File "SL-paper1.html"
Path: /IB QUESTIONBANKS/4 Fourth Edition - PAPER/HTML/Mathematics SL/Topic 2/SL-paper1html
File size: 189.31 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/ib-qb-46-logo-683ef0176708d789b2acbf6ece48c55de4cd5ddac781fb455afe3540d22d050e.jpg" alt="Ib qb 46 logo">
</div>
</div>
</div><h2>SL Paper 1</h2><div class="specification">
<p class="p1">Let \(f(x) = {(x - 5)^3}\), for \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({f^{ - 1}}(x)\).</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"><span class="s1">Let \(g\) </span>be a function so that \((f \circ g)(x) = 8{x^6}\)<span class="s1">. Find \(g(x)\).</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 \(f(x) = {\log _p}(x + 3)\) for \(x > - 3\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/folly.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph passes through A(6, 2) , has an <em>x</em>-intercept at (−2, 0) and has an asymptote </span><span style="font-family: times new roman,times; font-size: medium;">at \(x = - 3\) .</span></p>
<p> </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 <em>p</em> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is reflected in the line \(y = x\) to give the graph of <em>g</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the <em>y</em>-intercept of the graph of <em>g</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Sketch the graph of <em>g</em> , noting clearly any asymptotes and the image of A.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(f\) is reflected in the line \(y = x\) to give the graph of \(g\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(g(x)\) .</span></p>
<div class="marks">[4]</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;">Let \(f(x) = 6 + 6\sin x\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/abba.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The shaded region is enclosed by the curve of <em>f</em> , the <em>x</em>-axis, and the <em>y</em>-axis.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Solve for \(0 \le x < 2\pi \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(6 + 6\sin x = 6\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(6 + 6\sin x = 0\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the exact value of the <em>x</em>-intercept of <em>f</em> , for \(0 \le x < 2\pi \) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The area of the shaded region is <em>k</em> . Find the value of <em>k</em> , giving your answer in </span><span style="font-family: times new roman,times; font-size: medium;">terms of \(\pi \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to the graph of <em>g</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of this transformation.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to the graph of <em>g</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(\int_p^{p + \frac{{3\pi }}{2}} {g(x){\rm{d}}x} = k\) and \(0 \le p < 2\pi \) , write down the two values of <em>p</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {x^2}\) and \(g(x) = 2{(x - 1)^2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> can be obtained from the graph of <em>f</em> using two transformations.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of each of the two transformations.</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><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is translated by the vector \(\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 2}<br>\end{array}} \right)\) to give the graph of <em>h</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The point \(( - 1{\text{, }}1)\) on the graph of <em>f</em> is translated to the point P on the graph of <em>h</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Find the coordinates of P.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider \(f(x) = {x^2} + qx + r\). The graph of \(f\) has a minimum value when \(x = - 1.5\).</p>
<p class="p1">The distance between the two zeros of \(f\) <span class="s1">is 9</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the two zeros are 3 and \( - 6\)<span class="s1">.</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 class="p1">Find the value of \(q\) and of \(r\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the functions \(f(x)\) , \(g(x)\) and \(h(x)\) . The following table gives some values </span><span style="font-family: times new roman,times; font-size: medium;">associated with these functions.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/omt.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows parts of the graphs of \(h\) and \(h''\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/jls.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a point of inflexion on the graph of \(h\) at P, when \(x = 3\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(h(x) = f(x) \times g(x)\) ,<br></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;">Write down the value of \(g(3)\) , of \(f'(3)\) , and of \(h''(2)\) .</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;">Explain why P is a point of inflexion.</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><span style="font-family: times new roman,times; font-size: medium;">find the \(y\)-coordinate of P.</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><span style="font-family: times new roman,times; font-size: medium;">find the equation of the normal to the graph of \(h\) at P.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 2x - 1\) and \(g(x) = 3{x^2} + 2\) .</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 \({f^{ - 1}}(x)\) . </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><span style="font-family: times new roman,times; font-size: medium;">Find \((f \circ g)(1)\) . </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;">The diagram below shows the graph of a function \(f\) , for \( - 1 \le x \le 2\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/rachel.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \(f(2)\).<br></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of </span><span style="font-family: times new roman,times; font-size: medium;">\({f^{ - 1}}( - 1)\) .</span></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><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \({f^{ - 1}}\) on the grid below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/milly.png" alt></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 \(f(x) = \sqrt x \) . Line <em>L</em> is the normal to the graph of <em>f</em> at the point (4, 2) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In the diagram below, the shaded region <em>R</em> is bounded by the <em>x</em>-axis, the graph of <em>f</em> and </span><span style="font-family: times new roman,times; font-size: medium;">the line <em>L</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ring.png" alt></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;">Show that the equation of <em>L</em> is \(y = - 4x + 18\) .</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;">Point A is the <em>x</em>-intercept of <em>L</em> . Find the <em>x</em>-coordinate of A.</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><span style="font-family: times new roman,times;"><span style="font-size: medium;">Find an expression for the area of <em>R</em></span> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The region <em>R</em> is rotated \(360^\circ \) about the <em>x</em>-axis. Find the volume of the solid formed, </span><span style="font-family: times new roman,times; font-size: medium;">giving your answer in terms of \(\pi \) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3{(x + 1)^2} - 12\) .</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;">Show that \(f(x) = 3{x^2} + 6x - 9\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For the graph of <em>f</em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) write down the coordinates of the vertex;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) write down the <em>y</em>-intercept;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) find both <em>x</em>-intercepts.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Hence</strong> sketch the graph of <em>f</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><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^2}\) . The graph of <em>f</em> may be obtained from the graph of <em>g</em> by the following two transformations</span></p>
<p style="margin-left: 60px;"><span style="font-family: times new roman,times; font-size: medium;">a stretch of scale factor <em>t</em> in the <em>y</em>-direction,</span></p>
<p style="margin-left: 60px;"><span style="font-family: times new roman,times; font-size: medium;">followed by a translation of </span><span style="font-family: times new roman,times; font-size: medium;">\(\left( \begin{array}{l}<br>p\\<br>q<br>\end{array} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down \(\left( \begin{array}{l}<br>p\\<br>q<br>\end{array} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">and the value of <em>t</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 7 - 2x\) and \(g(x) = x + 3\) .</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 \((g \circ f)(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Write down \({g^{ - 1}}(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \((f \circ {g^{ - 1}})(5)\) .</span></p>
