaldzmQkRVYjt6pWlm76yljt8v1S0hmk+M9/8+L9vXfxJn1UvvDg+szGjSy6ncJuE8SocZ9VozELs7o8/ygmnSETmGh51H6TCjZkV/8Kh+1v+UVfsg98iG0Q2/mLRclM55lA6zkGPD/VchhGiZxvuDU2xBN2m5uE3CeZQOs+mb5JDXXo87w2shMun7btwm4TxKB8xq0mupeV3nNgmHUTpgJlPMxzE87DxKB8xkimxROudROmAmU2Sr0WgwPOwwSgfMZLq9SSidwygdMJPpLqSqSoTbJJxE6YCZTDccx/CwwygdMJPpmkXpHEbpgJlM16zMUaZQKMzjeDASpQNmMt3GJNwm4TBKB8xkuquoeV0/yeVsPxhchtIB05t6A2GGhx1G6YDpTR0sSucwSgdMb+pgNRqN1WjU7sPBpSgdML1CobCXTk/3s9wm4SRKB0xvlkuoqhJpt9v2Hg8uQ+mA6c2yezDDw06idMD0UslkuVye+mcpnWMoHTC9WWrFbRJOonTA9Faj0Wq1Ot3PcpuEkygdMKVyuTzL9dO8ru8kU+zd5AxKB0ysXC7HNS2uabOUzjTNe//+P6hKJHOUoXfzRumACbTb7cxRZjUaLRaLpmluJxJTv5ScOm40Gr0XtPE4MYTSARPYTiS2Ewk5BzfjHV39t0mYpkns5orSAePK63ovc0KIYrE49Q0SUv/it1qtsuX6/FA6YFyr0Wj/TMnsF0+HbpPIHGXYymlOKB0wrqHrD3ldn+JTwXoajcZ2ItGfznK5zAYnc0LpgHENbS9cLBa3EwlViaSSSXlbmNk1dENrtVqVzxeLxbyu76XT8rrtXjrd/4InuRzndHNC6YBxZY4yF9+Ya7fbpmnKPYRTyaR8qEqk/7Eajcrn99LpvK4XCoWLd1a02+2h1TFsROmAcbXb7bimzbJivYxcyXJCNz+UDpiATFIqmZz6JrCLisXiajRK5uaK0gGTabfbeV1fjUb30umpNzKRr1MsFuW9Fixa543SAdOQvYtr2mo0mjnKFIvFMc/yTNOUOxXLKxJMCzuD0gEzqVarslzycqpc28orD/LRu1LR+wZZRjYcdhKlA+z00UcfPX78uFwu90rXmz6x8a09TIrSAXbaTiRmvEUM80DpANsUi0XuXfUmSgfYQ47+MiziTZQOsMdJLrcajXKdwZsoHWCDRqOhKhE+AcezKB1gg1QyKfcfrlarXGP1IEoHzMo0TVWJyPscMkeZWbZcx5xQOmBWcU3rTZasRqOsYT2I0gEzKRQKvckS+bmITJl4EKUDpicnS3p7rLN09SxKB0xPfoBhb7KEpatnUTpgSvLTvHqbkcilK/N03kTpgCnJHUp6/zpy73V4BKUDpiHP4Pp30OSjqb2M0gETu/iBEvLefpaunkXpgInJ3dX7p0n20mmWrl5G6YDJNBqN/skSIUS73e6/NAEPonTAZDJHmbim9T/D0tX7KB0wAXmL69BHgrF09T5KB0xgaLJEsHT1CUoHjEuuUoc2ZZKfS83S1eMoHTAWeYtr/2SJtJdOX3wSXkPpgLHIyZKhcze5dB162w4eROmA6122eXqj0Rh62w7eROmA6+2l00OTJfAXSgdco3/zdPgUpQOusZ1IsET1O0oHXKV/83T4F6UDLiUnS05yObcPBLOidMClTnI5poKDgdIBo8nJEm7zCgZKB4yWSib5oK/AoHTACBc3T4evUTpghLimsRFTkFA6YNjFzdPhd5QOGCAnS/o3T0cAUDpggNw8ncmSgKF0wLlqtcpkSSBROuDcxc3TEQyUDuiQm6czWRJIlA4QQoh2ux3XNPZJDypKBwhxyebpQ05yOYbsfIrSAaLRaIwzWRLXtIsbrMMXKB3QmSy5+nvk/WGME/sUpUPYyc3Tr/18r8xRhhv+/YvSIezG3Dx9NRpl6epflA6hJidLqtXq1d/G0tXvKB3Ca/zN01m6+h2lQ3iNM1kisXT1O0qHkJKbp4/TL7l05Z5/X6N0CKm9dPrayRIpc5RhYNjvKB3CSE6WjHmL62o0yu4mfkfpEEZxTRtzzxJ5cZalq99ROoROoVAYf2RkL51m6RoAlA7hMtHm6e12m405g4HSIVxOcrkxJ0sES9cAoXQIkUk3T2fpGhiUDiGSSibHv9WBpWuQUDqEhRwAHn/z9GKxOP46Fx5H6RAWU2yezi39gUHpEAryFlfKFVqUDsE30WQJAonSIfjk5um84xZmlA4BJ29x5RJqyFE6BFwqmRzzFlcEGKVDkI25eToCj9IhsNrt9hSTJQgkSofAGn/zdAQepUMwNRoNJkvQQ+kQTONvno4woHQIIDlZUi6X3T4QeAWlQwBtJxJMlqAfpUPQyMkSbnFFP0qHQJG3uJ7kcm4fCLyF0iFQmCzBSJQOwdFoNFQlUigU3D4QeA6lQ3CkksmpJ0sKhQK7AAQYpUNAyMmS8TdPHxLXNMaMA4zSISDimjb153jJzwzjcm2AUToEQaFQmCVVJ7nc+J8ZBj+idPC92TdPj2sa1zGCjdLB9zJHmVkmS+SnI7J0DTZKB3+Tb7HNctk0c5Rh6Rp4lA7+Nvvm6avRKEvXwKN08DG58Jx6skSwdA0NSgcfm33zdJauIUHp4FfyFtcZT8dYuoYEpYMv2bJ5uly6sh1AGFA6+FLmKBPXtBkjlTnKTH1bBfyF0sF/7No8fTUa5a7+kKB08J/ZJ0tEd2tilq4hQengM7JQ1Wp1xtfZS6dZuoYHpYOftNvt2SdL5OvMeGcF/IXSwU/s2jydpWvYUDr4htw83Zb9MhuNBid0oULp4Bt76fTUm6cj5Cgd/GHGzdMRcpQO/rCdSMw+WYLQonTwAXkBgR1HMDVKB6+Tm6ef5HJuHwh8jNLB605yOVsmSxBmlA6eJidL2FgJM6J08LRUMslOmZgdpYN3MVkCu1A6eFdc07gJH7agdPCovK4zWQK7UDp4kZwsseUWV0BQOniTLZunAz2UDp5TrVbZPA72onTwHFs2Twf6UTp4i/xkQiZLYC9KBw+xa/P0IaZpcg035CgdPERunm57leKaxgYBIUfp4BWNRmMekyXy+sbsnyUGX6N08Ao5WWL7y57kcuzJDkoHT5C3uJbLZdtfOa5pTCCD0sET5jRZIpeuXI4ApYP75Obp83gr7SSXY9MnCEoH18lbXG2fLJFWo1F28YSgdHCdnCyZxy2ucgiZpSsEpYO75rp5euYow9IVEqWDm/bS6fmNgLB0RQ+lg2vmunk6S1f0o3RwzXYiMb89S1i6oh+lgzsKhcJcz7lYuqIfpYML5GTJ/O66l0tXtixGD6WDC05yuTlNlkiZowwfKoZ+lA5Ok5Ml89s8XZ4wsjk7+lE6OC2VTM71WoG8t4ylK/pROjjKgc3Ty+Uy+25iCKWDo+KaxjtocB6lg3PmtHk6cC1KB4fICwVsiglXUDo4RG6ezoUCuILSwQly718mP+AWSgcnzGnzdGBMlA5zJwfc5jpZAlyN0mG+2u12XNPmtHk6MCZKh/ma3+bpwPgoHeao0WgwWQIvoHSYIzlZ4vZRAJQOcyM3Ty+Xy24fCEDpvOusfprZXIioSkRVYjvvntati19aXE+9V6qf9b5g1YuHG8uqElHXUrlS30+4Ya6bpwMToXTeZLVKP9pM6WbLElb98dv3l5TF9f3TVudLh+tK7A2jZlk1Yze2tKXXZNJapwdrN9d3H9ats7rx1vrC6/na2ZW/ZY7kZEm1WnXrAIB+lM6bnhn7Py61uudkVuU4tqje2C+97PzzUspoyq+Y2TvKyo7xTIgXZvaeurBrNK3ej/S+zWHz3jwdmBSl84Wn+Y2bMluWmb2j3NzUn3a+0iuarOGGXu98YTB8XVa9lN+/v3Tjh8anHxxsLKvK4npKN5u1x2/fX1IiqjxVnPlwmSyB11A6P2gaOwvyxM1qGrtLyu2DUqv7taf5jZtqLGs+N3YWIrf2Sy87z7+s66+pyr1j80XfCz3Nb9xUlYiqLG++XaxbZzX99SXl5vrGD/NmU1if57eWL8ZxUnLzdD6XC55C6bzvrKa/fqvzZpzs14XS3dgvPdU3lYul6//Owe+X31fXN5XFO9mKJYQQrdL+bVV5LV9/KWaQSiadnCyJaxpVxbUonddZNT0RO+y+Z3dJ6WJZszqydEPndGJU6Xo/ZUPp5GSJY7e4yi1SuO6Ba1E6b2s9OU7tG/1zJOeXIIQQQrwsHdyI9N6n67sE0Srt3x61FJ1v6RzePP0kl2MyGeOgdB5m1Yz/8uZxZfDy6azXXudYukKhoCoRJzdPj2saV3gxDkrnVcOZO6sbhwfGM/m23dLC1nGlKVqVfOp8ns6q6YmF5c3sk5Zomvru6Hk6eYm2G0TLzN4ZLt3FBe9YnN88naUrxkfpPMmqGbsxVYkMPM6Xok1T311XIp0Zkd7YnbBapr6ztqgqEXVtN29eOJ97WTq40XvB136i/9Wt7ovf2jd+tX+7+6WL1zGud5LLOTxZwtIV46N0sIErm6fHNY1dUjAmSgcbpJLJ7UTCyd8o28oHKmJMlA6zKpfLzm+efpLLOdxW+Bqlw6xc2Tx9NRplYBjjo3SYibzF1eFVpDyLZOmK8VE6TM/5yRIpc5Rh6YqJUDpMT26e7vyeJSxdMSlKhynJW1wdniwRLF0xFUqHKaWSSVc2T2fpiilQOkzDxc3TWbpiCpQOE2u3265Mloju0pXdjDEpSoeJubh5+kku5+SuUAgMSofJNBoNVyZL+g/ArV8N/6J0mMxeOs0OIvAdSocJyMmScrns9oEAk6F0mMB2IuHKZAkwI0qHccnJEt4mgx9ROoxF3uLKhzbApygdxuL85umAjSgdrtdoNFQlwp0J8C9Kh+s5v3k6YC9Kh2vIyRKHN08H7EXpcI24pnEDFvyO0uEqhUKByRIEAKXDpdzaPB2wHaXDpTJHGSZLEAyUDqPJj452fvN0YB4oHUZza/N0YB4oHUaQW/syWYLAoHQYwa3N04ec5HLUFragdBgmN093fbJE3oLmyofyIHgoHQa4vnl6T17X2dwYdqF0GJA5ysQ1zQuTJXFNY5Mo2IXS4Zx3Nk+XMy4sXWEXSodz3pksOcnlWLrCRpQOHXLzdI+cRrF0hb0oHYQQot1ue2SyRLB0xRxQOgjRnSzxwoUIwdIVc0Dp0Jlc88JkiRTXNO8cDIKB0kHspdPeOYeSS1fX55YRMJQu7Ly2efpJLsdnVsB2lC7sthMJj0yWSKvRKB9CBttRulDz2ubpcg8V7xwPAoPShZfcPN1TY2uZowxLV8wDpQuvk1zOO5MlEktXzAmlCyk5WeKpzdNZumJ+KF1IpZJJr60TWbpifihdGHltskQyTdNrh4TAoHRhFNe0vXTa7aMAnEPpQiev67wdhrChdOEiJ0u4qxRhQ+nCxTubpwNOonQhIm+e99RkCeAMShci3tk8HXAYpQsLOZfLGAfCidKFgqc2TwecR+lCQW6ezmQJQovSBV+j0WCyBCFH6YJPTpa4fRSAmyhdwMlbXMvlstsHAriJ0gUckyWAoHTBViwW+YhoQFC6AJO3uDJZAghKF2BysoRbXAFB6YJKbp7u5c9kKJfLVBiOoXTBtJdOe3myRIaYK8JwDKULIG9unt6vUCisRqNuHwVChNIFUFzTPD5Zsp1IeOpzZhF4lC5oCoWCxzdPl0tXZl/gJEoXKL7YPD2v615+DxGBROkC5SSX8/5kSVzTWLrCYZQuOOSq0OObp8sd3lm6wmGULjhSyeR2IuH2UVzjJJdj6QrnUbqA8Mvm6Sxd4QpKFxBxTdtLp90+imuwdIVbKF0Q+GXzdJaucAul8z1fTJZIcU3zxXEieCid78nN0z0+WSK6S1fvn3gikCidv8l8eHyyRDrJ5bx/aRhBRen8zUebp69Go17eRQrBRul8TG6e7v3JEtEdgmHpCrdQOr9qt9txTfPL5umZowxLV7iI0vmVvzZPz+u6L95MRFBROl9qNBp+mSwBvIDS+ZLHN08HvIbS+Y/cPJ0PYQDGR+n8ZzuR8MtkCeARlM5n5GQJ4xrARCidn8hbXNn1CJgUpfMTf02WAN5B6XxDbp7ODVXAFCidb6SSSSZLgOlQOn+QkyW+uMUV8CBK5w++2Dwd8CxK5wOFQoHJEmAWlM7rfLR5OuBZlM7rTnI5JkuAGVE6T/PR5umAl1E6T0slk2xgCcyO0nmX3JGcyRJgdpTOu3y0eXo/eS9HtVp1+0CAc5TOo+Qtrn6cLCkUCqvRqNtHAQygdF7k68mS7UTCj6eiCDZK50WZo0xc0/w4WSKXruyHDK+hdJ4jb3H16WQJS1d4E6XznFQy6d/N07cTCTYKhQdROm+Rm6f79MIlV13hWZTOQ9rttk8nS6S8rrODHryJ0nmI3zdPj2saS1d4E6XzCrn08+lkiejeosvSFd5E6bxiL5329dLvJJfz9fEj2CidJwRg83SWrvAySucJ24mEfydLBEtXeB6lc5+cLPHjLa49LF3hcZTOZfIWV7+v++Ka5t9rKQgDSueyAGyeLpeuvj4nReBROjfJyZJCoeD2gczkJJdjY2R4HKVzUzA2T49rmt9jjcCjdK4JwGRJj69X3wgDSuea2L/+5to3vsHbW4ADKJ07CoWCqkT+7Z/+6Wo0ytIPmDdK54L+zdPzuq4qkVQyyQIQmB9K54LMUaZ/sqRarcY1bTUa5R4DYE4ondPk9NnQ5untdnsvnfbvpuqAx1E6p12xeXrmKEPsgHmgdI4ql8tXT5bIe2CJHWAvSueocTZPJ3aA7Sidc+Tm6eMM0MkZFGIH2IXSOaTRaPQmS8Yhr89yNRawBaVzSOYoE9e0iYbmUsnkpD8CYCRK5wR5i2u5XJ7op+SA8V46PaejAsKD0jnhismSq02XSABDKN3cyWupU7/jFoCtOgHXUbr5kivQaydLrn6FcWZTAFyB0s2XnCyZ8YxsxrNCAJRujmzcPH3qd/rsJe/PZU89+A6lm6O9dHrKzwb8w//75K/f2v3rf2j9ofOERzYoLhQKq9Gou8cATIHSzcslbWqaxoODjWVViagbel0IIYRVLx7KZ9ZSuVLdEuJl6X/95w+qjQ/+22Gp1fvJ7UTC9XfrvHAMwBQo3bxsJxLD681WJZ+KqUpsJ/t3pfpZ98nTg7Wb67sP69ZZ3XhrfeH1fO3M+qcP/uub/+OdN//7B/901vtp1z8AWy7GGXmBH1G6uZA3rg5Uyfo8v7WsLmwdV5p93/jCzN5TF3aNpiWEEFblOLa4lDKa4g8vW41nrZdDL+vuVuwsXeFflM5+crLkJJfre+6spr++pCwn9M+t/m+1Ksexxd4ydjh8F5zkclO+8WcHlq7wL0pnvxGzvrJoC/d/uHd/SYmoyvLm28W6JUTT2FmI3NovdU/eXtb111Tl3rH5YuQry/2KXRk3YekKX6N0NpNFGNpwyTKzd5TF9dR7pfqZsOqP376/pCzeyVasur6pXCzd7YO+CxFD3PoY6UKh4OLpJDAjSmezVDK5nUgMPfmytH9LWdkxnnX+vXV6sLaoxrJmdWTpLj2nE0JkjjKuDNZtJxKD63HATyidnS6behsunXia37ip3tgvnfUuQUit0v7tK96nE90rsPM4+CvIE1Vu0oB/UTo7xTVt9CZLTWNnIdJXtKf5jZtLKaN56bXXS7kSnbyus3SFr1E628jPqL5k3k1ee429YdQsIefmOuMmVk1PLCxvZp+0RNPUd+U83dW/aDUadXjj9bimsXSFr1E6e8jJkis3T2+a+u66Eum/F0IIIYTVMvWdtUVViahru3nzivO5jlQyOf4u7bNz8YIvYBdKZ48pNk+f2kku5+RFCXeH+ABbUDobyLMex1aUeV13snQsXREAlM4GDm+pVCwWHbsri6UrgoHSzapcLju8n5KcZXHmd7F0RTBQupm4svW5k6Vz65YMwF5fUuXVQB48ePAI7oNzOgDBR+kABB+l84m6vqlE1M4Od2f108zmQkS9cb43AIArUDofeV7av6vGflw6fWcn++TSfZ0AXEDpfKWubyqRpS29duleJwBGoHS+YlWOY8ub+lO3jwPwGUrnK2Ns6wTgIkrnI2c1/a92Un+xdOVWnQAuonR+YbUq7+28fdpqGjsL947NZk0/zF2+CTuAfpTOfu12O6/rozcfnkr3A3d0s2UJ8cxIragL9w9P63ad17Xb7b102uHdPQEnUTqb5XV9NRp19yOop5A5yqhKJK5p9A6BROlsUywW45omdx52ZktOezUaDXqHoKJ0NpCNU5VI5ijjx8b16/UulUw6uRUVMFeUblZyD6VAPlLJ5CUfAAT4DKWzR/9pnX/rYJpmKpnkhA7BQ+ns5N/e0TgEG6Wzn+ydjzYll5+WTeMQYJRuXvx1TuevowUmRekABB+lAxB8lA5A8FE6AMFH6QAEX+hK5/qNBxcfbv+VAMEXxtLd+MrSp598cvFLn37yyY2vLP3tT35y2Ze+++3vXPxSs9n85trtkV8SQnz329/55trtZnPEJsHf/fZ3bnxlidIBDghj6byTuU8/+YTSAQ4IY+kuPulW5i47HgD2onRuZm7k8QCwXdhL527mLh4PgHkIdelcz5ygdIAjwls6L2ROUDrAESEtnUcyJygd4Igwls47mfvbn/yE0gEOCGPpvJM5JocBZ4SxdN7JHJPDgDPCWLqLT7qVucuOB4C9KJ2bmRt5PABsF/bSuZu5i8cDYB5CXTrXMycoHeCI8JbOC5kTlA5wREhL55HMCUoHOCKMpfNO5pgyAZwRxtJ5J3NMDgPOCGPpvJM57gYDnCda9P4AAAZ3SURBVBHG0l180q3MXXY8AOxF6dzM3MjjAWC7sJfO3cxdPB4A8xDq0rmeOUHpAEeEt3ReyJygdIAjQlo6j2ROUDrAEWEsnXcyx+Qw4Iwwls47mWNyGHBGGEvnncx999vfoXSAA8JYupHPu5K5K44HgI0onRDuZe6y4wFgL0rnZuZGHg8A24W9dO5m7uLxAJiHUJfO9cwJSgc4Iryl80LmBKUDHBHS0nkkc81mk9IBDghj6byTuW+u3aZ0gAPCWDrvZI7JYcAZYSyddzIneJ8OcEQYS3fxSbcyd9nxALAXpXMzcyOPB4Dtwl46dzN38XgAzEOoS+d65gSlAxwR3tJ5IXOC0gGOCGnpPJI5QekAR4SxdN7JHPN0gDPCWDrvZI57JABnhLF03skc970Czghj6bz2cPuvBAi+0JUOQAhROgDBR+kABB+lAxB8lA5A8FE6AMHnydK1TOOn+5s3do2mNaffYNX0xMLdg9Lz677v8/zWyvr+aWtOxzGNs5r++tLaYak1r78cIHg8WLpnRmpFVSLqAqUbidIBE/Ng6YQQL8zsvdlKd1Z7VPxsDimwah8/+uyF069mPX306B8JGzC1gJaudXqwlTPtb8Pz0n7q2LSrdGO+mtUq/SiRrVA6YGqBLN3z0v5dNZa1u3RWq3S4rtyzqXRjv1rr9GDt5h1KB8zA3dJZLfODg41lVYmoC/cPHprdt8MulK5lfrh/f0mJqMry5v4HZmvU828X65boZK57S+lSymh2r298WPnl4cby0kau0npuGg8ONmI7xrORR3J4Wrc6Tz762f79Wymj2QlT92bVheTPfppc6vzrosyQZWbv9H7p0B+0VcmnYqqyvLmvP370cbn5cvDV5J/0rF56b2dtUVUiqhLb0SstITO32P3jrOwYz0ZdrmmaRrbzg/1/jfI772bNpvztkaWNXIV39xBKLpbOapXe2enkyWpVcpsLK930DJaudXqYeudx/UwIIVpPjjeWOylpnR6s3X3DqFmdKwyL3ROfF2b2Xvecrnd94/7haa1ZOlxf2/8wLyPV+3VWq3S4vvaWUT/rvN8vT7Waxs7CQLksM3un7yzMqj98Y+Bs66ympxLZJxcuX7wws1sJ/XNLfs/7Dx41rRGvZmbvLLyer50J0axkt5Z6X7Iqx7Heb7l4ueaspr9+q5OwpqnvrivLm9knrd53rr128O7HdUuIprFz/lcEhIuLpev+p9j36Dalv3RW09hdGvw2dWHXaL5sGru3emdPVv3x21t3OhdJ+0vX/UWDa2HLzN45L90zI3W3d35n1YuHG3/RuSxrVY5ji5eVrnNsvVe2Ksd33xyx4rYqx7GVzX29JGM9cAxDpeu81MCXBkonhv830DR2zv8PITqntJ2vPjNSK+d/D4N/FiBU3Ctd09hZuOxdqv7/mJ8ZqT8edSbyzEitXPLf7YSlG45Fn2tKN/AmmmVm/93o45HniRF14f7BTx/1lt4jXk0Iq176+U/3Nxci6lilk/8buBDfzh+N0gEd7pbussXUUOlWRl1eeGakVtTRY2VTlG5x9NDctaWTFVvYNZq/N1KvXX554axe+rvjVExVImpniXrh1az647fv39rY/9n7pVplzHO6/q51XoXSARe5vnqN7bzbef9fiOdl47eXrF4X11PvdVd/Vus3vyw1/9nM3lOVxfWU3j1Lks9bE5fOqhzHFs8vAvQfyfWl657W7f3wB3evv9pr1YuH3fcZB19t4JgnXL32Z7p/QU3pgA53r0j0XX9UIqrSu2lh+IpE3/XHiKp0/8O+7PnzH2+U3/9lzbr2fbrLj2RE6VZ2jN+3fvOBUeu96SYXp8vdaw4XPXukf9x9P/Hz/NZK37Xa3qvVzk9RrXrp3dS6srLzsPTo0T9avWNo/fbnxlNr+MK0vHzRu1b75HjjdvdIKB3Q4e6USd9cxfl4RP+Vis55jVU/zcml3+CUSd/zsZ2s0ff8wzfWFtW1t4zqb49jsoYr62v3j80Xg5c4esvnviNZSx0bZkuI3rVXVYl0emHVjN2YqsTkBd9zTWPnqrt0G2Xj7z/Uf7yztjhwDjv4ap1jln/A2sOdhd7p6lndeGtdWVzffVi3RvzlCNE0Hx5uykPtHXznRLV7Aefpw+E/CxAm3pwc9pnLr0UA8ARKNzPr8/zWln23iAGwH6Wb2vnNGEtbeo31IOBhlG5qnQsRA3enAfCk/w8Ws5gyDlRwFQAAAABJRU5ErkJggg==" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The cost of laying the cable along the sea bed is US$80 per metre, and the cost of laying it on land is US$20 per metre.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find, in terms of <em>x</em>, an expression for the cost of laying the cable.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>x</em>, to the nearest metre, such that this cost is minimized.