<div class="marks">[2]</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;">The following diagram shows the graph of \(y = f(x)\), for </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\( - 4 \le x \le 5\).</span></span></span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_3.png" 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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(f( - 3)\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of </span>\({f^{ - 1}}(1)\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 domain of \({f^{ - 1}}\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On the grid above, sketch the graph of \({f^{ - 1}}\).</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;">The diagram below shows part of the graph of \(f(x) = (x - 1)(x + 3)\) .</span></p>
<p><span style="font-family: TimesNewRomanPSMT;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/wash.png" alt></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;">(a) Write down the \(x\)-intercepts of the graph of \(f\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find the coordinates of the vertex of the graph of \(f\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the \(x\)-intercepts of the graph of \(f\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find the coordinates of the vertex of the graph of \(f\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \sqrt {x + 2} \) for <em>x</em> ≥ 2 and <em>g</em>(<em>x</em>) = 3<em>x</em> − 7 for \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <em>f </em>(14).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\left( {g \circ f} \right)\) (14).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find g<sup>−1</sup>(<em>x</em>).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 8x + 3\) and \(g(x) = 4x\), for \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down \(g(2)\).</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 class="p1">Find \((f \circ g)(x)\).</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 \({f^{ - 1}}(x)\).</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;">Let \(f(x) = 4x - 2\) and \(g(x) = - 2{x^2} + 8\) .</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 \({f^{ - 1}}(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \((f \circ g)(1)\) .</span></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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = 3{x^2} - 6x + p\). The equation \(f(x) = 0\) has two equal roots.</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;">Write down the <strong>value </strong>of the discriminant.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">Hence, show that \(p = 3\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 graph of \(f\)has its vertex on the \(x\)-axis.</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 coordinates of the vertex of the graph of \(f\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(f\) has its vertex on the \(x\)-axis.</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;">Write down the solution of \(f(x) = 0\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 graph of \(f\) has its vertex on the \(x\)-axis.</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 function can be written in the form \(f(x) = a{(x - h)^2} + k\). Write down the value of \(a\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d(i).</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 graph of \(f\) has its vertex on the \(x\)-axis.</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 function can be written in the form \(f(x) = a{(x - h)^2} + k\). Write down the value of \(h\).<em><br></em></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d(ii).</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 graph of \(f\) has its vertex on the \(x\)-axis.</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 function can be written in the form \(f(x) = a{(x - h)^2} + k\). Write down the value of \(k\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d(iii).</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 graph of \(f\) has its vertex on the \(x\)-axis.</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 graph of a function \(g\) is obtained from the graph of \(f\) by a reflection of \(f\) in the \(x\)-axis, followed by a translation by the vector \(\left( \begin{array}{c}0\\6\end{array} \right)\). Find \(g\), giving your answer in the form \(g(x) = A{x^2} + Bx + C\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider a function \(f\). The line <em>L</em><sub>1</sub> with equation \(y = 3x + 1\) is a tangent to the graph of \(f\) when \(x = 2\)</p>
</div>
<div class="specification">
<p>Let \(g\left( x \right) = f\left( {{x^2} + 1} \right)\) and P be the point on the graph of \(g\) where \(x = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down \(f'\left( 2 \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f\left( 2 \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the graph of <em>g</em> has a gradient of 6 at 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>Let <em>L</em><sub>2</sub> be the tangent to the graph of <em>g</em> at P. <em>L</em><sub>1</sub> intersects <em>L</em><sub>2</sub> at the point Q.</p>
<p>Find the y-coordinate of Q.</p>
<div class="marks">[7]</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;">Let \(f(t) = a\cos b(t - c) + d\) , \(t \ge 0\) . Part of the graph of \(y = f(t)\) is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/evening.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When \(t = 3\) , there is a maximum value of 29, at M.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> When \(t = 9\) , there is a minimum value of 15.</span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the value of <em>a</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(b = \frac{\pi }{6}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) Find the value of <em>d</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iv) Write down a value for <em>c</em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a(i), (ii), (iii) and (iv).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The transformation <em>P</em> is given by a horizontal stretch of a scale factor of \(\frac{1}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, followed </span><span style="font-family: times new roman,times; font-size: medium;">by a translation of \(\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 10}<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let \({M'}\) be the image of M under <em>P</em>. Find the coordinates of \({M'}\) .</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><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is the image of the graph of <em>f</em> under <em>P</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(g(t)\) in the form \(g(t) = 7\cos B(t - c) + D\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is the image of the graph of <em>f</em> under <em>P</em>.</span></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of the transformation that maps the graph of <em>g </em></span><span style="font-family: times new roman,times; font-size: medium;">to the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = a{(x - h)^2} + k\). The vertex of the graph of \(f\) is at \((2, 3)\) and the graph passes through \((1, 7)\).</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><span>Write down the value of </span></span><span style="font-family: 'times new roman', times; font-size: medium;"><span>\(h\)</span></span><em style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 21px; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-style: normal; line-height: normal;"> </span></span></em><span style="font-family: 'times new roman', times; font-size: medium;"><span>and of </span></span><span style="font-family: 'times new roman', times; font-size: medium;"><span>\(k\).</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </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: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(a\).</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;">The diagram below shows the graph of a function \(f(x)\) , for \( - 2 \le x \le 3\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q5.jpg" alt> </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;">Sketch the graph of \(f( - x)\) on the grid below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q5a.jpg" alt></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to obtain the graph of <em>g</em> . The graph of <em>g</em> is </span><span style="font-family: times new roman,times; font-size: medium;">shown below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q5b.jpg" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function <em>g</em> can be written in the form \(g(x) = af(x + b)\) . Write down the </span><span style="font-family: times new roman,times; font-size: medium;">value of <em>a</em> and of <em>b</em> .</span></p>
<p align="LEFT"> </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;">Let \(f(x) = {x^2} + 4\) and \(g(x) = x - 1\) .</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 \((f \circ g)(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">The vector \(\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 1}<br>\end{array}} \right)\) translates the graph of \((f \circ g)\) to the graph of <em>h</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the coordinates of the vertex of the graph of <em>h</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><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">The vector \(\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 1}<br>\end{array}} \right)\) translates the graph of \((f \circ g)\) to the graph of <em>h</em> .</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(h(x) = {x^2} - 8x + 19\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">The vector \(\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 1}<br>\end{array}} \right)\) translates the graph of \((f \circ g)\) to the graph of <em>h</em> .</span></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The line \(y = 2x - 6\) is a tangent to the graph of <em>h</em> at the point P. Find the </span><span style="font-family: times new roman,times; font-size: medium;"><em>x</em>-coordinate of P.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Let \(f(x) = m - \frac{1}{x}\), for \(x \ne 0\). The line \(y = x - m\) intersects the graph of \(f\) in two distinct points. Find the possible values of \(m\).</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 2{x^3} + 3\) and \(g(x) = {{\rm{e}}^{3x}} - 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(g(0)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \((f \circ g)(0)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \({f^{ - 1}}(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = {x^2}\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_17.08.23.png" alt="M17/5/MATME/SP1/ENG/TZ2/10"></p>
<p>The line \(L\) is the tangent to the graph of \(f\) at the point \({\text{A}}( - k,{\text{ }}{k^2})\), and intersects the \(x\)-axis at point B. The point C is \(( - k,{\text{ }}0)\).</p>
</div>
<div class="specification">
<p>The region \(R\) is enclosed by \(L\), the graph of \(f\), and the \(x\)-axis. This is shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_17.07.29.png" alt="M17/5/MATME/SP1/ENG/TZ2/10.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down \(f'(x)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the gradient of \(L\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle ABC, giving your answer in terms of \(k\).</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>Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the graph of a quadratic function <em>f</em> , for \(0 \le x \le 4\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q8.jpg" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph passes through the point P(0, 13) , and its vertex is the point V(2, 1) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function can be written in the form \(f(x) = a{(x - h)^2} + k\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>h</em> and of <em>k</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(a = 3\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f(x)\) , giving your answer in the form \(A{x^2} + Bx + C\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Calculate the area enclosed by the graph of <em>f</em> , the <em>x</em>-axis, and the lines \(x = 2\) and \(x = 4\) .</span></p>
<div class="marks">[8]</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;">The following diagram shows part of the graph of a quadratic function <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tent.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>x</em>-intercepts are at \(( - 4{\text{, }}0)\) and \((6{\text{, }}0)\) , and the <em>y</em>-intercept is at \((0{\text{, }}240)\) .</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;">Write down \(f(x)\) in the form \(f(x) = - 10(x - p)(x - q)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find another expression for \(f(x)\) in the form \(f(x) = - 10{(x - h)^2} + k\) .</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;">Show that \(f(x)\) can also be written in the form \(f(x) = 240 + 20x - 10{x^2}\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\) , at time <em>t</em> seconds is </span><span style="font-family: times new roman,times; font-size: medium;">given by \(v = 240 + 20t - 10{t^2}\) , for \(0 \le t \le 6\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the value of<em> t</em> when the speed of the particle is greatest.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the acceleration of the particle when its speed is zero.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^2} - 4x + 5\).</p>
</div>
<div class="specification">
<p class="p1">The function can also be expressed in the form \(f(x) = {(x - h)^2} + k\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the equation of the axis of symmetry of the graph 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"><span class="s1">(i) <span class="Apple-converted-space"> </span></span>Write down the value of \(h\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of \(k\).</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;">Let \(f(x) = \cos 2x\) and \(g(x) = 2{x^2} - 1\) .</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 \(f\left( {\frac{\pi }{2}} \right)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \((g \circ f)\left( {\frac{\pi }{2}} \right)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Given that \((g \circ f)(x)\) can be written as \(\cos (kx)\) , find the value of <em>k</em>, \(k \in \mathbb{Z}\) .</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;">Let \(f(x) = \ln (x + 5) + \ln 2\) , for \(x > - 5\) .</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 \({f^{ - 1}}(x)\) .</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;">Let \(g(x) = {{\rm{e}}^x}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find \((g \circ f)(x)\) , giving your answer in the form \(ax + b\) , where \(a,b \in \mathbb{Z}\) .</span></p>
<p> </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 \(f(x) = k{\log _2}x\) .</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;">Given that \({f^{ - 1}}(1) = 8\) , find the value of \(k\) .</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;">Find \({f^{ - 1}}\left( {\frac{2}{3}} \right)\) .</span></p>
<div class="marks">[4]</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;">Let \(f(x) = \frac{{{{(\ln x)}^2}}}{2}\), for \(x > 0\).</span></p>
</div>
<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 \(g(x) = \frac{1}{x}\). The following diagram shows parts of the graphs of \(f'\) and <em>g</em>.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; min-height: 25px; text-align: center; margin: 0px;"><img src="images/maths_10b.png" alt></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 graph of \(f'\) has an <em>x</em>-intercept at \(x = p\).</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;">Show that \(f'(x) = \frac{{\ln x}}{x}\).</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: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">There is a minimum on the graph of \(f\). Find the \(x\)-coordinate of this minimum.</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(p\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(g\) intersects the graph of \(f'\) when \(x = q\).</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 value of \(q\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(g\) intersects the graph of \(f'\) when \(x = q\).</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 \(R\) be the region enclosed by the graph of \(f'\), the graph of \(g\) and the line \(x = p\).</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;">Show that the area of \(R\) is \(\frac{1}{2}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = 1 + {{\text{e}}^{ - x}}\) and \(g(x) = 2x + b\), for \(x \in \mathbb{R}\), where \(b\) is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \((g \circ f)(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3\), find the value of \(b\).</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;">Let \(f(x) = lo{g_3}\sqrt x \) , for \(x > 0\) .</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;">Show that \({f^{ - 1}}(x) = {3^{2x}}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Write down the range of \({f^{ - 1}}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {\log _3}x\) , for \(x > 0\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(({f^{ - 1}} \circ g)(2)\) , giving your answer as an integer.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 3\sin \left( {\frac{\pi }{2}x} \right)\), for \(0 \leqslant x \leqslant 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Write down the amplitude of \(f\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the period of \(f\).</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">On the following grid sketch the graph of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_05.59.27.png" alt="M16/5/MATME/SP1/ENG/TZ1/03.b"></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>f</em> be the function given by \(f(x) = {{\rm{e}}^{0.5x}}\) , \(0 \le x \le 3.5\) . The diagram shows the </span><span style="font-family: times new roman,times; font-size: medium;">graph of <em>f</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/infinity.png" alt></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;">On the same diagram, sketch the graph of \({f^{ - 1}}\) .</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;">Write down the range of \({f^{ - 1}}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \({f^{ - 1}}(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = 5x\) and \(g(x) = {x^2} + 1\), for \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({f^{ - 1}}(x)\).</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 \((f \circ g)(7)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of a function \(f\), with domain \( - 2 \leqslant x \leqslant 4\).</p>
<p><img src="images/Schermafbeelding_2018-02-11_om_09.13.25.png" alt="N17/5/MATME/SP1/ENG/TZ0/03"></p>