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the curve \(y = \frac{{\cos x}}{{\sqrt {{x^2} + 1} }},{\text{ }} - 4 \leqslant x \leqslant 4\) showing clearly the coordinates of the <em>x-</em>intercepts, any maximum points and any minimum points.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the gradient of the curve at <em>x </em>= 1 .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the normal to the curve at <em>x </em>= 1 .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A jet plane travels horizontally along a straight path for one minute, starting at time \(t = 0\) , where \(t\) is measured in seconds. The acceleration, \(a\) , measured in ms<sup>−2</sup>, of the jet plane is given by the straight line graph below.</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find an expression for the acceleration of the jet plane during this time, in terms of \(t\) .</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><span style="font-family: times new roman,times; font-size: medium;">Given that when \(t = 0\) the jet plane is travelling at \(125\) ms<sup>−1</sup>, find its maximum velocity in ms<sup>−1</sup> during the minute that follows.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that the jet plane breaks the sound barrier at \(295\) ms<sup>−1</sup>, find out for how long the jet plane is travelling greater than this speed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A particle, A, is moving along a straight line. The velocity, \({v_A}{\text{ m}}{{\text{s}}^{ - 1}}\), of A <em>t</em> seconds after its motion begins is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{v_A} = {t^3} - 5{t^2} + 6t.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \({v_A} = {t^3} - 5{t^2} + 6t\) for \(t \geqslant 0\), with \({v_A}\) on the vertical axis and <em>t</em> on the horizontal. Show on your sketch the local maximum and minimum points, and the intercepts with the <em>t</em>-axis.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the times for which the velocity of the particle is increasing.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the times for which the magnitude of the velocity of the particle is increasing.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">At <em>t</em> = 0 the particle is at point O on the line.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for the particle’s displacement, \({x_A}{\text{m}}\), from O at time <em>t</em>.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A second particle, B, moving along the same line, has position \({x_B}{\text{ m}}\), velocity \({v_B}{\text{ m}}{{\text{s}}^{ - 1}}\) and acceleration, \({a_B}{\text{ m}}{{\text{s}}^{ - 2}}\), where \({a_B} = - 2{v_B}\) for \(t \geqslant 0\). At \(t = 0,{\text{ }}{x_B} = 20\) and \({v_B} = - 20\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \({v_B}\) in terms of <em>t</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>t</em> when the two particles meet.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = {({x^3} + 6{x^2} + 3x - 10)^{\frac{1}{2}}},{\text{ for }}x \in D,\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where \(D \subseteq \mathbb{R}\) is the greatest possible domain of <em>f</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the roots of \(f(x) = 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence specify the set <em>D</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the coordinates of the local maximum on the graph \(y = f(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Solve the equation \(f(x) = 3\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Sketch the graph of \(\left| y \right| = f(x),{\text{ for }}x \in D\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Find the area of the region completely enclosed by the graph of \(\left| y \right| = f(x)\)</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined on the domain [0, 2] by \(f(x) = \ln (x + 1)\sin (\pi x)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Obtain an expression for \(f'(x)\) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graphs of <em>f</em> and \(f'\) on the same axes, showing clearly all <em>x</em>-intercepts.