<p>The points \(( - 2,{\text{ }}0)\) and \((4,{\text{ }}7)\) lie on the graph of \(f\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of \(f\).</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>Write down \(f(2)\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down \({f^{ - 1}}(2)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the grid, sketch the graph of \({f^{ - 1}}\).</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;">Let \(f(x) = \frac{1}{2}{x^2} + kx + 8\) , where \(k \in \mathbb{Z}\) .</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 values of <em>k</em> such that \(f(x) = 0\) has two equal roots.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Each value of <em>k</em> is equally likely for \( - 5 \le k \le 5\) . Find the probability that </span><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 0\) has no roots.</span></p>
<div class="marks">[4]</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;">Let \(f(x) = 3x - 2\) and \(g(x) = \frac{5}{{3x}}\), for \(x \ne 0\).</span></p>
</div>
<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 \(h(x) = \frac{5}{{x + 2}}\), for \(x \geqslant 0\). The graph of <em>h </em>has a horizontal asymptote at \(y = 0\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({f^{ - 1}}(x)\).</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;">Show that \(\left( {g \circ {f^{ - 1}}} \right)(x) = \frac{5}{{x + 2}}\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the \(y\)-intercept of the graph of \(h\).</span></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 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;">Hence, sketch the graph of \(h\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 the graph of \({h^{ - 1}}\), write down the \(x\)-intercept;</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d(i).</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;">For the graph of \({h^{ - 1}}\), write down the equation of the vertical asymptote.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d(ii).</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;">Given that \({h^{ - 1}}(a) = 3\), find the value of \(a\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = p + \frac{9}{{x - q}}\), for \(x \ne q\). The line \(x = 3\) is a vertical asymptote to the graph of \(f\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the value of \(q\).</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 class="p1">The graph of \(f\) has a \(y\)-intercept at \((0,{\text{ }}4)\).</p>
<p class="p1">Find the value of \(p\).</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">The graph of \(f\) has a <em>\(y\)</em>-intercept at \((0,{\text{ }}4)\).</p>
<p class="p1">Write down the equation of the horizontal asymptote of the graph of \(f\).</p>
<div class="marks">[1]</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = p{x^3} + p{x^2} + qx\).</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(f'(x)\).</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: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(f'(x) \geqslant 0\), show that \({p^2} \leqslant 3pq\).</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;">Let \(f(x) = 8x - 2{x^2}\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: TimesNewRomanPSMT;"><br><img src="images/spin.png" alt></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 <em>x</em>-intercepts of the graph.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the equation of the axis of symmetry.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the <em>y</em>-coordinate of the vertex.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The velocity <em>v</em> ms<sup>−1</sup> of a particle at time <em>t</em> seconds, is given by \(v = 2t + \cos 2t\) , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le t \le 2\) .</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;">Write down the velocity of the particle when \(t = 0\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When \(t = k\) , the acceleration is zero.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(k = \frac{\pi }{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the exact velocity when \(t = \frac{\pi }{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When \(t < \frac{\pi }{4}\) , \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} > 0\)</span><span style="font-family: times new roman,times; font-size: medium;"> and when \(t > \frac{\pi }{4}\) , \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} > 0\) </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;">Sketch a graph of <em>v</em> against <em>t</em> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>d</em> be the distance travelled by the particle for \(0 \le t \le 1\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down an expression for <em>d</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Represent <em>d</em> on your sketch.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = p{x^2} + (10 - p)x + \frac{5}{4}p - 5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the discriminant of \(f(x)\) is \(100 - 4{p^2}\).</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 values of \(p\) so that \(f(x) = 0\) has two <strong>equal</strong> roots.</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 quadratic function. Part of the graph of \(f\) is shown 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="images/ally.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The vertex is at P(\(4\), \(2\)) and the <em>y</em>-intercept is at Q(\(0\), \(6\)) .</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;">Write down the equation of the axis of symmetry.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function <em>f</em> can be written in the form \(f(x) = a{(x - h)^2} + k\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>h</em> and of <em>k</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><span style="font-family: times new roman,times; font-size: medium;">The function <em>f</em> can be written in the form \(f(x) = a{(x - h)^2} + k\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>a</em> .</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;">The following diagram shows part of the graph of <em>f</em> , where \(f(x) = {x^2} - x - 2\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/gary.png" alt></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 both <em>x</em>-intercepts.</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;">Find the <em>x</em>-coordinate of the vertex.</span></p>
<div class="marks">[2]</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 \(f(x) = \ln ({x^4} + 1)\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The second derivative is given by \(f''(x) = \frac{{4{x^2}(3 - {x^4})}}{{{{({x^4} + 1)}^2}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The equation \(f''(x) = 0\) has only three solutions, when \(x = 0\) , \( \pm \sqrt[4]{3}\) \(( \pm 1.316 \ldots )\) .</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 value of \(f(0)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find the set of values of \(x\) for which \(f\) is increasing.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f''(1)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong>, show that there is no point of inflexion on the graph of \(f\) at \(x = 0\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There is a point of inflexion on the graph of \(f\) at \(x = \sqrt[4]{3}\) \((x = 1.316 \ldots )\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(f\) , for \(x \ge 0\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{x}{{ - 2{x^2} + 5x - 2}}\) for \( - 2 \le x \le 4\) , \(x \ne \frac{1}{2}\) , \(x \ne 2\) . The graph of \(f\) is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q10.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(f\) has a local minimum at A(\(1\), \(1\)) and a local maximum at B.</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;">Use the quotient rule to show that \(f'(x) = \frac{{2{x^2} - 2}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Hence find the coordinates of B.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that the line \(y = k\) does not meet the graph of <em>f</em> , find the possible values </span><span style="font-family: times new roman,times; font-size: medium;">of <em>k</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \({2^m} = 8\) and \({2^n} = 16\), write down the value of \(m\) and of \(n\).</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">Hence or otherwise solve \({8^{2x + 1}} = {16^{2x - 3}}\).</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;">Let \(f(x) = p(x - q)(x - r)\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/glee.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph passes through the points (−2, 0), (0, − 4) and (4, 0) .</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;">Write down the value of <em>q</em> and of <em>r</em>.</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><span style="font-family: times new roman,times; font-size: medium;">Write down the <strong>equation</strong> of the axis of symmetry.</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><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>p</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Let \(f(x) = 6x\sqrt {1 - {x^2}} \)</span>, for \( - 1 \leqslant x \leqslant 1\)<span class="s1">, and \(g(x) = \cos (x)\)</span>, for \(0 \leqslant x \leqslant \pi \)<span class="s1">.</span></p>