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the <em>x</em>-coordinates of the two points of inflexion on the graph of <em>f</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the normal to the graph of <em>f</em> where <em>x</em> = 0.75 , giving your answer in the form <em>y</em> = <em>mx</em> + <em>c</em> .</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the points \({\text{A}}\left( {a{\text{ }},{\text{ }}f(a)} \right)\) , \({\text{B}}\left( {b{\text{ }},{\text{ }}f(b)} \right)\) and \({\text{C}}\left( {c{\text{ }},{\text{ }}f(c)} \right)\) where <em>a</em> , <em>b</em> and <em>c</em> \((a < b < c)\) are the solutions of the equation \(f(x) = f'(x)\) . Find the area of the triangle ABC.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the graph of \(y = x + \sin (x - 3),{\text{ }} - \pi \leqslant x \leqslant \pi \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph, clearly labelling the <em>x</em> and <em>y</em> intercepts with their values.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of the region bounded by the graph and the <em>x</em> and <em>y</em> axes.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the curve with equation \(f(x) = {{\text{e}}^{ - 2{x^2}}}{\text{ for }}x < 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the point of inflexion and justify that it is a point of inflexion.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Below is a sketch of a Ferris wheel, an amusement park device carrying passengers </span><span style="font-family: times new roman,times; font-size: medium;">around the rim of the wheel.</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) The circular Ferris wheel has a radius of 10 metres and is revolving at a rate of 3 radians per minute. Determine how fast a passenger on the wheel is going vertically upwards when the passenger is at point A, 6 metres higher than the centre of the wheel, and is rising.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) The operator of the Ferris wheel stands directly below the centre such that the bottom of the Ferris wheel is level with his eyeline. As he watches the passenger his line of sight makes an angle \(\alpha \) with the horizontal. Find the rate of change of \(\alpha \) at point A.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A particle moves in a straight line with velocity <em>v </em>metres per second. At any time <em>t </em>seconds, \(0 \leqslant t < \frac{{3\pi }}{4}\), the velocity is given by the differential equation \(\frac{{{\text{d}}v}}{{{\text{d}}t}} + {v^2} + 1 = 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It is also given that <em>v </em>= 1 when <em>t </em>= 0 .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for <em>v </em>in terms of <em>t </em>.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of <em>v </em>against <em>t </em>, clearly showing the coordinates of any intercepts, and the equations of any asymptotes.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the time <em>T </em>at which the velocity is zero.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the distance travelled in the interval [0, <em>T</em>] .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for <em>s </em>, the displacement, in terms of <em>t </em>, given that <em>s </em>= 0 when <em>t </em>= 0 .