<p class="p2">Let \(h(x) = (f \circ g)(x)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write \(h(x)\) in the form \(a\sin (bx)\), where \(a,{\text{ }}b \in \mathbb{Z}\).</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 class="p1">Hence find the range of \(h\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A quadratic function \(f\) can be written in the form \(f(x) = a(x - p)(x - 3)\). The graph of \(f\) has axis of symmetry \(x = 2.5\) and \(y\)-intercept at \((0,{\text{ }} - 6)\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(a\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line \(y = kx - 5\) is a tangent to the curve of \(f\). Find the values of \(k\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows part of the graph of a quadratic function \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_09.12.00.png" alt="M16/5/MATME/SP1/ENG/TZ2/01"></p>
<p class="p1">The vertex is at \((3,{\text{ }} - 1)\) <span class="s1">and the \(x\)</span>-intercepts at 2 and 4<span class="s1">.</span></p>
<p class="p2">The function \(f\) can be written in the form \(f(x) = {(x - h)^2} + k\).</p>
</div>
<div class="specification">
<p class="p1">The function can also be written in the form \(f(x) = (x - a)(x - b)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the value of \(h\) and of \(k\).</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">Write down the value of \(a\) and of \(b\).</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 \(y\)-intercept.</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;">Let \(f(x) = \frac{{ax}}{{{x^2} + 1}}\) , \( - 8 \le x \le 8\) , \(a \in \mathbb{R}\) .The graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bernie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The region between \(x = 3\) and \(x = 7\) is shaded.</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;">Show that \(f( - x) = - f(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Given that \(f''(x) = \frac{{2ax({x^2} - 3)}}{{{{({x^2} + 1)}^3}}}\) , find the coordinates of all points of inflexion.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">It is given that \(\int {f(x){\rm{d}}x = \frac{a}{2}} \ln ({x^2} + 1) + C\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the area of the shaded region, giving your answer in the form \(p\ln q\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of \(\int_4^8 {2f(x - 1){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3 + \frac{{20}}{{{x^2} - 4}}\) , for \(x \ne \pm 2\) . The graph of <em>f</em> is given below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/fajita.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>y</em>-intercept is at the point A.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the coordinates of A.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(f'(x) = 0\) at A.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The second derivative \(f''(x) = \frac{{40(3{x^2} + 4)}}{{{{({x^2} - 4)}^3}}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> . Use this to</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) justify that the graph of <em>f</em> has a local maximum at A;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) explain why the graph of <em>f</em> does <strong>not</strong> have a point of inflexion.</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;">Describe the behaviour of the graph of \(f\) for large \(|x|\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of \(f\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve \({\log _2}x + {\log _2}(x - 2) = 3\) , for \(x > 2\) .</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 the graph of a function \(f(x)\) , for \( - 2 \le x \le 4\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/999.png" alt></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;">Let \(h(x) = f( - x)\) . Sketch the graph of \(h\) on the grid below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chubby.png" alt></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 align="LEFT"><span style="font-family: times new roman,times;"><span style="font-size: medium;">Let \(g(x) = \frac{1}{2}f(x - 1)\) </span><span style="font-size: medium;">. The point \({\text{A}}(3{\text{, }}2)\) on the graph of \(f\) is transformed to the </span><span style="font-size: medium;">point P on the graph of \(g\) . Find the coordinates of P.</span></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 \(f(x) = 3{(x + 1)^2} - 12\) .</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;">Show that \(f(x) = 3{x^2} + 6x - 9\) .</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><span style="font-family: times new roman,times; font-size: medium;">For the graph of <em>f </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) write down the coordinates of the vertex; </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) write down the <strong>equation</strong> of the axis of symmetry; </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) write down the <em>y</em>-intercept; </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (iv) find both <em>x</em>-intercepts. </span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b(i), (ii), (iii) and (iv).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Hence</strong> sketch 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><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^2}\) . The graph of <em>f</em> may be obtained from the graph of <em>g</em> by the two transformations: </span></p>
<p style="margin-left: 60px;"><span style="font-family: times new roman,times; font-size: medium;">a stretch of scale factor <em>t</em> in the <em>y</em>-direction</span></p>
<p style="margin-left: 60px;"><span style="font-family: times new roman,times; font-size: medium;">followed by </span><span style="font-family: times new roman,times; font-size: medium;">a translation of \(\left( {\begin{array}{*{20}{c}}<br>p\\<br>q<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Find \(\left( {\begin{array}{*{20}{c}}<br>p\\<br>q<br>\end{array}} \right)\) and the value of <em>t</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">The equation </span><span style="font-family: TimesNewRomanPS-ItalicMT;">\({x^2} - 3x + {k^2} = 4\)</span><span style="font-family: TimesNewRomanPSMT;"> has two distinct real roots. Find the possible values of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">k </span></em><span style="font-family: TimesNewRomanPSMT;">.</span></span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^2} + x - 6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the \(y\)-intercept of the graph of \(f\).</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 class="p1">Solve \(f(x) = 0\).</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">On the following grid, sketch the graph of \(f\), for \( - 4 \le x \le 3\).</p>
<p class="p1"><img src="images/Schermafbeelding_2015-12-13_om_16.03.12.png" alt></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = {x^2} - x\), for \(x \in \mathbb{R}\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.25.10.png" alt="N17/5/MATME/SP1/ENG/TZ0/08"></p>
<p>The graph of \(f\) crosses the \(x\)-axis at the origin and at the point \({\text{P}}(1,{\text{ }}0)\).</p>
</div>
<div class="specification">
<p>The line <em>L</em> is the normal to the graph of <em>f</em> at P.</p>
</div>
<div class="specification">
<p>The line \(L\) intersects the graph of \(f\) at another point Q, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.27.48.png" alt="N17/5/MATME/SP1/ENG/TZ0/08.c.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f’(1) = 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of \(L\) in the form \(y = ax + b\).</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 \(x\)-coordinate of Q.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graph of \(f\) and the line \(L\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows the graph of a function \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_05.56.49.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({f^{ - 1}}( - 1)\).</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 \((f \circ f)( - 1)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On the same diagram, sketch the graph of \(y = f( - x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \({\log _2}40 - {\log _2}5\) .</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;">Find the value of \({8^{{{\log }_2}5}}\) .</span></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;">Let \(f(x) = \sqrt {x - 5} \) , for \(x \ge 5\) .</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 \({f^{ - 1}}(2)\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g\) be a function such that \({g^{ - 1}}\) exists for all real numbers. Given that </span><span style="font-family: times new roman,times; font-size: medium;">\(g(30) = 3\) , find \((f \circ {g^{ - 1}})(3)\) .</span></p>
<div class="marks">[3]</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;">The equation \({x^2} + (k + 2)x + 2k = 0\) has two distinct real roots.</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 possible values of \(k\).</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3\ln x\) and \(g(x) = \ln 5{x^3}\) .</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;">Express \(g(x)\) in the form \(f(x) + \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is a transformation of the graph of <em>f</em> . Give a full geometric </span><span style="font-family: times new roman,times; font-size: medium;">description of this transformation.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider a function <em>f </em>(<em>x</em>) , for −2 ≤ <em>x</em> ≤ 2 . The following diagram shows the graph of <em>f</em>.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <em>f</em> (0).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <em>f </em><sup>−1</sup> (1).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of <em>f </em><sup>−1</sup>.</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>On the grid above, sketch the graph of <em>f </em><sup>−1</sup>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{1}{2}{x^3} - {x^2} - 3x\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sheldon.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a maximum point at A and a minimum point at B(3, − 9) .</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 coordinates of A.</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><span style="font-family: times new roman,times; font-size: medium;">Write down the coordinates of</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) the image of B after reflection in the <em>y</em>-axis;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the image of B after translation by the vector \(\left( {\begin{array}{*{20}{c}}<br>{ - 2}\\<br>5<br>\end{array}} \right)\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) the image of B after reflection in the <em>x</em>-axis followed by </span><span style="font-family: times new roman,times; font-size: medium;">a horizontal stretch with scale factor \(\frac{1}{2}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f'(x) = \frac{{6 - 2x}}{{6x - {x^2}}}\), for \(0 < x < 6\).</p>
<p class="p1"><span class="s1">The graph of \(f\) </span>has a maximum point at P<span class="s1">.</span></p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The \(y\)</span>-coordinate of P is \(\ln 27\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the \(x\)-coordinate of <span class="s1">P</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 class="p1">Find \(f(x)\), expressing your answer as a single logarithm.</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"><span class="s1">The graph of \(f\) </span>is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\). The image of P under this transformation has coordinates \((a,{\text{ }}b)\).</p>
<p class="p1">Find the value of \(a\) and of \(b\), where \(a,{\text{ }}b \in \mathbb{N}\).</p>
<div class="marks">[[N/A]]</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 \(f(x) = 2k{x^2} - 4kx + 1\) , for \(k \ne 0\) . The equation \(f(x) = 0\) has two equal roots.</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 value of <em>k</em> .</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><span style="font-family: times new roman,times; font-size: medium;">The line \(y = p\) intersects the graph of <em>f</em> . Find all possible values of <em>p</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows part of the graph of a quadratic function \(f\)<span class="s1">.</span></p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_09.59.14.png" alt></p>
<p class="p2">The vertex is at \((1,{\text{ }} - 9)\)<span class="s2">, and the graph crosses the <em>y</em>-</span>axis at the point \((0,{\text{ }}c)\).</p>
<p class="p1">The function can be written in the form \(f(x) = {(x - h)^2} + k\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the value of \(h\) and of \(k\).</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 the value of \(c\).</p>
<p class="p1"> </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>Let \(g(x) = - {(x - 3)^2} + 1\). The graph of \(g\) is obtained by a reflection of the graph of \(f\) in the \(x\)-axis, followed by a translation of \(\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right)\).</p>
<p><br>Find the value of \(p\) and of \(q\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <em>x</em>-coordinates of the points of intersection of the graphs of \(f\) and \(g\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the probability distribution of a discrete random variable \(A\), in terms of an angle \(\theta \).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_09.10.36.png" alt="M17/5/MATME/SP1/ENG/TZ1/10"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\cos \theta = \frac{3}{4}\).</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>Given that \(\tan \theta > 0\), find \(\tan \theta \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(y = \frac{1}{{\cos x}}\), for \(0 < x < \frac{\pi }{2}\). The graph of \(y\)between \(x = \theta \) and \(x = \frac{\pi }{4}\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[6]</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;">Let \(f(x) = {x^2}\) and \(g(x) = 2x - 3\) .</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 \({g^{ - 1}}(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \((f \circ g)(4)\) .</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;">Part of the graph of a function <em>f</em> is shown in the diagram below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/rugby.png" alt></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;">On the same diagram sketch the graph of \(y = - f(x)\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = f(x + 3)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(g( - 3)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Describe fully the transformation that maps the graph of <em>f</em> to the graph </span><span style="font-family: times new roman,times; font-size: medium;">of <em>g</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the equation \({x^2} + (k - 1)x + 1 = 0\) , where <em>k</em> is a real number.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the values of <em>k</em> for which the equation has two <strong>equal</strong> real solutions.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write the expression \(3\ln 2 - \ln 4\) in the form \(\ln k\), where \(k \in \mathbb{Z}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, solve \(3\ln 2 - \ln 4 = - \ln x\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f\left( x \right) = p{x^2} + qx - 4p\), where <em>p</em> ≠ 0. Find Find the number of roots for the equation \(f\left( x \right) = 0\).</p>
<p>Justify your answer.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <em>f</em>(<em>x</em>) = <em>ax</em><sup>2</sup> − 4<em>x</em> − <em>c</em>. A horizontal line, <em>L</em> , intersects the graph of<em> f</em> at <em>x</em> = −1 and <em>x</em> = 3.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The equation of the axis of symmetry is <em>x</em> = <em>p</em>. Find <em>p</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that <em>a</em> = 2.</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>The equation of <em>L</em> is <em>y</em> = 5 . Find the value of <em>c.</em></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 \(f(x) = {{\rm{e}}^{x + 3}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \({f^{ - 1}}(x) = \ln x - 3\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the domain of \({f^{ - 1}}\) .</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;">Solve the equation \({f^{ - 1}}(x) = \ln \frac{1}{x}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Three consecutive terms of a geometric sequence are \(x - 3\)<span class="s1">, 6 </span>and \(x + 2\).</p>
<p class="p1">Find the possible values of \(x\).</p>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of a function \(f\), for −4 ≤ <em>x</em> ≤ 2.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the same axes, sketch the graph of \(f\left( { - x} \right)\).</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>Another function, \(g\), can be written in the form \(g\left( x \right) = a \times f\left( {x + b} \right)\). The following diagram shows the graph of \(g\).</p>
<p><strong><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbIAAAGACAYAAADIwee6AAAgAElEQVR4Ae2dC3RV1bnv/zswUHoAvaD3NAEuWhiJPVePIoitMGoETGQ4BA4UX5WHYHttRalckwCtm5uIPIIXq8AZKCYCXqkHSxraHhEKEcaF9sIBweoZslNsfSXhHoFCzBWlzV53zA072VnZj7n3XnOvOdf6rzEysh7z8f9+39z729+a6xGwLMsCFxIgARIgARIwlECeobop2zcE/oaWuocRCAQQCBTgtmcOoi1iexitDYtQUDAXdU3nfUODhpIACXQnwEDWnQn3aEWgJ/KnrIPV/j5qSoA9v30PzWEhMA99vnkbZhX9BWf+X2SHVqophgRIIHcEGMhyx5o9ZUMgbxCuv70I+OA0Pr8Yt/L+fjCGfeNmXF/QK5uWWZcESMBwAgxkhjvQP/J7om//K4EPjuOjz/4G4Dya6l/GwTsnYXgfDmP/jANaSgLdCfAboDsT7tGSQE/0vbx/p7K2o9hy6BY8OXkIOIg7sXCNBPxIgN8BfvS6kTbn4e8u7498hPCnT46jofq3GPLIBAzkCDbSmxRNAk4S4NeAkzTZlkICefjaZf3xNZzEkbXrsKd4FiYP5NyYQuBsmgSMIcBAZoyrKDSvb38MBdDz1odQPnYgTylySJAACUQIMJBxIBhEoAcGLVyF5bOuRZ8Uqk+fPo3y8nJ8+umnKUryMAmQgOkEeppuAPX7hEDbQTxXk4eyJ8chX+Ln1/Lly7Fy5Ur85S9/wfr1630CiWaSgD8JBPiIKn863gir2w7imbseR8v0xzEodBLfefIHGCFxqf3Ro0cxfPjwDhO3bt2KKVOmdGxzhQRIwFsEJH7bestgWmMQgXAbPgv9GW+HzuE7P54tFcTOnTuHH/zgBygrK4sY+vTTT0NkZ+JUIxcSIAFvEmBG5k2/+taqNWvW4NFHH8Unn3yCwYMH44svvsCtt96K4uJiVFdX+5YLDScBLxNgIPOyd31mW2NjI4qKihA9lSgeNCxe7rB//36MGTMG+/btw+jRo31GheaSgPcJMJB538e+sFCcUrzvvvsittbX10f+RwOZ2BBXMO7Zswd79+5F7969fcGERpKAXwhwjswvnva4ndu3b8e2bdsSnj5csGBBhMCzzz7rcRI0jwT8R4AZmf987jmLxYUcAwYMwMaNGzFjxowO+2IzMrGzrq4OU6dORSgUQmFhYUc5rpAACZhNgBmZ2f6jeiByqlAEsWnTpiXlIS7BF+W4kAAJeIsAMzJv+ZPWxBCwZ2Qxh7hKAiTgIQLMyDzkTJpCAiRAAn4kwEDmR6/TZhIgARLwEAEGMg85k6aQAAmQgB8JMJD50eu0mQRIgAQ8RICBzEPONN+U82g5uBYzCwIIBApwW0UdGtvC5ptFC0iABJQSYCBTipeNyxMIo+3IduzEFLzcbKG9+XVMPLEYxU/tQat8IyxJAiTgQwIMZD50upYmhz/CobM3Yvqo/Mibn/Pyv405M+8CXtmFQ63MyrT0GUWRgCYEGMg0cYTvZeRdjeLiwZEgdoHFeZz48CMUzZ+MUf04TH0/PgiABJIQ4Buik8DhIZcItDWiYcsGbDw+C5uXjEIfl2SwWxIgATMIMJCZ4SefqAyjteGnuGbcMrQIi4uBScdvxpTCfj6xn2aSAAlkQoDnbDKhxjqKCOSh39ilaLa+QvOhTSjHBkwtXoS6pvOK+mOzJEACXiAg/azFxYsXR+ytrKxEdD0RAHsZ+3a8evYy9m3WiUcAsHOyb8erZS9j39aljnXqCA6sbUbb9Nsw/uqvdciqqqrqWE+1EgwG4xax25xqWzSSqoz9uBfr2GHabbZv28vLMGGdeARSj794tez+sG87UUe06fpiSS7BYFCypPvFTNIqaJmkN6da29+3akpuscp3f5bRoALEC6LNWXLKNkss1JolwATVTeIqTNBFr3RG5nrEpQD/EWhtQMWNdbh57ypMGdgrbfv59Pu0kbECCRhJQHqOLNXpRJ2sN0mr4GaSXlVaw011mFNQioqNB9EibhsLN6Fh+TqcWPQwSjIIYjqNR1ktqtjK9p9OOWpNh5Z8WZO4Cqt00SsdyORdwZIkkD6BvMsKMfr2ZlTPuhkFPQIoePDnODP1Obw8+1pefp8+TtYgAV8R4OX3vnK3xsb2uRazN76L2XyBs8ZOojQS0JMA58j09AtVOUCAc2QOQGQTJGAAAelTi7qcC5VhapJWYY9Jek3SKjNWdCpjEltqVTNyTOKq03eXdCBT4za2SgIkQAIkQALZEWAgy44fa5MACZAACbhMgIHMZQewexIgARIggewIMJBlx4+1SYAESIAEXCbAQOayA9g9CZAACZBAdgQYyLLjx9okQAIkQAIuE2Agc9kB7J4ESIAESCA7AtKBTItH9WdnK2uTAAmQAAl4kIB0IDPtRj0P+oomkQAJkAAJxCEgHcji1OUuEiABEiABEnCdAAOZ6y6gABIgARIggWwISAcyzpFlg5l1SYAESIAEVBGQDmScI1PlArZLAiRAAiSQDQHpQJZNJ6xLAiRAAiRAAqoIMJCpIst2SYAESIAEckKAgSwnmNkJCZAACZCAKgIMZKrIsl0SIAESIIGcEGAgywlmdkICJEACJKCKAAOZKrJslwRIgARIICcEGMhygpmdkAAJkAAJqCIQsCzLUtU42yUBNwkEAgFweLvpAfZNArkhIJ2RmXRDtElahZtN0muS1tx8hJzrxSS21Oqc32NbMomr0K2LXulAFgub6yRAAiRAAiSgCwEGMl08QR0kQAIkQAIZEeAcWUbYWMkEApwjM8FL1EgC2ROQzsh0ORcqY7JJWoU9Juk1SavMWNGpjElsqVXNyDGJq07fXdKBTI3b2CoJkAAJkAAJZEeAgSw7fqxNAiRAAiTgMgEGMpcdwO4FgTDaGnfgmZnXQcxrBQKlqNh4EC1h0iEBEiCB1AQYyFIzYgnFBMJNb2L9vwIT174Dy/oKzQcm4sTCybh/1UG0Ke6bzZMACZhPgIHMfB8absGX+PMf++PueaUo7COGYy/kj5qDnywZgz2r6nGwlWmZ4Q6mfBJQTqCn8h7YAQkkJXAphhZ/y1aiF75+1TDk2/ZykwRIgATiEWBGFo8K92lB4GsTRqIokqVpIYciSIAENCUgHcgqKys1NYGyvEfgNA7taMLDPxqLgdIj1HsUaBEJkIAcAemvCdNu1JMzn6X0IxBG2+HN2HjlI3h4xOX6yaMiEiAB7QhIP6IqGshEZhZdT2SNvYx9O149exn7NuvEIwDYOdm349Wyl7Fvu1qn7E6Uli7DsO9chyt6BbrZF9VWVVUVXU36PxgMxj1ut9m+Ha+SvYx9W+c68bTa96XaFvalKmM/rnsdu8/s+u3b9vIy9nm9jmDk+iLeRyazBINBmWJalDFJqwBmkl6lWts/tOpXvWa9/3m7I+MIEK8jM2dRytZhDNTqMNCLzZnEVUjWRa/0qUUtoq7rYZ8ClBEIN6Hhue3oe+8/4ZroBR5t72Fj7X60KuuUDZMACXiBgHQgS3U60QswaINLBNqOoW7hbIyb/0OMK7jk4tM9Agj0LcFmDEAfl2SxWxIgATMI8D4yM/zkXZXhj1A3bxqm1r4Xx8YxuGfMVZD+tRWnBe4iARLwPgEGMu/7WG8L84ZgSs27sGr0lkl1JEAC+hLgj119fUNlJEACJEACEgQYyCQgsQgJkAAJkIC+BBjI9PUNlZEACZAACUgQYCCTgMQiJEACJEAC+hJgINPXN1RGAiRAAiQgQUD6EVUSbbEICWhFQLxt2rIsrTRRDAmQgPMEpDMyk26INkmrcKlJek3S6vzHRW2LJrGlVjVjwSSuOn13SQcyNW5jqyRAAiRAAiSQHQEGsuz4sTYJkAAJkIDLBDhH5rID2L06ApwjU8eWLZOATgSkMzKTzt2apFUMBpP0mqRVpw+ajBaT2FKrjEfTL2MSV2GdLnqlA1n6LmENEiABEiABElBPgIFMPWP2QAIkQAIkoJAAA5lCuGyaBEiABEhAPQEGMvWM2QMJkAAJkIBCAgxkCuGyaRIgARIgAfUEGMjUM2YPJEACJEACCgkwkCmEy6ZJgARIgATUE5AOZJWVlerVsAcSIAESIAESSJOAdCDT5ca3NO1jcRIgARIgAY8TkA5kHudA80iABEiABAwlwEBmqOMomwRIgARI4AIB6UDGOTIOGRIgARIgAR0JSAcyzpHp6D5qIgESIAESkA5kREUCJEACJEACOhJgINPRK9REAiRAAiQgTYCBTBoVC5IACZAACehIgIFMR69QEwmQAAmQgDQBBjJpVCxIAiRAAiSgIwEGMh29Qk0kQAIkQALSBBjIpFGxoFoCYbQ17kHdL57BzGGL0NAaVtsdWycBEvAMgYBlWZZnrKEhxhIIN9ZiYuVeXPbWJmzGQuw+tgRj+2X3OysQCIDD29ghQeEkIE1A+pvCpBuiTdIqPGWSXlVa8wpn4zevvoDFS6ZJD16vFVTFVgUnalVB1azvAkFAl3EgHcjUuI2tkgAJkAAJkEB2BBjIsuPH2iRAAiRAAi4T4ByZyw5g97EEvkRj7QwU/XQY58hisXCdBEggKQHpjEyXc6FJrbl40CStQrJJek3SKjNWdCpjEltqVTNyTOKq03eXdEaWDmDxypfY8vbteEPAXsa+zTrxCAB2TvbteLXsZezb7tX5G7435I8o+nELpv/oNlx9SaCbfVFtVVVV0dWk/4PBYNzjdpvt2/Eq2cvYt3WuE0+rfV+qbWFfqjL247rXsfvMrt++bS8vY5/X6whGri/i8nuZJRgMyhTTooxJWgUwk/Sq1XrOCtVMs5C/0Np9tj3rsQSIq+/NWdSydZYDtTrLM9qaSVyFZl30Sp9adD3iUgAJkAAJkAAJxCGQ1qlFLVLIOEZwl1cI8GIPr3iSdpBALgl4MiOLnZ/LJcxM+zJJrzKtrQ2oKOiNojmvAy3LMO6yQSitPQY/PahKGdtMB2aSetSaBE4Wh0ziKszURW/PLJizKgk4R6DfWKxotrDCuRbZEgmQgE8IeDIj84nvaCYJkAAJkAAA6UDG+TGOFxIgARIgAR0JSAcyXc6F6giRmkiABEiABNwjIB3I3JPInkmABEiABEggMQEGssRseIQESIAESMAAAtKBjHNkBniTEkmABEjAhwSkAxnnyHw4OmgyCZAACRhAQDqQGWALJZIACZAACfiQAAOZD51Ok0mABEjASwQYyLzkTdpCAiRAAj4kwEDmQ6fTZBIgARLwEgEGMi95k7aQAAmQgA8JMJD50Ok0mQRIgAS8RICBzEvepC0kQAIk4EMC0i/W9CEbmmw4gUAgAMuyDLeC8kmABFIRkM7ITLoh2iStwkEm6TVJa6rBr9txk9hSq5rRYxJXnb67pAOZGrexVRIgARIgARLIjgADWXb8WJsESIAESMBlApwjc9kB7F4dAc6RqWPLlklAJwLSGZlJ525N0ioGg0l6TdKq0wdNRotJbKlVxqPplzGJq7BOF73SgSx9l7AGCZAACZAACagnwECmnjF7IAESIAESUEiAgUwhXDZNAiRAAiSgngADmXrG7IEESIAESEAhAQYyhXDZNAmQAAmQgHoCDGTqGbMHEiABEiABhQQYyBTCZdMkQAIkQALqCUgHssrKSvVq2AMJkAAJkAAJpElAOpDpcuNbmvaxOAmQAAmQgMcJSAcyj3OgeSRAAiRAAoYSYCAz1HGUTQIkQAIkcIGAdCDjHBmHDAmQAAmQgI4EpAMZ58h0dB81kQAJkIA5BHbu3Im6ujrHBUsHMsd7ZoMkQAIkQAK+ItDW1oapU6di8uTJaGxsdMx2BjLHULIhEiABEiCBZASmTJmCUCgUKVJUVIRNmzbh3LlzyapIHZN+saY4tch5MimmLKQJAb5YUxNHUAYJ2AiI4LV9+/ZIdnbTTTfhxRdfxA033GArJb/pyYzMtPk8k/SapFX+Y6BHSZPYUquaMWMSV0EgU729e/eGyM4++eQTXH/99Rg+fDiWLl2K06dPZwTWk4EsIxKsRAIuERC/TsUE+IkTJ1xSwG5JwB0CgwYNwvr167Fjxw7U19fjjjvuwNGjR9MWI31qUZym4UICJEACJEACKgls3bo1kq2l00daGZllWTDhTwAwQafQKJZgMGiUXpPY6qj1iy++gPiwirkB8SfWo8u+ffuMGAtCr45s42miVnXf29myFZ+FjRs3Rob/pEmTIheCiFOO6S5pZWRikJiwmDTJb5JW4XuT9OqmNTrBvXz58sjHaMGCBZgwYQLEfIHQunr1auzatStyikX3z5lubJPxotZkdLI7lg1bcfl9eXk5tm3bFglm06ZNi3wWMlGUVkaWSQesk5pAphOmqVtmCR0IROfAbr31VoggJgLY3r17I6dPRBCLLh9//HHkQ71///7oLv4nAc8REJ8HcWGHuPz+yiuvjFzwMWPGjIyDmADEQOa5YUKDkhM4j5bDr6DitgIEAtdh5rP70RJOXiPTo7IBLNq+CGoiK1u5cmV0F/+TgKcIiAs5xA86cWGHOKUuLvQQF3xkuzCQZUuQ9Q0iEEbb4bW4/4k/oXTzh7Da38TMz5bh/lUH0eagFekGsNiu77//fmZlsUC47ikCb7zxBoqLi/Hmm2+mfUFHUhCW5HJhbleysMvFqFWdA4xm2/6+VVNyi1W++7NOQGd3W+X506ya0LnOfRmuffHFF9bWrVutm266KfIn1sU+mSWW6+rVq61JkybJVHOtTKxe10RIdkytkqAyKOYs24+trTOGigsxLOQ/Ym399CvLam+2DqyaYeUD1tCVh6y/JtAorjySWpwVLNVlxoVM0xoMBjO2NdcVTWMby6c9VGOVwB60PrN2l99ildS8b7XHFk5jPZsAFu1GcI2Og1OnTkU+zPv27Yse1u6/yeNAO5gxgkziKmSr0Nv+6VZrdj6s/PLXrJ2r/odV8/7ZGELxV3lqMWm+yoPeIfAlju97Ezvzh+Gqr/eymfUVdv7L73A8zbmybE4h2gR02ezfvz/nyroQ4YafCOQN/A6+N30EWqqX4LUhszDrmn4pzZcOZOJeJ1MWk7SacktD1PcmsY1qvvC/DZ+G/gRcNwyD+kgP+65NxGyJ11GkugoxpnjKVfs40H2uzKRxQK0ph59mBfpjZGkJ8vFNjL7276WuSOypmQWUQwJaExCXxourCpubmyOX0UfvA3NadGxWNnr0aKebZ3sk4AgBtT8S9uFf9n2IWYXXpAxm2f80dQQHGyEB1QT6YFDRN4B3j+PTtjTPIQIQAUy8Q2nMmDEYP3583PvAnLZA96zMaXvZHgkIAuGW3XhmRzvuKwbeDTVLXVHMQMax4xMCl2LYmDtQYrc2fBIfHj2DkntuwbA4nwZ7ADt16hTmzp2b1c2bdgmJtmOzskRluJ8EPEUg/BHqK3fhhsd+gv82fQxaXtmFQ03vYOOzb6Apye/POB9dT2GhMSTQQSBv2C2457q92HEo5lURbc0IvXsj7hlzVZfTF4kCmAguuVyYleWSNvuSJxBGa8MiPFtVFXm8mnhUVaC0Fo1Jgk3StsPHUFtagMC4F4D5P8GUgX1QcP23UdyyAU89/wG+/f07MDBJtEpyKGm3kYPhpjrMKbgbtY1fpi7sSonzaDm4FjMLAggECnBbRR0aMzitlBvprWj87bMXtQYQuK0CGw+3INNxoV5zGG2Ne1D3i2cwc9giNLTqq7SDRV4h7l76XRx8ai0aWs4D4SY0LF+Fg/OfwN2Fl0aKuRLA2hrRULcZv6t/FRUNJzvkihW9srLOMVolvsB0H6PhFhx8dubFL9tSVNQdkzpN1cUBbm2EP0LdnJEorT2m53dA+BPsevXX+LyDT37CsxodRZKt5F2D2TuaYb21FFMKxVWKeegz4nG8ZTXjrRVTUJjiAi3pQNbt7dAiBQwuRm1LMnVuHguj7ch27MQUvNxsob35dUw8sRjFT+1Bq5uy4vbdjqZtr+BfcSfWNluw2ptxYOIJLBz5faw6fCZuDbd3hhs34N7Kl7H1sTJs+sJtNbL9iw/HI9i84gpsHHEJAj1mY0dREJvnj8I7tjmw6ClE5RmY+CV670+xcesy7PpDfJB6ZGXnu4zR4JPzNR+jZ3Ckfj9w73o8HvwJmg9MxIm5D+Ap2w8F2ZGT23Ln0VS/EnNrm3PbrXRv4ru1Di9c8RwqOt7c0Ywds1NflCHdRZoFpQNZ1wfbnsHhVSux/4p/RH6aHeasePgjHDp7I6aPyo+cMsrL/zbmzLwLEOdcdcserFb88bKJmHd7IfoIQHn5GDVvIZaUvI1VW97WMPACeYWz8ZtXX8DiJdNy5lJnOuqF/FGPYKP4wWDtwMRhf8UPp3RexJGzABY1RvwS/c0WvLz4cQyN7rP91yIrCzd1HaOBPlqP0fAH/46zN0/EqHxxz2AP5I+6BzOnA6/s+IOWn6dOl1toO7wOi/Zfgtu1/XI9jYNbXsXO6hX430eOoaHR/dRAOpB1ghbPq9uE5/EA5pcO6dyt21re1SguHhwz73EeJz78CEXzJ2NUvwzMVmlf4D/ZtIpgdgWuuqFAZa++btuVU4hZEHc9K+v2edJ7jOYNHY3igTE3vkcu6hmM+XffiNS312bhqGyrnm/GuueBx+bfga9n25ai+uHGX2FF9WEAO/G7X2/BuKJprp+2Tf8bve3QBdAPj0RfRaAcb1bMQdRW4enjsyKnkSJZj+OdqGhwACbcPPRClqaieR+2aVoAi7pIi6wsKqbLf93HaBjnT/0ZtQt/huMVazB/xOVd1Ou1cQYth94DHpuBEX176CUtRo04G7PDEtM1h3DPXbegGDtRPfUJrHNxGkQ6kF2YIzuDw+s2XwCdYvItxm4XVy9cWVPQtwjj5izDpt+/hd8fdz8NlgLS+gfsOHoHflQSm1VK1WShOARMDWCxprielcWKEevaj9GTaKgYheVrX8Gc6l/h9zsO4Li2F3tdONN1ANfiYa2DbecgyMsfgaLh47G7eScWFrs7DSIdyBYvDqLt8GvY+o3HNf9V0wlaXPnSb+xSNFtfofnQJpRjA6YWL0Jd0/nYQhqun8HhF3+FK5c+iBFG/GDQEOFFSV4IYFG6emVlXxowRq/A2BWHIC5MObRhOlA9FcXz6pPejxRlnfP/bYewfusQjPv2QOPOwOTlj8OCJ2e5ev1BHqLX74v7ABL+FeDV3x3Guq35eGTykJh5p1y7+0s01t6dROcFG6pePWK7n6EX8kdMx7IXlqCk5f/gQCg3WVm4sRalCZlGed+NI6f+FgNS/DJ7DVv6P5TjX2aZso2RrtGqeIGfWKJP4sj5RRyKWOiRlYVxvuU9F8ZohlADfTBiZiVeqJmGlu2HENIuKzuDw+sbMOSRCegbyNBGV6vloc+gYbjOoeeYZmJKHqLX71viKq5Ef5/iu/mtWLVsMgb1iH4B98Bl45ahBa9jTlHv7G6Gk1Z+KQpnb0mi84L+4PeGozBOrhm5IbbkEunesi0YPZecmKvQuwXDB3Q+8jLc9CbWvzcGT86+Nse/zLJjmy0rp+o3NjaivLwcw4cPjzTplQAW5aNDVibG6Nv/8V9cGKNRCpn8T/Bkl0yacrpO69vYsnIhpg66BJH788SP38vGobqlBTvnfBM9AjrfqytghNH26X9gRMXEuN+7TuOK116cr/v4xS65eiyauwS6dpzdvRD5mIaa0DlYO2a7ZkQ8xXH3iac4fDAcNxfped1SuKUBz225FPdOjwaxMNqO/QK1e7reJBvXNp/vjAawoqIiXH755RABTCzii99ri5tZWXSMXvuP//niDy1Txqj4sj2ODyaMRJFup+v7jcWKyO0gFsRDeCM/fM/uRnl+Pkpq3ke7tQWzL96w7/5YPo+Ww29gW8fDGtrRcvBFPL1nBB4tvsI1eZKBzDV9GXd84akjpajYeBAt4qETkac4rMOJRQ+jJPay3Ix7cLKihbbGOiy8/wHMnz8OBTFZb99vbgEKzLnO0kkqMm3FBjBRPhQKYdGiRZ4MYFEe7mRl4kkunWN01VPRRxP1gH5j9Dya6uai4OKTR8TbH8WDaJevOI1FZeOTPuooypj/kxA4+2/42cgC9AgE8Gz9AexvG4snq25HvovRxMWuk4By4FDeZYUYfXszqmfdHAkMBQ/+HGemPoeXc37KLrUx1uchzCueiuo9cR6TUnIHxgy78Pik1C3lsERrAyoKeqNozutAyzKMu2xQTh+nEy+AVVdXo7CwMIcQnOhKXFk3Ej2K5uADtKB63JVSp+lznZWFm+oNGqM9cdk/3ITbQ9WYNbIAT1Wtw4Ovfo6pG57HbImXNDrhVe+20Qv5YxfjrYtn5x6ffAu+O/bigxzcNDr+i6O7742+hr37Ef32mKRV0DNJr9taQ6GQVVZWFnnFuvgvthMtKl7DnqgvJ/any3b16tXWpEmTnOg67TbS1Zp2Bw5WoFYHYdqa0oWtZzMyN38csG/nCXgnA3OOTa6zMueUsyUScJZAQARYZ5tkayTgHAERwF566aXIW5nLysrw0EMPSZ8+FLeTeH14r1mzBrt27UJ9fb1z0NkSCRhGQDoj6/rQYL2tNEmrIGmS3lxp9WMGlglbt7KyTLS69a1BrerI68JWOpCpQ8GWSaCTgB8DWKf16a+5cwVj+jpZgwRUEmAgU0mXbUsTYACTRtWtoFtZWTch3EECLhHgHJlL4NntBQLZzIGlYuiHObIoA86VRUnwvx8JSGdkupwLlXGSSVqFPSbpdUorM7DuIzkbtrnOyrLR2t1ytXuoVR1fXdhKBzJ1KNiynwicPn0aS5cuhXiUlFjEkzjMvJFZL69xrkwvf1BNbgkwkOWWt297EwFMnP4aMGAAzpw5EyeAiUcg7UHdL57BzGGL0NAqnivGJR0Cuc7K0tHGsiSgkgADmUq6bBuxAUzc77Rv3764GVi4cQPurXwZWx8rw6YvCC4TAszKMqHGOl4gwEDmBS9qaEO8ACZu2h09enRcteKVN1VXEWwAABkBSURBVL959QUsXjIt7nHulCPArEyOE0t5iwADmbf86bo16QYw1wV7TACzMo85lOZIEWAgk8LEQqkIMIClIpS748zKcseaPelBgIFMDz8Yq4IBTD/XMSvTzydUpJaAdCCrrKxUq4StG0WAAUxvdzEr09s/VOcsAelApsuNb86az9bSJXDu3LmOy+ijVyHGvYgjfAy1pQUQT9dI/FeQ05dxpmuryeWZlZnsPWpPl4D0I6rSCWQie4stb9+OJ9Jexr7NOvEIAHZO9u14texl7Nvx6sybNw8PPPAAtm/fHrmZuaamBjt37oxXtGOfvV37dkfBjpW/4XtD/oiiH7dg+o9uw9WXBLrZFy1aVVUVXU36PxgMxj1u12LfjlfJXsa+rVud8vLyyOtvHnzwQdTW1nb5TAqtdv2ptr1Yx+4zGQas05WAYOb6YnvhZ8JNXd4EmlBgzAGTtArZOus9deqUJd5ELN62LN5G/OCDD8aQdnr1nBWqmWYhf6G1+2x71o17/Q3RMoBUvUVa5zFr50KtdiLObevCVvrUohZR1/Ww7x8BiebABg8e7B8IHrCUc2UecCJNSElAOpDFnipM2SoLGEtAeg7MaQtbG1BR0BtFc14HWpZh3GWDOH/mAGPOlTkAkU1oT6Cn9gopMCcERAAT81/Lly9HQUFB5FFSiZ7CoURQv7FY0WxhhZLG/d2oyMoeffRR7N+/P+GTVfxNiNabToCBzHQPZqk/NoCJppYsWYKSkpIsW2V1nQjEZmU5/XGiEwRq8TQBBjJPuzexcfYAtmDBAkyYMAG9e/dOXIlHjCXArMxY11G4BAEGMglIXirCAOYlb8rbwqxMnhVLmkeAgcw8n2WkmAEsI2yeqsSszFPupDExBBjIYmB4cZUBzItezcwmZmWZcWMt/QkwkOnvo4wUMoBlhM3zlZiVed7FvjRQ+hFVvqRjoNEMYJ1OE894tCzxUBIusQTWrFkD8ZxM8YxMLiTgBQKevCHatJu3ndArAlhdXR1uvfXWyL1g4irEvXv3YsqUKY5eieiEVi98cFTYkCu2Iivbtm0bjh49mrEZudKascCYitQaA8PhVV3Y8tSiw47NdXPMwHJN3Pz+onNlmzdvxg033GC+QbTA9wQYyAwdAgxghjpOE9kiKxswYAAeeughFBYWaqKKMkggMwKcI8uMm2u1GMDk0XOOLDkrMVf28ccfo7q6OnlBHiUBzQlwjkwDB8meZxbPyrvvvvuUzoGlwiGrNVU7PN6dQK7Ziqxs5cqVaGxs7C4mxZ5ca00hJ+lhak2KJ6uDurCVDmRZWcvKWREQAWzy5Ml4/PHHMWPGDCUXcWQlkJWNJBCdK3vppZeM1E/RJBAlwEAWJaHh/2gAGzNmDMaPH48333zT8asQNTSbknJIIJusLIcy2RUJJCXAQJYUjzsH7QHs1KlTmDt3LsQvaC4k4CQBZmVO0mRbbhFgIHOLfJx+GcDiQOEu5QSYlSlHzA4UE2AgUwxYpvlPPvkkMgcWPYXIDEyGGss4RYBZmVMk2Y5bBBjI3CIPRN7YKy7iePnllyNzYAxgLjrD512Ll6lmegWjz9HRfA0IMJC54AT7KcSysjLOgbngB3bZSUDcFC3GIa9g7GTCNXMISAeyyspKc6zSVKk9gEUzML6VWVOH+UyWeMoHszKfOd0j5koHMl1ufDORe6IAxqsQTfSmdzUzK/Oub71umXQg8zoIFfYxgKmgyjZVEmBWppIu21ZFgIFMAVkGMAVQ2WROCDArywlmduIwAelAxjmy1OTFM+vEVYi8jD41K5bQlwCzMn19Q2XxCUgHMs6RxQco9ooAVl5ejqKiIl5GnxgTjxhCgFmZIY6izA4C0oGsowZXOgjEBjCxU9zYzEdJdeDhisEEmJUZ7DwfSmcgy8Dp9gAWCoUi73QaNGhQBq2xCgnoR4BZmX4+oaLEBKRfrClOLfp9nkwEMHHDqLjXRtw8Kn61ig88Fz0J8MWa2flFjHdxulz8UOM4z44la6sl4MmMzOn5vEQZmFMfbqf1qhwyJmlVyUFF27qxTZaV6aY1mT+oNRmd7I7pwtaTgSw713TWVh3AOnviGgnoSYBzZXr6haq6EmAg68ojssUAFgcKd/mSQLKszJdAaLSWBBjIYtzCABYDI6erYbQ17sAzM6+DmNcKBEpRsfEgWsI5FcHOEhBgVpYADHdrQ0D6Yg9tFCsQIgIYL+JQAFayyXDTG3huSw/c+f3bUdjnb2g5uB4LJj+Nj+fX49dPjEIfyXbsxXixh51I5tviPkmxVFdXZ94Ia5KAIgLSGZkuk3oyHGS16pKByeqVsV11Gee1fok//7E/7p5XisI+Yjj2Qv6oOfjJkjHYs6oeB1v9k5Y5z9a50WDPynTWareaWu1EnNvWha10IHPOdPdb0iWAuU9CBwWXYmjxtzCwy0jsha9fNQz5OsijhggBzpVxIOhMoMvXh85CndDGAOYExdy18bUJI1EUydJy1yd7SkzAnpUlLskjJJBbAr6YIzt9+jQ2b96MRx99lDcy53Z8ZdjbSTRU/He8ffdzeGLE5Rm2gciFI5ZlZVyfFbsT4FxZdybc4z4B6YxMl3OhMsiiWkUAW7NmDQYMGIB33nkn8oQCMVnt1I3MMlpkykT1ypR1u4x6rWG0Hd6MjVc+goezCGJuc8qkf/VsM1HVtU40KxPPFDVlMYFrlKVJWoVmXfRKZ2TpCBaPsootb9+OOi32v72MfTu2bHTdXiZ2+9y5c5FHSYlH7Nxyyy0YPHhwpFpsGbHDvh1tO/a/vYx9O7ZsdN1exr4dLRf7317Gvh1bNrpuL2PfjpaL/W8vY9+OLRtdt5exb0fLdfy3TmLIgV9izs6Wjl3dV/pi6F3Tcf/wKxC4eLCy7E6Uli7DsO9chyt6BRL6p6qqqntzcfYEg8E4e7v7PaU9ccaKSXXiabXvS7UtQIoyo0ePjjAdP358N//Y24jWSfZ94GYd++Cwa7Fv28vL2Of1OoKR64sluQSDQcmS7heLag2FQu6LkVAQ1StR1PUiSrW2f2jVr3rNev/zdkfsBOBIO7lqRClbB40QnyvBlp8vB6FebMqUMRC1XBe90qcWXY+4GQjQ7RRiBib4p0q4CQ3PbUffe/8J10Qv8Gh7Dxtr96PVPxSMsFR8rsRZDnHvJRcS0IGApwOZDoCpQYJA2zHULZyNcfN/iHEFl1x8ukcAgb4l2IwBGd8QLdEzi2RIYPjw4ZFT9+JKYC4k4DYBBjK3PeD3/sMfoW7eNEyt3hmHxBjcM+YqcJDGQePyLnEBlXiVEbMylx3B7iMEepIDCbhKIG8IptS8C6vGVRXsPAMC4gpGcTGV+M/T+BkAZBXHCPDHrmMo2RAJ+IuACF7Myvzlc12tlQ5kWlxiqStF6iIBnxKI3lfGuTKfDgBNzJYOZLH3gWiinTJIgARcJsCszGUHsPsIAelARl4kQAIkEI8As7J4VLgvlwQYyHJJm32RgAcJMCvzoFMNM0k6kHGOzDDPUi4J5JAAs7IcwmZX3QhIBzLOkXVjxx0kQAIXCTAr41Bwk4B0IHNTJPsmARLQnwCzMv195FWFDGRe9SztIoEcE2BWlmPg7K6DQFqvceE8WQc3rhhAIBAIiMffG6DUOxLF/WTiaR+hUIhP+/COW7W3xJMZmWnzeSbpNUmr9p8+m0CT2CbSqmNWlkirDb8WmyZpFcB00evJQKbFiKQIEvApAc6V+dTxLprNQOYifHZNAl4koGNW5kXOtKmTAANZJwuukQAJOESAWZlDINmMFAHpiz2kWmMhEtCIAC/2cNcZ5eXlEQHV1dXuCmHvnicgnZHpMqkn4xGTtAp7TNJrklaZsaJTGZPYymjVJSuT0arLODBJq07fXdKBTBdHUwcJkIAZBDhXZoafvKCSgcwLXqQNJKApAV2yMk3xUJZDBDhH5hBINqMfAc6R6eETzpXp4Qcvq5DOyEw6d2uSVjG4TNJrklbTPrgmsU1Hq9tZWTpa3R4zJmnV6btLOpC57WD2TwIkYCYBzpWZ6TeTVDOQmeQtaiUBQwm4nZUZio2yJQkwkEmCYjESIIHMCTAry5wda6YmwECWmhFLkAAJOECAWZkDENlEXAIMZHGxcCcJkIDTBJiVOU2U7UUJMJBFSfA/CZCAcgLMypQj9mUHDGS+dDuNJgF3CDArc4e713uVDmR8O7TXhwLtI4HcEGBWlhvOfupFOpCZdqOen5xIW0nAJALMykzylhlapQOZGeZQJQmQgAkEmJWZ4CVzNDKQmeMrbysNt+DgszNREAggEChFRd0xtHnbYl9bx6zM1+533HjpQMY5MsfZs8EOAmdwpH4/cO96NFtfofnARJyY+wCeajjZUYIr3iPArMx7PnXLIulAxjkyt1zk/X7DH/w7zt48EaPyewHohfxR92DmdOCVHX9Aq/fN962FzMp863rHDZcOZI73zAZJ4CKBvKGjUTxQBLGLS/gkPjw6GPPvvhH9ovv435MEmJV50q05N4qBLOfI2WFiAmG0NTagduHPcLxiDeaPuDxxUR7xBAFmZZ5wo+tGSL9YU5xa5DyZ6/7ysICTaKi4A+OqDwPIR3H5Grzw5GQU9sn8txZfrGnGcGlsbERRURFCoRBEYONCAukSyPxbIt2ecljetPk8k/Sq03oFxq44BKu9GYc2TAeqp6J4Xj2awjkcOC53pY6t84Y5qVV1VuakVudJdm3RJK1CuS56PRnIug4NbrlCIHwMtaUFEFlR4r8ClNYeQ5dYlZePETMr8ULNNLRsP4RQW5ejrpjCTtUT4FyZesZe7iGtU4uyIMQpyNhIbd+O1469jH2bdeIRQOR0rxdZW6eOYPOmv+DFT/di7/+s7GZ8VVVVt33xdgSDwXi7Pcstaqz982PfFuXs+1Jtq66za9euiPzx48fnTFuUV/S/DINo2eh/v9cR9ru+WJJLMBiULOl+MZO0Clom6c2N1nbr7O6F1tDZW61P2zMfTwAyr+xCzdywdcYwFVpDoZAlfCb+O7mo0Oqkvti2TNIqdOuiVzojcz3iUoBHCZxHU9183LT677DsmR9j+oh8oOW3+OmsrRj2XDVmX5P5Bfi82MO8IVNeXh4RXV1dbZ54KnaNgPQcWezpK9fUSnZsklZhkkl6ndfaE5f9w024PVSNWSML0CNwHR589XNM3fB8VkFMcqhoVcx5turMU6VVxVyZKq0q6JqkVdivi17pQKbCaWyTBIA89LlmJjY2W+I8ICzrXWx8YgpGRJ7yQT5+I6D6Cka/8fSLvQxkfvE07SQBQwioyMoMMZ0yMyTAObIMwbGa/gQ4R6a/jxIp5FxZIjLcH4+AdEamy7nQeEbY95mkVWg3Sa9JWu3jQvdtk9iq1upkVqZaq5PjyiStOn13SQcyJ53FtkiABEggGQHOlSWjw2N2AgxkdiLcJgES0IKAk1mZFgZRhDICDGTK0LJhEiCBbAgwK8uGnr/qMpD5y9+0lgSMIsCszCh3uSaWgcw19OyYBEggFQFmZakI8bggwEDGcUACJKA1AWZlWrtHC3EMZFq4gSJIgAQSEWBWlogM90cJSAcyLR7VH1XN/yRAAr4iwKzMV+5O21jpQGbajXppk2AFEiABbQkwK9PWNVoIkw5kWqilCBIgAd8SYFbmW9enNJyBLCUiFiABEtCBALMyHbygpwbpQMY5Mj0dSFUk4CcCzMr85G15W6UDGefI5KGyJAmQgBoCzMrUcDW9VelAZrqh1E8CJOANAszKvOFHJ61gIHOSJtsiARJQToBZmXLExnUg/WJNcWqR82TG+dfXgvliTe+6v7GxEUVFRQiFQhCBjYu/CXgyIzNtPs8kvSZpNe2jbRJbt7Wmk5W5rTWdcWiSVmGXLno9GcjSGTgsSwIkYCYBzpWZ6TcVqhnIVFBlmyRAAsoJpJOVKRfDDlwlwEDmKn52TgIkkA0BZmXZ0PNOXemLPbxjMi3xCwFe7OEPT5eXl0cMra6u9ofBtLIbAemMTJdJvW4WxNlhklYh3yS9JmmNMzS03mUSW520psrKdNKaagCapFWn7y7pQJbKATxOAiRAAm4Q4FyZG9T16pOBTC9/UA0JkEAGBFJlZRk0ySoGEeAcmUHOotT0CHCOLD1eppfmXJnpHsxcv3RGZtK5W5O0CteZpNckrZl/LNypaRJbHbUmysp01JpohJmkVafvLulAlgg895MACZCADgQ4V6aDF9zRwEDmDnf2SgIkoIBAoqxMQVdsUiMCDGQaOYNSLhIIf4S6OSNRWnsMYUIhgTQIMCtLA5aHijKQeciZ3jDlPJrqV2JubbM3zKEVOSfArCznyF3vkIHMdRdQQCeBMNoOr8Oi/Zfg9vzOvVwjgXQIMCtLh5Y3yjKQecOP3rCi7RDWPQ88Nv8OfN0bFtEKlwgwK3MJvEvdMpC5BJ7d2gmcweF1m4HHZmBE3x72g9wmgbQIMCtLC5fxhaUDGd8ObbyvNTZAnFLchOdxPx4ecbnGOinNJALRrOzUqVMmyabWDAhIBzLTbtTLgAWruEWg7RDWbx2CpfNHoY9bGtiv5whEs7IjR454zjYa1JWA9COq0glkInuLLW/f7irhwpa9jH2bdeIRAOyc7NvxatnL2LcdqWOdxJADv8ScnS3xmru4ry+G3vVdFH/1MT7/r9/CNX17XLBnwQzs+ud6/N/i6Wj81VpULl7crY2qqqpu++LtCAaD8Xbryw1qfBrPx/Z9qbYFyFRl7MfdrjN37lysXbsWZWVlEK95if1eijcw7Prt26zTnYBg5PpiSS7BYFCypPvFTNIqaJmk13GtZ3db5fmwgER/06ya0LmMBpVo06TFcbYKjTdJa1lZmUISzjZtEldhuS56pU8tahF1XQ/7FOA4gX5jsaLZEhGn8+/sbpTn56Ok5n20W1swu/BSx7tlg/4h0Lt3b/8Y61NLpQNZqpTcp/xoNgmQAAmQgMsEpAOZyzrZPQmQAAmQAAnEJdAz7l7uJAE3CURON/IRVW66gH2TgEkE0rpqkfNkJrmWWvliTY4BEvAHAU+eWjRtPs8kvSZpNe0jbBJbalUzukziKgjooteTgUzNEGOrJEACJEACOhJgINPRK9REAiRAAiQgTYCBTBoVC5IACZAACehIQPpiDx3FUxMJJCPAiz2S0eExEvAOAemMTJdJPRn0JmkV9pik1yStMmNFpzImsaVWNSPHJK46fXdJBzI1bmOrJEACJEACJJAdAQay7PixNgmQAAmQgMsEOEfmsgPYvToCnCNTx5Ytk4BOBKQzMpPO3ZqkVQwGk/SapFWnD5qMFpPYUquMR9MvYxJXYZ0ueqUDWfouYQ0SIAESIAESUE+AgUw9Y/ZAAiRAAiSgkAADmUK4bJoESIAESEA9AQYy9YzZAwmQAAmQgEICDGQK4bJpEiABEiAB9QQYyNQzZg8kQAIkQAIKCTCQKYTLpkmABEiABNQTkA5kfDu0emewBxIgARIggfQJSAcyXW58S99E1iABEiABEvAyAelA5mUItI0ESIAESMBcAgxk5vqOykmABEiABADwocEcBp4lwIcGe9a1NIwEuhCQzshMmiMzSavwhkl6TdLaZaQbsGESW2pVM6BM4qrTd5d0IFPjNrZKAiRAAiRAAtkRYCDLjh9rkwAJkAAJuExAeo5MpLy8l8xlb7H7tAhwjiwtXCxMAsYS8GRGxvPM6sajaWzVkXC+ZZPYUqvz/hctmsRVJ72eDGRqhhhbJQESIAES0JEAA5mOXqEmEiABEiABaQIMZNKoWFA5gdYGVBQEIOa2LvzdjdrGL5V3yw5IgATMJiB9sYfZZlK9/gTOo6luPm6auhYtUbElNQhtn43CDH9u8WKPKEj+JwFvE5D+ijBpEtIkrWJ4maRXmda2o/j5C/3xv862w7KsC387Mg9iJn5slbFVAINaFUA17LtAp++unmrcwVZJIB0CYbQerMeqnRtQ9PRgnCm9FSVjC9EnnSZYlgRIwLcEpDMy3xKi4eoJhBvxixUb0IIW7Kn+AaaOK8ZdFXVobAur75s9kAAJGE+Ac2TGu9BDBoRbcPjXv8GWny1G9R6geGU9fv3EqIwzM86ReWhs0BQSSEJAOiPjOfEkFLM8RLYXAeblY8Sk72PF7n/D7oXXYc+qehxs9U9WxnGQ5QcpQXVyTQDGgd26sJXOyNIRLB5lFVvevh2Pn72MfZt14hFA5LFhWrK2TmLIgV9izs6OaxDjGNAXQ++ajvuHX4HAxaMdfv/qz9j1z7swpv4PuHTv6m51q6qquu2LtyMYDMbbrS83qPFpB9cYGvZ9qbZF1VRl7Md1rxODI7Jq12/ftpeXsc/rdQQjtxfpQOa2UPbvMwLhY6id8BSwugazCy/1mfE0lwRIIB0C0qcW02mUZUkgawJtzTg+Yg6+yyCWNUo2QAJeJ8BA5nUPG2BfuOUwtm07jJaL02Hhlv149um3Mf7RMehngH5KJAEScJcAA5m7/Nl7hMBpHPjZXSjoEUCgYCZW7f8r7nxyPsbm9yIfEiABEkhJgHNkKRGxAAmQAAmQgM4EmJHp7B1qIwESIAESSEng/wN5FG9WsY0KRgAAAABJRU5ErkJggg=="></strong></p>
<p>Write down the value of <em>a</em> and of <em>b</em>.</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;">Let \(f(x) = \sin x + \frac{1}{2}{x^2} - 2x\) , for \(0 \le x \le \pi \) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g\) be a quadratic function such that \(g(0) = 5\) . The line \(x = 2\) is the axis of </span><span style="font-family: times new roman,times; font-size: medium;">symmetry of the graph of \(g\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The function \(g\) can be expressed in the form \(g(x) = a{(x - h)^2} + 3\) .</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 \(f'(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \(g(4)\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of \(h\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of \(a\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(x\) for which the tangent to the graph of \(f\) is parallel to the </span><span style="font-family: times new roman,times; font-size: medium;">tangent to the graph of \(g\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\log _3}27\);</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\log _8}\frac{1}{8}\);</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\log _{16}}4\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(iii).</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 New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, solve \({\log _3}27 + {\log _8}\frac{1}{8} - {\log _{16}}4 = {\log _4}x\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Let \(f(x) = 3{\tan ^4}x + 2k\) and \(g(x) = - {\tan ^4}x + 8k{\tan ^2}x + k\)<span class="s1">, for \(0 \leqslant x \leqslant 1\), where \(0 < k < 1\)</span>. The graphs of \(f\) and \(g\) intersect at exactly one point. Find the value of \(k\).</p>
</div>
<br><hr><br>