</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, show that \(s = \frac{1}{2}\ln \frac{2}{{1 + {v^2}}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram shows the plan of an art gallery <em>a </em>metres wide. [AB] represents a doorway, leading to an exit corridor <em>b </em>metres wide. In order to remove a painting from the art gallery, CD (denoted by <em>L </em>) is measured for various values of \(\alpha \) , as represented in the diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfAAAAFPCAIAAAA9UgsRAAAVeUlEQVR4nO3d728b92HHcf4L2/6A7wGHNKEwAXHTFUHq6UoUARwNEZy0YAVk7eA5Xk21ToK0WF1SUNs5WUullZdgs9Z1nVRTxdpm6RFJ68I+e3TbgA8MCLkHe6Dcs8Bn6IFxCAQBFxgG8d2D029RP0je8Xvfu/cL98SyRH315J3LR6RZkACGqrPu/eE3cy+OCsMUhikmphda3r2b351prak+GXRXUH0AIFcerLZeGxcj47WFlrcmpZSd1WV77lzRMM/aq6oPB90RdGBoOuvLb40LY/S8fa+z7+MEHQMj6MDQ3G/VxkwxNt26v/dvOh81v/9Lr9Pti4BjI+jAsKy1pouGWZxprVFuJIKgA8Oyap8Thvn43PJD1SdBRhF0YFiioHOHjsQQdGBYOiuLEyOmeGHR+0T1UZBNBB0Ymgf37FdGuzzLRcrOvdabv1he584dAyHowBB17rVmJkwxMl772XvLqx0ppeysezcun//W4gqvK8KgCDowZA9Wl+3LZ580N14p+uS5OXt59YHqUyELCDoAZARBB4CMIOgAkBEEHQAyooegb/4Oh4uLi4trqBdB5+Li4srIlVTQj//5AI601GiYwnBdV/VBkFIEHdBGGIaT5XLJssIwVH0WpBFBB3TieZ4pjNl6XfVBkEYEHdBMNLx4nqf6IEgdgg5oJgzDkmVNlssML9iDoAP6cV3XFMZSo6H6IEgXgg5oabZeZ3jBHgQd0BLDC/Yj6ICuGF6wB0EHNMbwgp0IOqAxhhfsRNABvbXbbVMYTdtWfRCoR9AB7dWqVVMYvu+rPggUI+iA9oIgKFnWVKWi+iBQjKADWeA4DsMLCDqQEQwvIOhARjC8gKAD2cHwknMEHciUaHgJgkD1QaAAQQcyJQgCUxi1alX1QaAAQQeypmnbpjAcx1F9EAwbQQcyaKpSKVkWw0veEHQgg3zfZ3jJIYIOZBPDSw4RdCCzGF7yhqADmcXwkjcEHciyaHhxXVf1QTAMBB3IsjAMJ8vlkmXxDhh5QNCBjPM8zxTGbL2u+iBIHEEHsm+p0WB4yQOCDmQfw0tOEHQgFxhe8oCgA3nB8JJ5BB3Ii2h4mSyXGV6yiqADOeK6rimMpUZD9UGQCIIO5MtsvW4Kw/M81QdB/Ag6kC9hGJYsi+Elkwg6kDsML1lF0IE8YnjJJIIO5BHDSyYRdCCnGF6yh6AD+VWrVk1h+L6v+iCIB0EH8isIgpJlTVUqqg+CeBB0INccxzGF0bRt1QdBDAg6kHcML5lB0IG8Y3jJDIIOgOElIwg6ACkZXjKBoAOQUsogCExh1KpV1QdB/wg6gA1N2zaF4TiO6oOgTwQdwLapSqVkWUEQqD4I+kHQAWzzfZ/hRV8EHcAuDC/6IugA9mJ40RRBB7AXw4umCDqALqLhpd1uqz4IekDQAXQ3WS6XLIt3wNAIQQfQned5pjBm63XVB8FxEXQAB1pqNExhuK6r+iA4FoIO4EBhGDK8aISgAzgMw4tGCDqAIzC86IKgAzgCw4suCDqAo0XDy1KjofogOAxBB3Ass/W6KQzP81QfBAci6ACOJQzDkmVNlssML6lF0AEcl+u6DC9pRtAB9IDhJc0IOoAeMLykGUEH0BuGl9Qi6AB6xvCSTgQdQM+CIChZ1lSlovog2IWgA+iH4zimMJq2rfog2EbQAfSpVq2awvB9X/VBsIGgA+gTw0vaEHQA/WN4SRWCDmAgDC/pQdABDIThJT0IOoBBNW3bFIbjOKoPkncEHUAMpiqVkmUFQaD6ILlG0AHEwPd9Uxi1alX1QXKNoAOIB8OLcgQdQGwYXtQi6ABiw/CiVn9Bf7B6Z/5c0TDF8/W5bz0hxqZb9wd5XACZwfCiUB9BX1tZOD9afPGtO6ud9TuXT42YxZnWWmeQxwWQJdHwwjtgDF+vQe+sL781Lp6/vPyxlFLKu82zJ8yJBW9vzwk6kF+e55nCmK3XVR8kd3oMeuej5vknR2uttegDnZXFiROnF1b29ZygA7m21GiYwnBdV/VB8qW3oHe8hdPbi3l0t95lQO/1cQFkTBiGk+Uyw8uQ9Rb0tdbMqLjQXH0opZTr/7d49smuA3qvjwsgexhehq+PoD9/efnjzmr7rdqFc6dGzIk3mq0PVtnQAezD8DJkPW7o0V25GBmv2d76R82zJ0ZPXfivjV+Q9v+4ADKJ4WXIeGERgARFw8v8lXnVB8kFgg4gWfNX5k1heJ6n+iDZR9ABJCsMw5JlTZbLDC9JI+gAEue6rimMpUZD9UEyjqADGIbZep3hJWkEHcAwMLwMAUEHMCQML0kj6ACGh+ElUQQdwPBEw8tUpaL6INlE0AEMleM4pjCatq36IBlE0AEMW61aNYXh+77qg2QNQQcwbEEQMLwkgaADUIDhJQkEHYAaDC+xI+gA1GB4iR1BB6AMw0u8CDoAlaYqlZJlBUGg+iBZQNABqOT7vimMWrWq+iBZQNABKNa0bVMYjuOoPoj2CDoA9RheYkHQAajH8BILgg4gFRheBkfQAaQFw8uACDqAtIiGl9l6XfVBdEXQAaTIUqNhCsN1XdUH0RJBB5AiYRhOlssly+KtR/tA0AGki+d5DC/9IegAUofhpT8EHUDqMLz0h6ADSCOGlz4QdAApFQ0vnuepPog2CDqAlArDsGRZk+Uyw8sxEXQA6eW6rimMpUZD9UH0QNABpNpsvc7wckwEHUCqMbwcH0EHkHYML8dE0AFogOHlOAg6AA1sDS+qD5JqBB2AHtrttimMpm2rPkh6EXQA2qhVq6YwfN9XfZCUIugAtBEEQcmypioV1QdJKYIOQCeO4zC8HISgA9AMw8tBCDoAzTC8HISgA9APw0tXBB2AlqLhJQgC1QdJEYIOQEu+75vCqFWrqg+SIgQdgK6atm0Kw3Ec1QdJC4IOQGNTlUrJshheIgQdgMYYXnYi6AD0xvCyhaAD0B7DS4SgA9Aew0uEoAPIgqVGwxSG67qqD6ISQQeQBWEYTpbLJcvK81uPEnQAGeF5nimM2Xpd9UGUIegAsiPnwwtBB5Ad0VuPjp08mc/hhaADyIggCM6cOfMXf/bnpjB++IMfqD6OAgQdQBZ8+OGHn/70pwuFwo/n/qVWnc7n8ELQAWjv/fffLxQKjz762H//8u2bt25f+/31p7/w9Je++MW8DS8EHYDeopo/M/43v7t2/Q9/fP/mrds3b93+j5/+zBTGUqOh+nRDRdABaGxnzVu3/xTVPLrOf61iCsPzPNVnHB6CDkBXH3zwwVbNb966vXV7Hl3Xfn/9s098JlfDC0EHoKW7d++eOHHi0Ucf61rzfA4vBB2Als6cOfPYY8Xot6A3b93es7fkc3gh6AD08/Of/7xQKPyw/kZU7T/+qd215nkbXgg6AM3cvXu3UCj8/dkXD4p41+GladuqD544gg5AM6+++upjjxVt+92o1396/8Db863rK3/7VVMYvu+rPnuyCDoAnUTPU/ze9/7pmLfn0WU33/vsZ/7qa//wNdXHTxZBB6CTZ5999qnPnYye2bLj9vza1R+9PPGIYYpPnfy7Hy/+6EzxkTNz7/3vzqa/8cZc5ocXgg5AG9ETz998819334BfW7z4XFFYX73U+K3z60unrZNPjRS/8ua7+RteCDoAbXzzm9/ceuL51pNbblytfl586uTXf/LbW7dv3rr19qUvmeLUxas3cji8EHQAegiCoFAo/OO3v7P7iefvzn3lCfORL1/69c3tP45Vrzrdx/RsDy8EHYAerl+/XigUtl5JtPHSUKdxcexTWwPLjavVz4uRiUtv5/MZLwQdgB5effXVpz53clfNb92+eeOn3ygaf/n1n16/dfum8+tLpx83xeYfD37GiymM71y8qPoHih9BB6CBMAy77S2bG8tTL//77367ePG54iPPXfy31776+W/96PsvV7vN6NF16dLrpjAcx1H9Y8WMoAPQwIcfflgoFH7yk/+8ue/f4brx9pWXnn3cFIb51JlLV69F9+nF06//6oAZPbqef+75sc+dDIJA9U8WJ4IOQAPvvPNOoVDYen7L4Ncvf/U/2RteCDoADbz++uvRgH6cF/of88re8ELQAWjgpZdeKn95Mq6UZ3V4IegANFAoFC5ceDnG2/NMDi8EHYAG+vgHuXoaXtrttuofMQYEHYAGCoXCa6/9cxJBv/b7609/4emxk3+dgXfASDboXFxcXBm4+g5df1/Y91cd/uV7EHQuLq48Xn2Hru809/dVh3/5Hj0EHQA0EqWwZFlaDy8EHQC2b35n63XVZ+kfQQeAjRQuNRqmMFzXVX2cPhF0ANhIYRiGk+WyvsMLQQeQA2ut6eLI+Nyd9QP+fiuFnudFt+rDPF1cNn+Kh6v2hS6/OC2+ePmdP3jrneiTCToAHT24Z78yKgyzONNa63T9jJ33trP1uikMz/OGeMJ47LpD76wsToyYZ+3V6I/r3q25F0eFMXrevteRkqAD0FJnZXFi4tzZZ0wxNt263/VTdqYwDMOSZU2Wy9oNL7snl7vNsye2gy43Ey9eWPQ+kQQdgIY6a62Z0YmFlbt2pbh9f7rHnvXZdV0dh5cjgi7vt2pjBB2Avu63ahMV+6PO7pztsf/XiToOL4cGfc2zZ8aZXADoq+MtnN6YzjtrrZlRMXJ6YWX/Pfr+oOs4vHQJ+t7fi27/FoGgA9DLx8tzk9sFX79z+dRI11+Ndn3Cn3bDy2F36J3VZXvuXNEwi+cXV9YkQQegmbXWdHH/P3vS5VejBz2DOxpefN8fynEHddSGHv0/imFOLHgdgg5AJ594Cy+M1lprOz7U8RZOC2PPB+XBQQ+CoGRZU5VKogeNC0EHkEmd9ZXGueIzl5f3vJYoWpYnpu2VnX9xyGssHccxhdG07eTOGpddP0W0L20H/cHq8i+mT42Y4smK/VGHyQWALh4uzz2xMbCcOGff3fzw7t8T7rh7PfxF87VqVYvh5YhXigpj9Ozcb1pe9F8ygg4gmw4Pui7DC/+WCwAcnUIthheCDgDHSmH6hxeCDgDHSmH6hxeCDgDHTWHTtk1hOI4zhCP1gaADQA8pnKpUSpYVBEHSR+oDQQeAHlLo+74pjFq1mvSR+kDQAaC3FKZ2eCHoANBbCmVahxeCDgA9Bz2dwwtBB4Cegy5TObwQdADoJ+hyc3hJzztgEHQA6DPonueZwpit15M4Uh8IOgD0GXQp5VKjYQrDdd3Yj9QHgg4A/Qc9DMPJcjklwwtBB4D+gy7TNLwQdAAYKOgyNcMLQQeAQYOekuGFoAPAoEGXm8PL/JX5uI7UB4IOADEEXUo5f2XeFIbnebEcqQ8EHQDiCXoYhiXLmiyXVQ0vBB0A4gm6lNJ1XVMYS43G4A/VB4IOALEFXUo5W6+rGl4IOgDEGXSFwwtBB4A4gy7VDS8EHQBiDrpUNLwQdACIP+hbw0uMj3kkgg4A8QddSuk4jimMpm3H+7CHIOgAkEjQpZS1atUUhu/7sT9yVwQdAJIKehAEJcuaqlRif+SuCDoAJBV0OdzhhaADQIJBl0McXgg6ACQb9KENLwQdAJINuhzW8ELQASDxoEspa9VqybKCIEjuWxB0ABhG0H3fN4VRq1aT+xYEHQCGEXQpZdO2TWE4jpPQ4xN0ABhS0KWUU5VKcsMLQQeA4QU90eGFoAPA8IIukxxeCDoADDXoMrHhhaADwLCDHg0vs/V6vA9L0AFg2EGXUi41GqYwXNeN8TEJOgAoCHoYhpPlcsmyYnzrUYIOAAqCLqX0PC/e4YWgA4CaoMu4hxeCDgDKgh7v8ELQAUBZ0GWswwtBBwCVQZfxDS8EHQAUBz0Mw5JlTZbLAw4vBB0AFAddSum6rimMpUZjkAch6ACgPuhSytl63RSG53l9PwJBB4BUBH3w4YWgA0Aqgi4HHl4IOgCkJehysOGFoANAioI+yPBC0AEgRUGXm8NL07Z7/UKCDgDpCrqUslatmsLwfb+nryLoAJC6oAdBULKsqUqlp68i6ACQuqBLKR3H6XV4IegAkMagy96HF4IOACkNeq/DC0EHgJQGXfY4vBB0AEhv0OXm8BIEwZGfSdABINVBD4LAFEatWj3yMwk6AKQ66FLKpm2bwnAc5/BPI+gAkPagSymnKpWSZR0+vBB0ANAg6L7vHzm8EHQA0CDo8hjDC0EHAD2CLo8aXgg6AGgT9MOHF4IOANoEXW4OL+12e/9fEXQA0CnoYRhOlssly9r/DhgEHQB0CrqU0vM8Uxiz9fqejxN0ANAs6FLKpUbDFIbrujs/SNABQL+gdx1eCDoA6Bd02W14IegAoGXQ5b7hhaADgK5B3zO8bPwUq/Y5YZh7r5Hx2s+a7y6vdja+lqADyCZNgy43h5elRkPu+inut2pjprjQXH0opZTyweqyffnsk6YwRs82VtY7kqADyCp9gy6lnK3XTWFEZd/8KR6u2hd2BF1KKWXnXmtmwhQj43N31gk6gKzSOuhhGJYsa7JcPiLoUsq11nTRMIszrbUOQQeQTVoHXUrpuu7WXC6lPDDoG1PM2HTrPkEHkE26B11uDi9HBX19ee4ZU5w4Z98l6ACyKQNBD8Pw2Hfoz1xeXifoALIpA0GXu34KNnQAeZWLoPMsFwB5kLmgH/Q89JHxmZvRa4sIOoBsylTQD3mlaMtb3/xkgg4gmzIV9OMh6ACyiaADQEYQdADICIIOABlB0AEgIwg6AGREskHfeh7h43PLD4/+9L4RdABI/g69s7I4MTJaa60l+D0IOgAkH/SOt3BajE237if3LSRBBwCZeNA/8RZeiP5JrMS+hZQEHQBk0kGP9pZvL9x488VRYZhi4rute0mknaADQMJBj/7R2lMzTW9Nrt+5fGokobt1gg4AiQa9s9aaGRXPX17+WEq5+Z5B+996IgYEHQASDfr9Vm3MnFjwOjv+mMzzFwk6ACQZ9LXWdHHk9MJKZ8cfo7eYiB1BB4AEg/5wee4J8cKi94mUUsoH9+xXRouvNO89SOJ7EXQA6J7CIAiCIBjsgT/xFl7Y2Fs6q8tXa+PbY3r/wjD0fX//xwk6AOxNYRAE81fmTWFMVSqDPXBn3Wst1iY23zNooeXF8FpRx3FMYdSqVc/zdn6coAPAdgq3Um4KY/7K/MB36IkIw7Bp2yXL2pN1gg4AGyncWcmum0aq7M86QQcAue8tlTW+jvkjE3QA2aS8wgQdAOIRpVCLAX1Lu92eLJdNYUyWy+12W7KhA4DcncKdWW/attqDdeX7/p6URwg6AHRJoe/7s/X6/JV5VUc6RLvdrlWrO1MeIegAwHuKAkBWEHQAyAiCDgAZQdABICMIOgBkBEEHgIwg6ACQEQQdADKCoANARhB0AMiIHAb9/wEFDvLxGc1tewAAAABJRU5ErkJggg==" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">If </span><span style="font: 12.5px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\alpha \)</span> </span><span style="font-family: 'times new roman', times; font-size: medium;">is the angle between [CD] and the wall, show that \(L = \frac{a }{{\sin \alpha }} + \frac{b}{{\cos \alpha }}{\text{, }}0 < \alpha < \frac{\pi }{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"> </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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: arial, helvetica, sans-serif;"><span style="font-family: 'times new roman', times; font-size: medium;">If <em>a </em>= 5 and <em>b </em>= 1, find the maximum length of a painting that can be removed through this doorway.</span><br></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find, in terms of <em>k </em>, the maximum length of a painting that can be removed from the gallery through this doorway.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the minimum value of <em>k </em>if a painting 8 metres long is to be removed through this doorway.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of the region enclosed by the curves \(y = {x^3}\) and \(x = {y^2} - 3\) .<br></span></p>
<p> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"> </p>
<p> </p>
<p> </p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the graphs \(y = {{\text{e}}^{ - x}}\) and \(y = {{\text{e}}^{ - x}}\sin 4x\) , for \(0 \leqslant x \leqslant \frac{{5\pi }}{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) On the same set of axes draw, on graph paper, the graphs, for \(0 \leqslant x \leqslant \frac{{5\pi }}{4}\). </span><span style="font-family: times new roman,times; font-size: medium;">Use a scale of \(1\) cm to \(\frac{\pi }{8}\) </span><span style="font-family: times new roman,times; font-size: medium;">on your \(x\)-axis and \(5\) cm to \(1\) unit on your \(y\)-axis.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Show that the \(x\)-intercepts of the graph \(y = {{\text{e}}^{ - x}}\) sin 4x are \(\frac{{n\pi }}{4}\)</span> <span style="font-family: times new roman,times; font-size: medium;">, \(n = 0\), \(1\), \(2\), \(3\), \(4\), \(5\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) Find the \(x\)-coordinates of the points at which the graph of \(y = {{\text{e}}^{ - x}}\sin 4x\) meets </span><span style="font-family: times new roman,times; font-size: medium;">the graph of \(y = {{\text{e}}^{ - x}}\) . Give your answers in terms of \( \pi\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) (i) Show that when the graph of \(y = {{\text{e}}^{ - x}}\sin 4x\) meets the graph of \(y = {{\text{e}}^{ - x}}\) , </span><span style="font-family: times new roman,times; font-size: medium;">their gradients are equal.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Hence explain why these three meeting points are not local maxima of the </span><span style="font-family: times new roman,times; font-size: medium;">graph \(y = {{\text{e}}^{ - x}}\sin 4x\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) (i) Determine the \(y\)-coordinates, \({y_1}\) , \({y_2}\) and \({y_3}\), where \({y_1} > {y_2} > {y_3}\) , of the </span><span style="font-family: times new roman,times; font-size: medium;">local maxima of \(y = {{\text{e}}^{ - x}}\sin 4x\) for \(0 \leqslant x \leqslant \frac{{5\pi }}{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> . You do not need to show </span><span style="font-family: times new roman,times; font-size: medium;">that they are maximum values, but the values should be simplified.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Show that </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({y_1}\) , \({y_2}\) and \({y_3}\)</span> form a geometric sequence and determine the common ratio \(r\) .</span></p>
</div>
<br><hr><br>